Sigfrid Karg-Elert's Precepts on the Polarity of Sound and Tonality 9780980416213, 0980416213

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Sigfrid Karg-Elert’s

Precepts on the

Polarity of

Sound and

Tonality Complete Original German Text and Illustrations Reprinted with English Translation by Harold Fabrikant and Staffan Thuringer iii

Original German text supplied as a facsimile by Anthony Caldicott on behalf of the Karg-Elert Archive from an original print of the edition by F.E.C. Leuckart, Leipzig 1931 held in the British Library, London English Translation by Harold Fabrikant and Staffan Thuringer 2003 – 2007 Introduction by Harold Fabrikant 2006 Restoration of figures, layout and printing: Terry Truman

First published — October, 2007 By Dr Harold Fabrikant 37 Maxwell Grove Caulfield. Victoria 3162 Australia

National Library of Australia Cataloguing-in-Publication Data [Polaristische Klang- und Tonalitätslehre(Harmonologik) English] Sigfrid Karg-Elert’s precepts on the polarity of sound and tonality. ISBN 978-0-9804162-1-3

(CD-Rom version)

1. Polarization(Sound). 2. Tonality. 3. Harmony. 4. Music theory I. Fabrikant, Harold. II. Thuringer, Staffan. III. Title. 781.25

This work is also available in a printed edition complete with recorded examples CDs ISBN 978-0-9804162-0-6

Please note that the English translation in this book is not subject to copyright. On the condition that use of, or reference to, the translation is suitably acknowledged by reference to the title of this work, and to the translators, we welcome the use of this material.

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Foreword

It is a great privilege to be asked to provide a Foreword to this monumental work, the result of many years' intensive study by Dr Harold Fabrikant, with the invaluable assistance of Staffan Thuringer and Terry Truman. Ever since I received the gift of Karg-Elert's Theory of Polarity photocopied and bound through the good offices of the British Library, the concept of its being made available to the English-speaking world has been an objective of the KargElert Archive. Such a project also received the enthusiastic support of our late Vice-President, Dr Felix Aprahamian. As Karg-Elert's 'Letters to his Australian Friends' (The Harmony of the Soul, published by Dr Fabrikant in 1996) make clear, the composer regarded an English version as 'an act of culture'. Many years later, Karg-Elert's Australian friend and advocate, Dr Arthur Nickson, wrote an essay on the composer in which he referred to the Polarity book as 'a fully documented volume on Modern Harmony, which would be of priceless value to musicians acquainted with the author's original ideas on Tonality, Modulation, Enharmonic systems, and the Whole Note and the Twelve Note experimental scales'. (The Harmony of the Soul, p. 96) Those who admire, study and play the music of Karg-Elert are already indebted to Dr Fabrikant for his outstanding work in the compilation of the composer's letters in The Harmony of the Soul and also for completing the translation of all the correspondence to Godfrey Sceats (Your Ever Grateful, Devoted Friend, 2000) as well as that of the composer and his daughter during his North American tour of 1932 (Everyone is Amazed, 2002). The light which these collections shed on Karg-Elert's character and work has now been even more fully enhanced by an achievement in musicology which confirms the composer's own assertion: 'I know that I have, by this book, gained a place in the history of music'. Anthony Caldicott Chairman, the Karg-Elert Archive

v

Contents

Acknowledgements ...................................................................................... ix Introduction ................................................................................................... x Some Explanatory Notes ........................................................................ xviii Karg-Elert’s Foreword ................................................................................. II Main index ........................................................................................... IV – V Alphabetical Index of composer’s musical texts ............................... VI - VII First Part – Fundamental Introduction .......................................................... 1 Second Part – The System of Harmony in Polaristic Perception Fifth relationship: Diatonality ........................................... 69 Third Part – The System of Harmony in Polaristic Perception Third and Seventh Relationship: Chromatic Tonality ....... 173

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Acknowledgements Several generous people have been essential in this prolonged project. Anthony Caldicott of the Karg-Elert Archive was the first, showing me the photocopies of the entire text held by the Archive in London. That was late in 2002 and I expressed dismay over my difficulty in understanding the text and the exceedingly complicated illustrations. I resolved to make a start but this came to very little. Only much later, as the work unfolded in 2003— 04 and it seemed likely completion would ultimately occur, did I ask Anthony Caldicott to photocopy the complete book from the Archive’s copy and post the sheets across. This he did, with great courtesy. Extra copies were made here, as there were now three people needing access, all in different places. A copy of the facsimile edition by Peter Ewers was purchased still later, a most fortunate acquisition. In 2003, I was introduced to Staffan Thuringer in Melbourne and was delighted to find him interested, even enthusiastic. Staffan, of Swedish origin, and also fluent in everyday spoken German, has been a tower of strength. Without his invaluable assistance, the translation would have been permanently abandoned. Terry Truman in Adelaide represents another essential ingredient, responsible for restoring all illustrations from second or third generation photocopies, a stupendous undertaking, sufficient to test the patience of Job. He has also handled printing, layout and binding. The magnificent appearance of the final work is all due to his care and attention to the finest detail. Proof-reading has been my responsibility. I am completely indebted to all these good friends. Here was truly a chain in which all links had, by necessity, to withstand an equally high load and to do so over several years. Special terms posed additional difficulties. I have been fortunate in the ready assistance received from the staff of the Music Library at the University of Melbourne, where I have been a frequent visitor. Of particular help has been the “Dictionary of Terms in Music”, K.G. Saur, 1981, which the kindly librarians allowed me to retain for months at a time. My own Collins German-English Dictionary, 1980, was of course in perpetual use. Many passages have still posed great difficulty in translation. At times of uncertainty, I have taken refuge in a more literal change into English, trying always to avoid corruption of K-E’s intention in seeking something which seems more readable. I am aware that mistakes must inevitably remain and for these I take sole responsibility. Readers are invited to make comments in our wish to improve the work. H.F., August 2006

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Introduction Sigfrid Karg-Elert (‘K-E’ hereon), 1877—1933, was a supreme authority on harmony in music. His numerous compositions furnish all the evidence needed in making such a sweeping statement, for which I offer no apology. He sought the analysis of harmonies and their sequences, representing them as symbols (or ‘grammalogues’, as he put it). This implies his ability to pass such ideas over to others and, at first glance, suggests an excellent teacher. That this was not the case will however unfold. The notion of symbols was not entirely new but he extended it to an extreme degree. At the same time he developed concepts of harmony which evolved from the foundations laid by 19th century German musicologists; this formed a new science, their domain. He acknowledges his debt to some of them in the body of the text, but immediately denigrates their contribution compared with his present level of understanding! Moreover, his preface (pp. II—III) makes no mention of their essential role, which will be presented in some detail later in this introduction. Instead, K-E claims that in the early 1900s, intuition revealed to him the basic proposition of polarity in music, viz., that the major and minor chords are natural phenomena of equal importance. (By ‘natural’ he means the sound of the human voice or acoustic musical instruments, not their synthetic counterpart.) Major and minor are different from, and influence, each other but the names are misleading in suggesting one is more important than the other. In 1919, K-E became teacher of composition and theory at the Leipzig Conservatorium, which was founded in 1843. In 1924 it was renamed Leipzig State Conservatorium and in 1927, the year he attained the age of 50, he was made a member of the Conservatorium Senate. However, it was not until 1932 that he was elevated to professor. By then it was too late to be more than a symbolic gesture for he was desperately ill, following an ill-advised concert tour of North America in the first three months of that year. His decline continued and death occurred in April 1933, aged only 55. There was no shortage of German textbooks on harmony by the 1920s. But by then K-E had already worked out his concepts of polarity sufficiently that this is what he taught. He planned to publish these ideas but was greatly delayed, so that the preface (referred to above) is dated Autumn 1930. It seems reasonable to assume that the content of the published work and his lectures were at least similar, if not identical. Throughout the 1920s, K-E was very close to his pupils. He was an extremely colourful, outspoken eccentric and several pupils became his friends (even lovers); they were frequent visitors at his home over long evenings, despite the needs of his faithful wife Minna Louise (“Liesel”) Kretzschmar, 1890—1971, whom he married in 1910, and their beloved daughter Katharina (“Katie” or “Kitty”), 1914—84. K-E’s enormous letter to the Nickson circle in December 1923 claims that his students no longer have any trouble with their theoretical studies(!): “The giant province of Harmony, Modulation, Tonality and Atonality, and the Phenomena of Sound from the many thousand-fold dissonance-colours, lie before us like an open book — and everyone can read it and understand. I have experienced it with my students when they arrive stuffed up with the dust of Jadassohn’s dead documents — unload the whole mess within 10 minutes and — new and free of prejudice climb into the Philosophy of Sounds of Cells and of Cosmic Polarism. In only 14 days they are as if transformed and Palestrina, Gesualdo and Monteverdi are as fluent for them as Debussy, Scriabin or Schönberg. All the Masters simply obey a divine dictate, a cosmic compulsion which in the case of us relatively egocentric beings cannot express itself otherwise than through polarity.” (The Harmony of the Soul, 1996, p. 56) K-E was however deluded. The students were so horrified by his hieroglyphics (even more than his concepts) that some found the courage to petition the Conservatorium Senate, seeking to have his teachings excluded from their examinations. Equally remarkable: the

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Senate agreed and K-E’s material was never examined! (Ref. Sigfrid Karg-Elert und seine Leipziger Schüler; von Bockel Verlag, Hamburg 1999, p. 31) Meanwhile, K-E was committing his thoughts on polarity to paper. As early as 1923, according to the letter mentioned above (p. 57), he planned an English edition, to be done under his authority by Greta Bellmont in Australia, Nickson’s chief translator. He already envisaged approaching Novello for publication, for they had put out the Lake Constance Pastels, Op. 96, that year. Bellmont made only one return trip to Germany, in 1926, and visited K-E when the subject of ‘polarity’ arose: “The big book which I am to translate is not yet finished by the printer. I can see already how very interesting it will be.” (Harmony of the Soul, p. 81) Bellmont’s diabetes became uncontrolled whilst she was in Germany and a stay in hospital was necessary. She returned to Australia and died about 12 months later (presumably from the same disease or its complications). The idea of an English version died with her. K-E had already entered a contract with the Leipzig published Leuckart but as the book grew in size and complexity, the proprietor, Martin Sander, decided to withdraw from the agreement! K-E wrote to Bellmont in July 1926 of his misery: “But this work — my life’s goal — has had ominous misfortune. 380 engraving plates, on which the International Firm C.G. Röder has worked for exactly a whole year, and which are ready for printing (they look ravishingly beautiful!), still lack about 80 pages on Atonality and 4ths and 6ths respectively — and now that the work becomes considerably larger than originally agreed upon, my publisher C.F. Leuckart (Martin Sander) has annulled the contract, thus declaring it invalid. Since 1902, I have worked unswervingly, day and night, on this gigantic work which on each page opens completely new spheres and which is without doubt the Precepts of Harmony of the future, — and now . . . . . . I stand before nothing! I do not know what will happen: I cannot offer the work to another publisher as 380 plates are already complete and belong to the publishing-house. But the publisher is unable to start doing something with the torso and furthermore he will not. The engraver and lithographer is demanding however his approximately 5,000Mk manufacturing costs. He cannot be made good by me but instead by the publisher, who again makes me liable for having exceeded the limits of the contract. Well, a lawsuit in our dear Fatherland can run for 2—3 years. Lawyers also want to live . . . . . ” (Harmony of the Soul, pp. 82—83) One can easily see what has happened. Sander agreed to publish a work which was probably merely a plan or sketch, and then found he had a far larger, more mysterious volume on his hands than business sense could accept. The engraver had had (was still having) a frightful time in setting the type (all by hand in those days) and wanted payment. The merest glance at this book reveals how ‘impossible’ was the task: not the text, but the endless diagrams and legends with their hieroglyphics which would stun the readers. Misprints there are, not surprisingly, even for such meticulous workers, some of which have been corrected here wherever recognition has made this possible. The dispute did come to the law courts where settlement was made; how K-E could find the money for this is amazing. It seems likely that neither side came out as ‘winner’, more a compromise allowing the work’s completion. K-E wrote to Sceats in March 1931 of that happy conclusion: “On the 15th January I sold my Organ Symphony to C.F. Peters, and finally my child of sorrow Polaritätslehre (Harmonologie) was published. I have worked on this without pause since 1902. It is unfortunately an expensive publication and will therefore have a limited circulation! But my work will not go with me into the tomb. I know that I have, by this book, gained a place in the history of music!!” (Your Ever Grateful, Devoted Friend, 2002, p. 31) One is left to speculate how many copies were printed: perhaps a few hundred? Economic conditions were appallingly bad in Germany and this ‘difficult’ text could never be a best-seller in musical circles. To compound the position, two major events followed: (a) the death of K-E in April 1933, leaving the way clear for rivals, especially Hermann Grabner (v. infra) and (b) the inclusion of K-E’s name in the Nazi’s scurrilous pamphlet: ABC of Jews

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in Music. K-E may have looked the part but his father was Catholic and his mother Lutheran; he was brought up Lutheran but swung towards Catholicism in the 1920s and occasionally produced his own anti-Semitic remarks in his letters. Baptised Siegfried Karg, he added his mother’s maiden name (Ehlert) with slightly altered spelling to give the hyphenated Karg-Elert in 1902, when a piano instructor at the Magdeburg Conservatorium, on the advice of the Director. A Jewish merchant in that city had the name Karger and they were apprehensive the young musician might be confused with him; worse still, might be thought to be Jewish himself. (Anti-Semitism was then well-entrenched in Germany; the Nazis had merely to exploit it.) The inclusion of K-E’s name in the Nazi pamphlet was later acknowledged to be wrong and it was removed, but irreparable damage may already have occurred. Whether copies of the book were actually destroyed by the Nazis is uncertain. Few came to English-speaking countries, e.g., a copy is held in the Library of Congress, Washington, D.C. and the copy in the British Library, London, is probably the only one in the U.K. Not one has been found in Australia. Nickson, writing in the 1950s, thought this work, among others, had been lost: “Mention should be made of the loss of much material left by the composer after the World Wars, particularly of a fully documented volume on Modern Harmony, which would be of priceless value to musicians acquainted with the author’s original ideas on Tonality, Modulation, Enharmonic systems, and the Whole Note and the Twelve Note experimental scales.” (Harmony of the Soul, p. 96) Comments such as Nickson’s, assuming the worth of K-E’s ‘polarity’ text, also come from other quarters, their common features being a complete faith in K-E’s knowledge of harmony, an assumption of his capacity to write a lucid text on the subject and ignorance of the book’s actual content. Those who did see it in Germany in the 1930s were not so enamoured. The work was judged unreadable and that remains the belief of some contemporary Germans (Sigfrid Karg-Elert und seine Leipziger Schüler, p. 31). A major difficulty lies in the symbols, which K-E, we must presume, found simplicity itself. Indeed, they pose no problem where the harmony is straightforward (examples will be provided below) but they become intolerable, meaningless to most of us as the complexity increases. There is also a distressing degree of inconsistency. Most terms are defined but even these may be difficult to follow. K-E attempts a format analogous to the great Oxford English Dictionary, viz., explain the meaning of a word and then illustrate its use over the decades, even centuries, by various writers. Thus an analytical text on harmony, not a practical handbook for budding composers. A prodigious number of musical examples is provided, many of great interest in themselves. K-E advances right up to atonality but includes relatively few extracts from his own works, choosing instead his more famous contemporaries and preceding composers. At the other end of the scale, Grabner’s texts were standard reference works, evidently much easier to grasp; some came out in multiple editions over several decades whilst K-E’s books were ignored. In 2004, an enormous service has been carried out by the release of a facsimile edition of ‘Polarity’ by the publisher Peter Ewers of Paderborn; a large preface is given by Dr. Thomas Lipski (with English translation). Their source is a library copy in Weimar and the volume is entitled: Sigfrid Karg-Elert — Die theoretischen Werke, ISBN 3-928243-16-0. Two other works by K-E are included. Those fluent in German thus have had ready access from 2004 and with the present translation, English readers may also avail themselves of the work. That was of course K-E’s wish. He hints that readers should think deeply about his concepts and argue a contradictory case if they feel so moved. Whatever we think of the work, it represents the highest development of the polaristic concept and, if only for that reason, is important historically. Of all the pupils of K-E, Paul Schenk (1899—1977) became his leading advocate. He taught at the Leipzig State Conservatorium for several decades and wrote some texts. Even Schenk, however, seems to place himself at a distance from the polaristic viewpoint; in any case, he took it no further. K-E’s preface to ‘Polarity’ is rather rambling and suggests his ideas were evolved on philosophical grounds. There is, however, no possibility that he was unaware of the great

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musicologists still alive in Leipzig and of their fundamental work on polarity. In a footnote (p. V) he refers us to a second text: Acoustic Regulation of Sound and Determination of Function, Leipzig, 1930. He labels it ‘supplement’ or ‘complement’. Ewers gives a facsimile of it in his work of 2004. Much shorter than ‘Polarity’ (96 pages vs. 327) it is given as an exquisitely hand-written text with equally complicated diagrams and musical examples. In the place where a preface is expected, a succinct summary is given, thus — “Instead of a preface: The number is the essence of all things (Pythagoras). Relations in music truly seem to me the basic relations of nature. The concept of mathematics is the most valid witness of idealism in nature; the deep interrelation, the sympathy of the universe is its basis. Pure mathematics is the notion of reason as universe. It appears in music formally as revelation, as creative idealism (Novalis).” To these, Lipski adds: “The whole is more than the sum of its parts (Aristotle).” {Pythagoras, ca. 582—ca. 500 B.C.; Novalis: pen-name of the German poet Friedrich von Hardenberg, 1772—1801; Aristotle, 384—322 B.C.} K-E thus wishes harmony to be placed on a scientific basis. He refers in the text to Gioseffo Zarlino, 1517—90, whose work on harmony (published in 1558) stated “sine scientia ars nihil est” — “without science, art is nothing”. Musicology arose in Germany during the 19th century; the major proponents even lived and taught in Leipzig. Here is some background detail which unquestionably forms the basis of K-E’s work: Hermann von Helmholtz, 1821—94, pursued medical studies in Berlin and graduated in 1842, but also studied mathematics, physics and philosophy. He came to hold professorships in anatomy, physiology and physics. Admittedly, the sum of knowledge then was small compared with the present, but this was still an amazing achievement. His research activity was maintained over a broad range. He invented the ophthalmoscope (still in everyday use), created the fields of physiological optics and acoustics, and some aspects of the science of electricity. He studied the anatomy of the human ear and founded works on the physiology of hearing. Helmholtz explored the role of harmonics in timbre and that of beats in consonance vs. dissonance. He published a text in 1863: Die Lehre von den Tonempfindungen, embodying his principle studies on acoustics, which was translated into English (1875) under the title: On the Sensation of Tone. In it was his nomenclature of pitch, still the standard in Germany and employed by K-E in his ‘Polarity’, reproduced below. The English equivalent is shown, plus the modification by Sir James Jeans (Science & Music, 1953) which combines the German treble designation with the English bass and, because of this union, is used in the English text here: Organ pipe length

32’

16’

8’

4’

2’

1’

6’’

3’’

11/2’’

Frequency (c.p.s.)

16

32

64

128

256

512

Germany

C’’

C’

C

c

c1

c2

c3

c4

c5

England

CCCC

CCC

CC

C

c

c1

c2

c3

c4

Jeans

CCCC

CCC

CC

C

c1

c2

c3

c4

c5

1024 2048 4096

In trying to explain harmony on scientific principles, Helmholtz immediately came into difficulty with the minor chord, which he pronounced ambiguous and acoustically impure; from that he concluded minor chords were inferior to their major counterpart. This however struck him as nonsensical, obliging him to produce excuses in his writings. These failed to solve the problem. Gottfried Weber, 1779—1839 (q.v., p. 61), German composer and, more important, theorist, whose published works appeared ca. 1820—30. He refined the work of earlier

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writers to produce a simple system of chord representation, still in use. Thus capital (upper case) letters denote major keys, small (lower case) letters the minor. Roman numerals denote the scale degree (‘step theory’) on which a chord is based, e.g., v = the dominant. Major/minor qualification of the step is represented by upper/lower case letters. Thus, the following sequence: “D: ii iv V I” represents, in D major: E minor triad, G minor triad, A major triad and finally, D major triad. Weber was aware of the same triad appearing on different steps in different keys, e.g., C major triad is C:I or a:III or G:IV, referring to this as “Mehrdeutigkeit” (multiple meaning or ambiguity). This would be expanded by later writers. Ernst Friedrich Richter, 1808—79, (q.v., p. 61) theorist, teacher and composer. He came to Leipzig University in 1831 to study theology but soon turned to music. He became a founding teacher of harmony and counterpoint at the Leipzig Conservatorium (1843) and was also Thomas-Cantor from 1868, succeeding Hauptmann, when he was elected professor at the Conservatorium. Richter’s Textbook of Harmony (1853) was adopted as the official text at Leipzig. By 1953 it had gone through 36 editions and been translated into 9 languages! Richter followed closely Weber’s theories, adding refinements, e.g., to denote augmented and diminished chords. Moritz Hauptmann, 1792—1868, theorist, composer and teacher, born in Dresden. Thomas-cantor from 1842, foundation teacher of theory and composition at the Leipzig Conservatorium (1843) and a foundation member of the Bach Gesellschaft (1850). His text of 1853 was not a practical but a philosophical account of musical events: major elements are termed ‘unity’, ‘opposition’ or ‘re-union’ (= higher unity), depending on their degree in the scale and whether major or minor. His source may have been Hegel’s philosophy but, in any case, his theory is often labelled ‘Hegelian’. In it he conceives major and minor triads as opposites, the major represented as ascending from the root and the minor descending from the fifth (as if suspended from it). This concept is the very basis of ‘polarity’. Arthur von Oettingen, 1836—1920, came from Dorpat, Livonia (now Tartu, Estonia). His university training in Dorpat and Berlin was in physics, mathematics and physiology, from 1853 to 1863! An associate professor of physics by 1865, he was already professor the next year. At the same time he was a capable amateur musician (e.g., orchestra conductor). His academic life was distinguished. In 1894 he retired from official university work and settled in Leipzig where he was hon. prof. until 1919. Apart from his scientific articles, he published a text on music theory in 1866 which won acclaim: System of Harmony in Dual Development. It opposed the work of Helmholtz (1863) regarding the latter’s concept of consonance vs. dissonance. Oettingen’s text was expanded further in 1913 as The Dual System of Harmony (or Harmonic Dualism). Even in 1866, Oettingen had developed further the idea of opposing major and minor triads sharing a common note called “tonica” e.g., the major triad C-E-G, designated c+, is mirrored by the minor triad F-A -C, designated co, where the two chords share a common C. This gave rise to the terms Oberklang (upper sound) for the major triad and Unterklang (lower sound) for the minor; in this example; ‘C Oberklang’ vs. ‘c Unterklang’. He also postulated a “tonic major scale” (e.g., c1 to c2), mirrored by a “phonic scale” (in this instance, e2 down to e1, which is the “common” [or melodic] minor scale of A minor from the dominant. This aspect was not taken up by K-E but the concept of mirrored major/minor triads is fundamental to his ‘polarity’, and that name will generally be used hereafter instead of ‘dualism’. In the body of the following text, ‘Oberklang’ is translated ‘upper sound’ or ‘upper triad’. ‘Unterklang’ is translated ‘lower sound’ or ‘lower triad’, unless the context requires otherwise, when ‘hanging triad’ is substituted. Hugo Riemann, 1849—1919, German giant of musicology, the pre-eminent scholar in music and teacher of his time. He was initially a capable pianist, trained in Berlin and Tübingen, who came to Leipzig in 1871 and became acquainted with Oettingen’s dualism. He left to take up various teaching posts elsewhere between 1876 and 1895, then returned to Leipzig and was elevated to professor in 1901. He produced about 20 textbooks and had an

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enormous influence over the next generation, not only in developing musicology but also piano performance, orchestration, etc. He taught Reger composition from 1890 and championed his work until a rift in 1905 developed into angry confrontation, never rectified. Riemann devoted his energy to the scientific explanation of “musical hearing”, using the fields of acoustics, physiology and psychology. The Nature of Harmony (1882) grew into Harmony Simplified (1893, English translation 1895) which was subtitled The Theory of the Tonal Functions of Chords. (There were also translations into other European languages and later editions, so warmly were these works received.) He had immediately taken up Oettingen’s dualism, then later developed ‘function’ theory, for which he is best remembered. The term is lifted from mathematics and refers to ‘harmonic identity’, which is determined by the relative frequencies of notes in chords, rather than their absolute values. It is irrelevant whether chords are in root position or an inversion in his notation. This is completely different from figured bass which specifies only the position of a chord and the specific notes, i.e., their absolute values. Riemann nominates three ‘principle’ chords, their roots separated by the interval of a 5th: the tonic chord (symbol: T) which may shift up a 5th to the chord based on the dominant (D), or conversely, down a 5th to give the chord on the subdominant (S) — the ‘5th-relatives’. He also recognises three basic harmonic transformations of a given tonic, dominant or subdominant ‘function’, viz., the variant (e.g., C major chord shifts to C minor chord or vice versa; in English parlance, the tonic major-minor shift); the parallel (triads shifted a minor 3rd, just as relative major-minor keys represent a shift of a minor 3rd); and the leading-note change (where triads are shifted by a major 3rd) — the latter two are the ‘3rd-relatives’. He introduces symbols to represent all these, which for the novice are rather bewildering. K-E took these up and expanded them even further, to the distress of his pupils. This is not ‘harmony simplified’ as the title announces, but a difficult subject made more intimidating. As early as 1875, under the spell of Oettingen’s text (1866), Riemann convinced himself that, just as he could hear the overtones of a major chord, he could also hear undertones emanating from a minor chord; and this was done at the piano! It was not just a matter of writing upright vs. hanging chords to represent the major vs. the minor, but hearing them in these opposing directions (more precisely, mathematically, in opposing sense). He wrote of this in 1877 (Musical Syntaxis), stating that even if others were incapable of hearing the undertones and confirming his finding, he could still hear something distinct! Oettingen’s dualism was for Riemann not merely an abstract idea but physical reality. It is of interest that the phenomenon of the Emperor’s New Clothes (Hans Christian Andersen, 1805—75) did not arise; other listeners did not agree that they could hear what was inaudible! Moreover, Riemann never wavered, even after audible undertones had been proven nonexistent. Given this, it is remarkable that his texts became, and remained, so important. Furthermore, he believed his theory of harmonic function was dependent on the acceptance of polarity. In latter decades, curiously, harmonic function has remained alive at the same time that polarity has been condemned and rejected outright, a separation Riemann would not accept. Some modern writers go so far to claim polarity is not just redundant to function theory but even contradicts it (Hugo Riemann and the Birth of Modern Musical Thought by Alexander Rehding; CUP, 2003). Two additional comments here: Riemann made his conclusions on functional harmony based on just intonation, not on an equally tempered scale; K-E follows that to a great deal, as much of his text shows. Functional harmony has generally not been taught in the English world, where one learns more of specific chords and their resolutions. Hermann Grabner, 1886—1969, trained in the law (doctorate, 1909) and simultaneously in music, becoming a pupil of Reger in composition from 1910 and his assistant at Meiningen in 1912. He held various posts from 1913 and returned to Leipzig in 1924 as Conservatorium lecturer in composition, competing with K-E; it seems that the relationship between these gentlemen was not particularly cordial and their students were also separated. Grabner became Conservatorium director in 1930 but not professor until 1932, the year of K-E’s elevation to that title. He moved to Berlin in 1938 and continued

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teaching until 1951. His textbooks were issued from 1920 to 1963, e.g., Reger’s Harmony (1920), Hugo Riemann’s Function-Theory and Its Significance in Practical Analysis (1923), General Music Theory (1924), Handbook of Functional Harmony (1944). Grabner’s theoretical works and teachings were highly valued. He rejected much of Riemann’s symbol notation and its basis in harmonic dualism, yet retained function theory in harmonic analysis. K-E’s text could never compete with this simpler style which remained popular.

Regarding Pitch and Temperament Our origins in the second half of the last century lead most of us to assume modern pitch and equal temperament long to have been the norm. Experts in early music practice know best how inaccurate this is and we can now easily understand why K-E spends so much time in his text discussing these vital issues. The diatonic scale seems to have arisen in Greece in the 6th century B.C. and was defined by Pythagoras in terms of wavelengths. (These could be observed on a string, when frequencies were unknown.) His octave is abundant in ‘pure’ tones, i.e., concords, giving no audible beat when sounded together: ‘just intonation’. From this the Greek ‘modes’ (scales) were derived, later to be adopted by the Church as the ecclesiastical (church) ‘tones’; this started in the 4th century but was not finalised until the 16th. A lack of variety was always their drawback as the absence of multiple semitones prohibited much modulation. The ancient modes could be extended infinitely up or down but the notes created by tuning in perfect 4ths or 5ths caused glaring errors which no ear could endure: G , for example, in the rising sequence was quite different (sharper) from A on descending. The appalling beats heard by sounding them together is called the ‘wolf’, on account of the howling! Nevertheless, such tuning remains ideal for singers or players of the violin family, where infinite variation of pitch over their range is possible, especially if they are operating independently. But instruments of ‘fixed’ pitch are at a prohibitive disadvantage because of the ‘wolf’. Mean temperament was adopted as a compromise to the Pythagorean scale and arose ca. 1500: the 5ths are tuned a little flat and the 4ths a little sharp, but the 3rds are pure (no beats). This provides a more beautiful triad than equal temperament allows. The ‘wolf’ however still amounts to 1/4 tone and limits the range of this system to keys of no more than 2 or 3 sharps or flats. In the 16th century, some organs even had split raised keys, sounding different pipes, to give the slightly differently tuned sharp or flat of a given note, an expensive burden and a difficulty for the player; this did at least allow the one instrument to expand the keys it could handle, without being retuned. More remote keys still had to be avoided. Various tuning modifications arose in the 18th century when musicians demanded greater flexibility; whence Bach’s Well-tempered Clavier (Book I, 1723) in celebrating the novel idea that one instrument, at any one time, could be played in any of the 12 keys, major or minor. These modifications were not accepted universally. As late as the mid 19th century, S.S. Wesley still favoured mean temperament. In any case, the modifications were still a far cry from equal temperament where the octave frequency ratio of 2:1 is divided exactly equally into 12 steps; each semitone has a frequency 1.05946 (the twelfth root of 2) times that of the note below. Meantone tuning gives a remarkably different affect for different keys, a major reason for its retention for so long when the disadvantage is plain. It is amazing to learn that equal temperament had been proposed by the Spaniard, Bartolo Rames, in 1482 and the frequency ratios calculated accurately by the French mathematician, Mersenne, in 1636. Apart from conservatism, the need to have different keys sounding apart, evoking a different emotion, seems the main reason for continuing mean temperament. Many of us — for no objective reason, it seems — find differing affect in equal temperament between G vs. G major, D vs. D major, etc., when all logic demands these are identical. One wonders whether it is the style of the music which dictates the altered sensation and has nothing to do with the pitch. That may well be true of the piano pieces by Chopin employing multiple flats, but the well-known Schubert Impromptu in G was also published in the ‘simpler’ G and that version sounds different to

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many of us on the modern piano (or is that a comment on listeners with absolute pitch, ‘expecting’ to hear the piece in G ?). On the other hand, equal temperament, although convenient, seems exceedingly crude: the octaves alone are in perfect tune; the 4ths and 5ths are barely out, the error hardly audible (sharp and flat respectively) but the 3rds are distinctly sharp (1/8 semitone). It is remarkable that the modern ear accepts this. Yet there is surely no choice. As composition developed, modulation has become ever more adventurous. If we are to play a single work (even a single movement!) without stopping for a retune to suit transitions into remote keys, the instrument must be tuned with equal compromise. The more adventurous the harmony, the more unsatisfactory modified meantone temperament is and the more impossible unmodified meantone temperament is. Even a conservative writer like Rheinberger modulates far and wide; K-E and later writers far more so. Equal temperament probably became the norm only late in the 19th century, hence K-E’s preoccupation with such matters. Standardisation of pitch was even slower. Open metal organ pipes (cone tuned) form a fairly reliable means of determining pitch historically and early builders were under no obligation to obey some standard, e.g., Silbermann (Strasbourg Cathedral, 1713): a1=393; Schnitger (St. Jacob’s, Hamburg, ca. 1690): a1=489, i.e., nearly 4 semitones higher! Father Smith (Trinity College, Cambridge, 1759): a1=395, virtually the same as Silbermann at Strasbourg; Schmidt (Durham Cathedral): a1=474, i.e., about 3 semitones above Trinity College. ‘Church pitch’ (Pretorius, Northern Germany, 1619) had a1=567, 6 semitones above Strasbourg. ‘Chamber pitch’, for secular music, was generally higher again, seeking greater brilliance (all figures quoted from Jeans, 1953). Early in the 18th century, an attempt was made to standardise pitch and it stayed at about a1 = 415—430 for around a century. In 1859, a French Government Commission recommended a1=435 and this became widely accepted on the Continent. The Covent Garden orchestra used a1=450 in 1879, and concert pitch in America went as high as a1=462, a1=440 later being accepted as standard. In England, where C was the usual reference, c2=522 was adopted, which matches a1=438.9. It was only in 1939 that an international conference in London adopted a1=440 as the world standard (at least for broadcasting!). No wonder K-E raises the subject in a book written during the 1920s.

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Some Explanatory Notes The following is a vain attempt to explain K-E’s terms and symbols, some taken directly from Riemann, others adapted, modified or expanded on the way. I should like to think it is better than nothing, and hope it may lighten the burden of readers unaccustomed to such specialised language. K-E’s examples on pp. 52—53 are very helpful. Harmonic function is described by terms which are often complicated and represented by symbols (letters in italics), sometimes compounds, even inconsistent. Starting with the simplest: the principal triads are denoted by capital letters, the italics sloping from left to right. They are the triads on the tonic, subdominant and dominant, thus: T, C, D. As major triads, these stand upright upon their root notes. (One change by K-E from Riemann: C, the contra(domina)nt, is placed in the octave below the tonic, so that its 5th = the tonic; Riemann placed his equivalent triad in the same octave as the tonic, labelling it S for subdominant.) A minor triad ‘hangs’ from its 5th, which becomes the root, for, based on Riemann’s hypothesis, one ‘hears’ the minor chord from above downwards (K-E provides arrows depicting this at times). The symbol representing this should be inverted, to oppose the upright symbol for the major triad. Indeed it is for T, which becomes but for readability, C and D are not inverted as they would appear virtually identical either way. Instead they are reversed, the italic character sloping from right to left. This has its own logic but is confusing in failing to match the inverted T. Moreover, chord modification has its function letter qualified by a superscript or subscript (sometimes a compounding of these); they are reversed and not inverted in the ‘lower sounds’, even when used in conjunction with T. Additionally, P must be differentiated from p and that is difficult in small font superscript or subscript, reversed or not. To return to the principal triads, in C major: the tonic triad T = C-E-G, the contradominant (=contrant) C = F-A-C, and the dominant D = G-B-D. Their polar counterparts are, in C major:

= A-C-E,

= E-G-B, and

= D-F-A.

The principals may form minor chords on the same roots, so-called variant shifts: (in C), t = C-E -G, c = F-A -C, d = G-B -D. Immediate confusion arises: although minor chords, they are still represented by upright letters! Moreover, the variants also have polar forms, giving them major triads yet retaining their ‘opposite’ symbols. The double dominant or dominant on the dominant (Riemann’s ‘ultradominant’, but K-E uses this older term also), compares the triad on a root with that two 5ths higher. For example, the triad on C (C-E-G) compared with that on D, a little over an octave higher, i.e., the 9th, viz., D-F -A. Mediant shifts are manifold. The simplest is a rise or fall by a major 3rd. The mediant function, represented by superscript M denotes a rise, whilst the counter-mediant, represented by subscript M denotes a fall, e.g., TM in C indicates a change from C-E-G to E-G -B whilst TM changes it to A -C-E . As the polar counterpart of C major is A minor, the inverse forms (‘lower sounds’) are the triads of C minor and F minor respectively, where each T is inverted ( ) but the function letters remain upright, if reversed in sub/superscript. The auxiliary mediant has a minor 3rd rise or fall in the root. Riemann defines this as parallel (retained by K-E), presumably on the basis that a minor 3rd separates parallel (= relative) keys, e.g., a minor 3rd separates E minor from its relative major, G. In C, TP = A major triad and Tp = A minor triad; the former shifts a minor 3rd but remains a major triad as dictated by P, whereas the latter, with its p indication, shifts a minor 3rd and also becomes minor. This may be altered further using variant shifts but the terminology becomes confusing, e.g., variant-parallel vs. parallel-variant: these are to be considered sequential changes, e.g., variant-parallel = make a major-minor change, then add a parallel shift (e.g., C major triad to C minor, then elevate a minor 3rd, giving E major triad finally, assuming function is tp — not for the new reader seeking instant solutions!).

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There are also compound characters, e.g., the mediant-parallel; using C again, where T represents C-E-G, TMp means: elevate the initial chord to the mediant, viz., E-G -B, then lower it a minor 3rd (i.e., the parallel triad to the mediant just formed), viz., C -E-G sharp. In this double shift, the root and 5th rise a semitone each but the 3rd stays the same, receiving the qualification, ‘same 3rd’ or ‘3rd alike’. The terminology is clear but it is a difficult way to give such simple instructions. The leading-note changed triad (from Riemann) is another major 3rd shift which alters specifically major to minor (or vice versa), e.g., in C, tonic C-E-G changes to E-G-B. This is the 1st inversion of the tonic triad except that the leading-note (B) is exchanged for the tonic (C). Its symbol is T l. The polar counterpart here changes the A minor triad to the F major triad (leading-note E exchanged for tonic F); inverted T but l merely altered to subscript. The collective leading-note change is an elaboration of this, where more than one note in a triad simultaneously makes a semitone shift to form a new chord. The outer voices always move outwards; the central voice is stationary or may shift a semitone up or down (p. 202); e.g., F major triad F-A-C shifts to C minor (1st inversion): E-G -C . Chromatic chords (lit. chromonants) are merely semitone shifts, up or down, of an entire triad, e.g., A to A major, which K-E represents as TM : CM or TM : TP or cP : TP. This refers to C major; it shows the flexibility of functional harmony but also the inherent complexity and likelihood of error. The Neapolitan 6th, viz., the major triad upon the flattened supertonic, may be represented as ‘contrant counter-mediant’ = CM, e.g., in C, the triad D -F-A . In these terms, the contrant F-A-C shifted downwards a major 3rd. Tritone chords (lit. tritonants) have their roots separated by three whole tones (= an augmented 4th); they may be functionally defined as a mediant broadening of the double dominant relationship (see p. 201). This may be carried out on either the 1st or 2nd chord, but not both simultaneously. In C, e.g., D to G major triads, explained thus: the double dominant relationship = C : D, i.e., F-A-C : G-B-D. Now modify the former by a countermediant shift, viz., to D -F-A . The final relation = CM : D. Or again in C, F to B major triads, still a tritone range; start with the same pair C : D but this time raise the latter by a mediant shift, viz., to B-D -F . The final relation is now C : DM. K-E gives as further example (p. 201), still in C, A to E major, represented as CM : DM or as TP : tP. {This is however printed TP : tP, which I believe is incorrect as TP in this setting means the E major triad.} Another example of the complexity of harmonic function. This is merely an outline! The pages are filled with terrifying symbols and often hyphenless compound terms; German readers seem content with such words which the English reader finds impossible. Here is one sample; even with added hyphens it is difficult to comprehend: dominant-variant-parallel-variant. {I presume he means: change the dominant chord to the tonic minor, modify it by a parallel shift (which will also make it major), then alter it back to the minor, but I have lingering doubt nevertheless! (p. 53, ex. 107).} H.F., August 2006

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Precepts on the Polarity of Sound and Tonality (The Logic of Harmony)

by

Sigfrid Karg-Elert (State Conservatorium in Leipzig) Copyright of the Publisher in all countries All rights reserved, including translations

Leipzig, Publishing House of F. E. C. Leuckart Founded 1782

Treatise on Polarity

FOREWORD This work, the origins of which reach back to 1900, I owe neither to my pupils (as A. Schönberg claimed for his “Harmony Teachings”), nor to my instructors (for they considerably restricted my view of essential elements at that time); just as little have mathematical calculations or geometrical comparisons of form had a primary influence on the fundamental emergence of my system of harmony, although I keep referring with emphasis to analogies between musical and mathematical (or more precisely, geometrical) proportions. Instead, the stimulus for the present work, which has become my life’s labour, I have received exclusively and intuitively through a special contemplative principle regarding living music. Without any influence from other quarters, I recognised thirty years ago that in tonal events, two differing tendencies are undoubtedly in effect, which stand in perpetual interaction and influence each other. That the major and minor chords are natural phenomena of equal value is confirmed without further ado through naïve tonal sense, through musical feeling and through the facts of the living harmonic event. But the theory which is exclusively based on the effective overtone — which is indeed no natural thing but only a partial consequence, and consequently a secondary manifestation adopting a priori a primary will of nature — brings forth the conclusion that the minor chord is either a primary major consonance pair (10:12:15) or else a degradation into an unnaturally deformed structure (a profoundly altered major third!). The consequences of the investigation by Stumpf into fusion phenomena are likewise completely untenable for current theories of music, for in no way do they confirm the logic of harmony — which is one of the acoustical and physiological senses of overriding musical wisdom. One feels like stressing the naturalness of the major consonance through an allusion to the “naturally given, simple proportions of frequencies” (whereby one only too willingly suppresses the fused natural seventh despite tangible evidence from the literature), and likewise, one does not remain silent over the “naturally given, simple proportions of wavelengths”; yet the wave is just as “natural” as the oscillation! The most zealous theories lack a clear presentation of the fundamental differences between harmony and chord. (I refer here to the first part of this work, in particular, to sections VIII and XII.) People have precepts on chords, as is their due, but fail to give their accompanying concepts (forms of transposition, inversions, doubling of voices, bass foundation) in the service of harmony, which merely has to do with the indivisible, abstract unity of sound and its relationships. This realisation and a second one: that the harmonic and melodic spheres point to completely different building units, and neither the system of harmony from the scale, nor that which results from this, constitutes from the beginning a fundamental requirement, the starting point of my teaching. At an early stage I was conscious that in the first regard of the theory of the nature of harmony, the interior, essential strengths which create the tonal analogues between major and minor are to be recognised and correspondingly valued, and it became apparent to me that functional corresponding natures always necessitate opposites in space. As already stated, in no way did I come to the construction of my teachings from mathematical, especially geometrical, postulates, but rather from pure, naïve experience. I have made clear and covered logarithmically in my “Acoustic Regulation of Sound and Determination of Function” the causes of comma differences, the fourfold mediant relationship, the seventh counter-tone, the manifold inflections of dissonances (beginning with the mixed seventh-chord) and many others up to the very latest,— but this objective, incorruptible proof follows first, since the interior nature of this formation had become obvious to me long before, through sensitive, naïve empathy for living, practical music.

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Inevitably and consistently am I not to recommend the elimination of contradictions in the practice of conventional theories of music up to now? And logically I have felt obliged to formulate a new nomenclature, which corresponds to my assessment, and is suitable in awakening ideas of association linked by necessity to slogans. I do not wish to foist on the cursory reader in unwarranted fashion a childish, vain intention, through enforced, new technical terms “interestingly attracting attention”. I have been exceedingly cautious in arranging the teaching material as intelligibly as possible,— yet on the other hand I believe I have no right to simplify in self-willed fashion what are in effect complicated phenomena in view of a most probably tested, original construction. Superficial, swift readers; sluggish thinking musicians (“feeling without thinking is luck” said Busoni); connoisseurs, who delight only in an interesting end-result; or obstinate know-alls, who have to believe in the strength of their counter-evidence merely to draw out the fact “that until now it went otherwise”; all these may well keep providence away from this reading matter, which has demanded intensive assistance from my students. The system of Polarity has been tried and tested, in all facets up to the practical, in over twenty years’ teaching activity (introduced in 1919 at the State Conservatorium in Leipzig) to well over a thousand students — amongst whom were those of moderate as well as exceptional talent, now working as pioneers in Germany and abroad; it has continued to arouse mounting interest and has furnished in manifest form the proof of a lively commencement. I recall in gratitude my late publisher Martin Sander (died on 14.3.1930), ideally disposed, who carried out his work at considerable sacrifice and through whom it is now at last possible for me — after numerous publishing mishaps — to be able to hand over my life’s work in so splendid a form. Leipzig, Autumn 1930. SIGFRID KARG-ELERT. To my friends and former pupils, K. von Rudloff, P. Schenk and E. Hennies, thanks for valuable assistance.

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INDEX Foreword ............................................................................................................................. II Index (summary of chapters) ............................................................................................ IV Alphabetical composers’ index of the musical examples ................................................. VI

FIRST PART Fundamental Introduction to the Complete Work I. II. III. IV. V. VI. VII. VIII. IX. X.

XI. XII. XIII. XIV. XV. XVI.

The Pythagorean System of Determining Intervals ........................................................... 3 The Didymean System of Determining Intervals (harmonic separation) ........................ 5 The Persian-Arabian System of Determining Intervals (arithmetic separation) ............ 8 The Determination of Intervals by Tonal Frequency ......................................................... 9 The Aliquots (Overtones) .................................................................................................... 11 The Perception of Polarity through Harmonic Illusion .................................................... 11 Acoustic Grammalogues ..................................................................................................... 12 The Erroneous Conception of the Minor Consonance ..................................................... 18 The Comma Difference and its Functional Significance .................................................. 19 The Cosmic Physics of Tones and Sounds in General ....................................................... 21 The Hypothetical Centre of Sound ............................................................................ 26 Central Extract (Chart) of 30 Different Tones .......................................................... 32 Analysis of this Table ................................................................................................. 34 New and Atonal Forms respectively ................................................................................. 45 Harmony as the Essence, the Chord as Manifestation ..................................................... 50 General Survey of (Consonant) Relationships in Sound .................................................. 52 Mobile Sounds in General and Harmonic Polarity .................................................. 55 & 58 Precepts of Figured Bass, Sound, Degrees and Function ................................................ 61 Grammalogues for Polaristic Functions in General ......................................................... 65

SECOND PART The System of Harmony in Polaristic Perception First Section (Fifth relationship: Diatonality) 1st chapter: 2nd chapter: 3rd chapter: 4th chapter: 5th chapter: 6th chapter: 7th chapter: 8th chapter:

The Principal triads ..................................................................................................... 71 The Tempered Contrants ........................................................................................... 73 The Ninth-chord Forms of the Dominants and Contrants ...................................... 77 The Ultradominants ................................................................................................... 82 The Diatonic Substitutes for Principal Triads .......................................................... 86 The Problematic Triad ............................................................................................... 99 The Neapolitan Sixth-chord .................................................................................... 104 The Comma-pure Modulation on the Basis of Principal Triads ............................. 112 Fundamental Guidelines .......................................................................................... 119 General Remarks on Chromaticism (which belong to the Second Section) ......... 134 The Concordance as a Means for Modulation ......................................................... 137 Pseudovariant Modulation ...................................................................................... 140 Ultradominant Chains ............................................................................................. 142 Epilogue to the Chapter on Modulation .................................................................. 147 Supplement to the First Section

9th chapter:

Twins and Triplets ................................................................................................... 156

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THIRD PART Second Section (Third and Seventh Relationship: Chromatic Tonality) 10th Chapter: 11th Chapter: 12th Chapter:

13th Chapter:

14th Chapter: 15th Chapter:

The Ecclesiastical Tone Systems (belonging in part to the First Section)......... 175 Variants as Metharmonic Means of Modulation ............................................... 188 General Comments on the Third Relationship ................................................... 199 Tritonants ....................................................................................................... 199 Collective Changed Sounds ............................................................................ 201 Chromatic Chords ......................................................................................... 202 A) The Style of the Mediant ................................................................................ 203 B) Acoustic and Functional Individuality of Mediants .................................... 207 C) Obvious Tonality Changes through Comma Differences.............................. 207 D) Metharmonic and Enharmonic Concepts..................................................... 209 E) Harmonological Tonal Interpretation............................................................ 211 F) Cadence Broadening........................................................................................ 213 G) Mediant Tonality-jumps, Shifts, Parentheses and Sequences ...................... 221 The Seventh Counter-sounds (Counter-concordances) .................................... 309 Abolished (Decentralised) Tonality..................................................................... 312 a) General Comments .......................................................................................... 312 b) Parallel Chord Shift ......................................................................................... 313 c) Atonal Complexes (see also XI in the Introduction, p. 45) ........................... 320 d) Consonant and Concordant Tonal Interpretation in the Freetonal Sphere and its Tendency towards Bi- and Poly-tonality respectively...................... .322

As Supplement to this Work: S. Karg-Elert: “Acoustic Regulation of Sound and Determination of Function (The 1,000-piece Octave)”, Carl Rothe Publishing House. Leipzig, 1930. Further: “The Evolution of Dissonance” (in preparation).

{The first of these works has not been translated for inclusion in this volume, except for a fragment used in the Introduction. We have no information whether the second work was completed.}

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Mahler, Gustav: 1860—1911 236. Marx, Joseph: 1882—1964 398, 399. Mendelssohn, Felix: 1809—1847 159, 656, 694. Monteverdi, Claudio: 1567—1643 242, 548, 550. Mozart, Wolfgang Amadeus: 1756—1791 211, 212, 247, 655. Moussorgsky (Mus[s]orgsky), Modest Petrovich: 1835—1881 169. Niemann, Walter: 1876—1953 168, 391, 714. Palestrina, Giovanni Pierluigi da: 1525 or 1526—1594 179. Pfohl, Ferdinand: 1862—1949 454. Rachmaninoff, Sergei Vassilievitch: 1873—1943 259. Reger, Max: 1873—1916 664—676. Rossini, Gioacchino Antonio: 1792—1868 250. Rubinstein, Anton Gregorovitch: 1829—1894 464. Schartel, Rudolf 406. Schönberg (Schoenberg), Arnold: 1874—1951 397.

Schop, Johann: active in 1610, died 1667 140. Schubert, Franz Peter: 1797—1828 251, 495, 573, 575—580, 582—584, 586. Schumann, Robert: 1810—1856 142, 213, 214, 253, 254, 378C, 382, 589—591. Schütz, Heinrich: 1585—1672 206, 553. Scott, Cyril: 1879—1970 426. Scrjabin, Alexander Nicolaievitch: 1872—1915 405, 427. Sgambati, Giovanni: 1841—1914 234. Strauß, Johann junior: 1825—1899 164, 385. Strauß, Richard: 1864—1949 677—680, 684, 685. (Tanzlied a. d. XIV. Jahrh.) [Dance-song from the 14th century] 178. Wagner, Richard: 1813—1883 110, 111, 118, 161, 162, 216, 217, 461, 496, 627—639, 643—652, 654, 695. Weber, Carl Maria von: 1786—1826 158, 249. Widor, Charles-Marie: 1844—1937 260. Wolf, Hugo: 1860—1903 657, 662, 663. Woyrsch, Felix: 1860—1944 458, 459.

Konstruktionsbeispiele (stilisiert) von S. K.-E. [(Stylised) Examples of Construction by S. K.-E.] 87, 88, 133—135, 172, 173, 177A, 177B, 306, 307, 311—315, 342, 343, 351—371, 407—418, 438—447, 455, 485, 487, 489, 490, 492, 494, 498, 499, 538—546, 600—603, 642, 687, 688, 717—725, 727—730, 737—746, 750—778.

VII

Treatise on Polarity

Dedicated in gratitude to my beloved wife

First Part

Fundamental Introduction to the Complete Work Ideational Pure Tuning (The naturally given 3-type system)

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Treatise on Polarity

The following pages are arranged with the original German text on the left hand page and the corresponding English translation on the facing right hand page.

All page numbers correspond with the original German publication, all page number references are accurately maintained.

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I The Pythagorean System of Determining Intervals Pythagoras (born in 582 B. C.), using the monochord, discovered the natural proportion of octave wavelengths = more precisely the ratio of string lengths 1:1/2 , and that for fifths 1/2:1/3 . Given, for example, a string with its fundamental tuned to ‘d1’, this yields both halves of the tone ‘d2 d2’ and the three thirds ‘a2 a2 a2’, each 2/3 against the deeper octave of the third, thus ‘a1’.

By continually lining up fifths, Pythagoras obtained a tonal range of infinite extent. In this connection, a constantly increasing differentiation of intervals [contraction] is obtained, provided that all fifths are brought into the compass of one octave. [Musical example 2 on p. 4] 4th 3rd 2nd 1st

1st 2nd 3rd 4th

*)

The numbers in the corners indicate the number of semitones in the compass: 3, 5, 7 etc. The dots above and below the names of notes are acoustic grammalogues introduced by me to indicate values (see later); they show their degree of relation to the specified root: e.g., 3 dots over the musical letter = 3rd fifth above it, 4 dots below it = 4th fifth underneath, to = distance of four fifths..

[* See NB # p.7] English - 3

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The chain of fifths as above in the compass of an imaginary octave [1:1/2] yields:

(* means: new entry into the prevailing compass)

Principal notes (rudiments) Pentatonic

Heptatonic (Diatonic) (1st )

(1st )

9-note compass (2nd )

(2nd )

11-note Chromatic Compass (3rd )

(3rd )

13-note Chromatic Compass 15-note Compass (4th )

(4th )

For the 3-note compass: the primitive tone sequence (e.g., from D: A—D, D—G) is of fundamental importance in harmony as it later gives principal keynotes. Whole-tone G : A = melodic basic element. For the 5-note compass: the two following whole tones C-D-E are added to the above sequence. The possibility of primitive melody is given! Original scale of the oldest music! (Greek and exotic pentatones, a large part of the Gregorian Chant.) The Pythagorean third occurs as a closed quadruple 5th shifted close together. For the 7-note compass: the diatonic semitones (leading notes) are added to the previous sequence. The scale runs through, without break (diatonic). % Common scale of all diatonic musical idiom! By gradual advances, all Greek and Occidental church tones result. The elements are natural, however the remodelling is artificial. For the 9-note compass: the leading notes are joined to the outer principal notes:

←

→

A : B || F : G. As with A and G, B corresponds similarly to F . At the same time, the first chromatic colours occur between the newly added and the previously given

←→ ←→

leading notes: B : B | F : F . This yields the important observation that leading notes “strive for” nature, whilst chromatic notes are “repelled by” nature. For the 11-note compass: chromaticism increases, yet the semitone scale is still not perfect, for between the end and the new commencement, G : A is still a gaping void. The

→

diminished third is recognisable as a pair of striving leading notes: C : D : E .

←

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For the 13-note compass: the whole-tone gap is closed by two overlapping chromatic notes . They form an enharmonic relationship. The breadth of the interval between these notes amounts to about a ninth tone (Pythagorean comma). Consequently it is to be understood that the leading note forms a gap smaller than a semitone (ca. 4/9 tone) = “Limma”, whilst the chromatic note gives a gap larger than a semitone (ca. 5/9 tone) = “Apotome”.

For the 15-note compass: the enharmonic concept develops further. Added to that given above is: ↑

↑

D : C | E : D . The bichromatic interval D : D appears. ↓

↓

Its size = ca. 11/9 tones (ca. 5/9 x 2). ____________ _ ____________ Summary: The Pythagorean tonal compass is set up through rigorous fifths. % It is exclusively to be understood from the viewpoint of linearity. Every melody not overpowered by harmony — no matter whether of diatonic or chromatic type — is by its interval’s values of a Pythagorean nature: i.e., all whole tones are equal, the leading notes equally narrow and smaller than chromatic intervals (D : E is smaller than, and D : D larger than a semitone).

II The Didymean System of Determining Intervals 500 years after Pythagoras, Didymos (born in 68 B.C.), discovered on the monochord the naturally given third-relationship in the proportions of wave lengths (string lengths) 1:1/5 , i.e., reduced to the octave 1/4:1/5 = 5:4. Tuning, e.g., a string, as basic unit, to d1 will produce any one of the 5 fifths of the tone 3 2 f , /5 of the tone f 2, 4/5 of the tone f 1.

The Pythagorean third is known as 64/81; as 4th fifth it has of course no value as a primary interval (5th of the quint is 5th of the quint). The Didymean third as 4/5 is a natural value of the first source (primary value). If one compares the two thirds, i.e., moves them into a common octave, then it will emerge that the Didymean third is slightly smaller than the Pythagorean one:

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The difference between the Pythagorean and Didymean third, and thus between the 4th fifth and the primary third, is called ‘Didymean or syntonic comma’; the size of the interval itself amounts to approximately a tenth of a tone. The note A blown as a third on an F-horn (notation: F-horn, note e1) differs by a tenth of a tone from the A on a D-horn as its fifth (notation: D-horn, note g1). The flageolet tone {i.e., the harmonic, obtained by pressing the string lightly} e1 on the viola’s C-string is a tenth of a tone lower than the e2 on the pure fifth open violin string. Against the tuning note a1, e2 is the first upper fifth, whereas C is the 3rd fifth below, which has a natural third exactly as in the harmonic e2. The Pythagorean creation of fifths can form a scale; (not until the 12th fifth do we get the enharmonic difference [Pythagorean comma]). This is not the case with the Didymean creation of thirds, where the enharmonic difference already appears with the 3rd third:. The 12th fifth will exceed the octave repetition of the tonic by about 1/9 tone. The 3rd third, however, is short of the octave repetition of the tonic by about one narrow 1/4 tone. The difference is called ‘small diesis’.

e.g.

In the Didymean perception, ‘C ’ is lower than ‘D ’ ‘G ’ is lower than ‘A ’ ‘D ’ is lower than ‘E ’, etc. An assessment, which completely contradicts the melodic sense of the instinctive musician! And in fact the naturally intended primary third is not a melodic but a harmonic interval. This has, as far as I know, been ignored by theory until now! The Greek, mathematically orientated theory of music counts the Pythagorean third as a dissonance. They were correct in that the concept of consonance and dissonance is of course a harmonic evaluation. The Didymean primary or natural third definitely sounds consonant, but in terms of the scale it is unnatural, as it requires broad leading notes, deep sharps and high flats.

!

) Explanation of acoustic grammalogues, see NB # on p. 7 English - 6

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e.g.

By means of the lowered third, the step from the pure (large) second to the third becomes narrower, as from the fundamental note to the second. It becomes as much narrowed as the leading note becomes broader. Von Oettingen often wrote about the “unbearable, unmusical mannerism of the singer who ‘tears’ his leading note to his target note, intoning G higher than A . . . . (!)” I regard these mannerisms as outspokenly instinctive, musically! The linearity of the melody has nothing to do with “large and small whole tones”, nor with comma differences, and things like that. In my opinion, the horizontal operates uniquely according to Pythagorean principles. The horizontal is the central track of the principal notes. (See later.) In the discovery of the ‘primary third’, Nature revealed her desire for harmony. Intended by Nature is the scale, dependent upon itself in the Pythagorean sense:

Intended by Nature is the Consonance (major triad), dependent upon itself in the Didymean sense:

1st fifth 4th fifth Fundamental

NB# Just as the fifth tones are designated by dots, all thirds are designated by slanting lines, which show the comma lowering, e.g. E; double third, e.g. G ; triple third, e.g. B (referring to fundamental ©). If, for example, © is the fundamental, its fifth = Ġ, the third of which = B, which has in turn its third = D , and so on. Calculation formulae for the fifth-added and third-added thirds, and summary tables for these, appear on pages 12 and following.

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By means of the Pythagorean-Didymean evaluation, the major consonance is immediately made clear as a primary form, intended by Nature. The minor consonance, however, appears as a very entangled, derived creation where the minor third can never be a nature-given primary form: it has a difference in Pythagorean value between the 3rd fifth and the over-sharp fundamental ; in Didymean value, a difference between the 3rd and the 5th tone . In no event here is the deeper note (x) a fundamental and the higher one a third in the sense of type. The problem over the minor consonance is thus not soluble by calculation of string length division . . . .

III. The Persian-Arabian System of Determining Intervals Messel’s Theory The Persian-Arabian system of determining intervals takes a small string measure (‘Messel’) as the unit and determines, on this basis, the deeper tones in terms of Messel units. Tonic D (Messel unit, root, source..) D is octave tonic /B is third below the octave G is fifth below the octave, however a fundamental note

Here Nature manifests spectacularly her desire for the minor consonance. If the major consonance 1 : 1/3 : 1/5 is brought into a common octave range 1/4 : 1/5 : 1/6 with the values of a ‘nature-given primary form’, then logically the same applies to the minor consonances 1 : 3 : 5 and 4 : 5 : 6 respectively! In terms of a wave division, the minor sound is an equally unnatural form, just as the major sound is in terms of a wave multiplication. Nature’s revelations are both forms, her elements are ordered by polarity, symmetricalreciprocal. The major chord is to be known as micro-cosmic, the minor chord as macrocosmic.

Fundamental

1/ 6 1/ 5 1/ 4

Zarlino pointed out this naturally intended polarity in his magnificent work “L‘istituzione di armoniche” {“The Realm of Harmony”} for the first time. (NB: The first original print of 1558 is in my possession.) The teachings of Rameau and his followers are also based on this extraordinarily important discovery.

____________ _ ____________

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IV

The Determination of Intervals by Tonal Frequency Sauveur (born in 1653) discovered the phenomenon of the relationships between oscillations. They correspond to the proportions of wavelengths in symmetrical opposition.

W=unit for wavelength S=unit for frequency

A string or pipe of length x (W) produces tone y, with a frequency of z (S). A string or pipe of length W = x/2 produces the upper octave of y, with a frequency of S = 2z. A string or pipe of length W = 2x produces the lower octave of y, with a frequency of S = z/2; and so on. The longer the wave, the smaller the frequency — i.e., the lower the note. The shorter the wave, the greater the frequency — i.e., the higher the note. But 2 is always the octave (multiplying) figure, 3 the figure for the fifth, and 5 that for the natural third. If, for example, ‘CC’ is set as S1, the consonant third c1 e1 becomes S=4:5. The same consonant third is W=4:5, when W1 is set as ‘E’ thrice struck.

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Each further octave range doubles the number of partial tones: Octave Range

etc. (No. of Partial Tones) With increasing distance from the original tonic, the intervals gradually become narrower and narrower. Each interval returns in its further octaves, unevenly divided: the larger part moves towards, and the smaller part away from, the original tonic.

returns as Natural + Minor 3rds

It is completely absurd to try to relate diatonic, chromatic or whole tone scales to arrays of partial tones. In scales, uniformity exists between all large and small steps, but in partial tone arrays, there is an obvious distinction between all succeeding intervals. [For example, the whole tones gradually merge into semitones (about 16), which imperceptibly merge into quarter tones (about 32), and these similarly into eighth tones (about 64) . . .] % The arrays of partial tones are seen as obvious natural manifestations in terms of sound, whereas the Pythagorean scales establish the tonal range for the linear absolute melody. The first octave contains only the sound-root. The second octave introduces an ‘underharmonic’ element: the rudimentary sound (neutral, without gender). In the third octave, the ripe and over-ripe forms (consonance and concordance) of harmony are developed. The fourth octave brings in a progressive escalation of ‘overharmonic’ elements (forms of dissonance); thus the partial tones 9 + 12 + 15 display the consonance of the 1st fifth, i.e., the naturally intended primary fifth-harmony, and thus the natural dominant.

Harmonic Leading Note to

In itself a consonance, it acts together with the consonance of the third octave as a primarily dissonant element. Nature’s desire for a dissonant form is obvious here. In 11 and 13, distinct fifth changes are proclaimed. They are replaced in the practically orientated theory by stylised approximations, but in such a way that the uneven relationship 11:12 accounts for wide, and 12:13 narrow, chromatic intervals respectively. See later G : G (wide chromatic : interval), G : G (narrow chromatic interval). :

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V The Aliquots (Overtones) These are the natural upper partials of a tone rich in colour. They appear in different groupings (only the first partials, only the even-numbered ones, only the odd-numbered stronger products, omitted products — thus, only prime-number related partials — etc.). Their way of involvement determines primarily the sound colour of the main tones (full, flat, pointed, hollow, nasal, shrill, . . . .). The organ incorporates the overtones as ‘mixture stops and choruses’ [Quint 51/3', 22/3', 13/5'; Tierce 31/5', 13/5', 4/5'; Septime 22/7', 11/7'; “Larigot”, “Scharf”, “Doublette”, mixtures and cornets in the various divisions, Carillon (with Septime), Cymbel, etc.]. The aliquot tones manifest themselves in the flageolet tones of stringed instruments {such as the violin, i.e., the ‘harmonics’, obtained by pressing the strings lightly}, harp and zither. They are (also) the result of overblowing woodwind and brass instruments. % It is in this regard a basic error of certain textbooks on harmony, which regard overtones as proof for the natural form of the consonance, as through this it is actually proven that the natural minor consonance does not exist. It would be equally one-sided to regard ‘the consonance’ as a naturally intended creation, based only on the primary values of wave units. This would then be a proof that the major consonance does not exist in a natural sense! As space can not be measured in time values, nor time in spatial units, it is equally impossible to measure the major interval, which can be simply expressed in units of time (oscillation phases measured in seconds), by means of spatial measurements (string- or wave-lengths, measured in units such as the foot or metre), nor the minor interval measured in time, without rather complicated fractions resulting. If the tetrad

is in “naturally-given” form (its elements point in S-fashion to

the first prime numerical values

), then logically the tetrad

is equally

“naturally given” (its elements point in W-fashion to the same prime numbers

).

Whether measured in time or space, the “formula” is the same.

VI The Perception of Polarity through Harmonic Illusion The naturally intended polarity of tonal genders can easily be proven, even without mathematical tools. And this proof is valid, even when one would like to deny the obvious connection between mathematical and acoustic values. We are dealing with the ideational calling up of consonant supplementary tones in the effect of realistic intervals. Each interval is without gender, neutral, gravely balanced:

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is understood as C major or minor chord (third missing!) is understood as F major or D minor chord (fifth missing!) is understood as E major or G minor chord (root missing!) is understood as F major or A minor chord (third and fifth missing!) is understood as B major or F minor chord (root missing!) is inderstood as a type of transfer, a) as D major or F minor chord, or b) in normal form as C major or G minor chord. How superficial, therefore, to call the isolated great third ‘major third’ and the isolated small third ‘minor third’! Each great third is a primary form; it belongs equally well to the minor and major consonance (see 2b!). Each small third is the difference between the great third and the fifth; it belongs equally well to the major and minor consonance. In addition there is also a ‘narrower’ small third (it levels out in the half-tone system) as the difference between fifth and seventh. in the sense of concordance, this is understood as belonging to the A major or G minor chord: All ideationally called up supplementary tones are symmetrically polar to realistic intervals!

VII Acoustic Grammalogues and their Conversion Formulae Fifths, thirds and sevenths can be fully regarded as incomparable values in their own right. They can never coincide at exactly the same tonal level. The greatest possible approximation gives the comma-difference. And what applies to the tones also applies in the same way to the sounds. If the 4th fifth is of essentially different value from the 1st third, then the 4th dominant is also of completely different functional sound value from the third on the tonic.

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The logic of the value of functions requires mathematical proof. But purely numerical assessments mean little to he who is solely a musician. What good is it to him to have the following: “The sequence F—E is valued at 256 : 243 in the melodic sense, at 16 : 15 in the consonant sense, and at 21 : 20 in the concordant sense?” Nothing directly! But perhaps indirectly through a re-interpretation of the mathematical relationships into musical functions. And this detour avoids my acoustic grammalogue, already thought of in 1905, which — retraced — first reports musical function-values, and only then, through calculation, discloses their mathematical nature. For purposes of general and absolute validity, standing for these alone or for purposes of special and relative validity, the grammalogues can have a tonal or chordal printed character added:

A.) The Fundamental Note (Sound root) or , , etc. { Notation: Acoustic numerical value: 1 , and its octaves 2, 4, 8, 16, 32, 64, 128, 256, 512 … respectively.

B.) The Fifths (1st type of relationship)

{

Notation: dot, or dots respectively, over the dash or note-character = upper fifth dot, or dots respectively, under the dash or note-character = lower fifth Acoustic numerical value: 3 , and its octaves 6, 12, 24, 48 . . . respectively for

= G, D, A, E ║ for

= A, D, G, C

2 dots = 3 x 3 = 9, 3 dots = 3 x 3 x 3 = 27, 4 dots = 3 4 = 81, etc.

{

C.) The Thirds (2nd type of relationship) Notation: oblique (‘slash’), thus: ”

”

”

1st upper third,

2nd upper third, e.g., E G sharp (for

1st lower third,

2nd lower third, e.g., C A (for

)

)

Acoustic numerical value: 5 , and its octaves 10, 20, 40, 80 . . . respectively 2 obliques = 5 x 5 = 25, and its octaves 50, 100 . . . respectively. = comma flattening

= comma sharpening

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D.) The Sevenths (3rd type of relationship)

{

Notation: wedge (acute angled), thus:

upper seventh (lowered), e.g., for lower seventh (raised), e.g., for

= F = B

Acoustic numerical value: 7 , and its octaves 14, 28, 56, 112 . . . respectively.

Combinations of the 3 types Referring to

↑

and

↓

= 1st upper third ( ) of the 1st upper fifth ( ), or the 1st upper fifth of the 1st upper third = B = 1st lower third ( ) of the 1st lower fifth ( ), or the

1st lower

fifth of the

1st

Dominant third in

C major A minor

lower third = F

In the musical sense, this form of notation is to some extent an addition of intervals:

or

But in the mathematical sense, in all similar cases (i.e., when 2 or more grammalogues appear on the same side), a multiplication value is introduced: means thus 3 x 5, or 5 x 3 =

Starting from Starting from

, ‘F’ as , ‘B’ as

15

means thus 3 x 5, or 5 x 3 = 15

can be understood as dominant seventh, calculation 3 x 7 = 21 can be understood as dominant seventh, calculation 3 x 7 = 21

Or: F and B are interpreted as counter fifths, i.e., (

) F. , (

)

.B

( 3 3).

The necessity of an accurate comparison between nominally identical tones appears here. My grammalogues show at once to the musician who is not conversant with numbers the difference between F (pure fifth) and F (lowered by a Leipzig comma: ca. 1/8 tone). The purely mathematical solution is carried out as follows: (1)

F = 3 ; instead of the 1, an octave number which is larger than the denominator is set as 4

numerator thus: 3 = 11/3, whereby the lower twelfth becomes upper fourth. Now once more this value (11/3) is multiplied by an octave number, so that the product comes into the general vicinity of the number with which it is to be compared. The 21 belongs to the octave 16—32, therefore 11/3 is to be multiplied by 16: 11/3 x 16 = 1616/3 = 211/3 .

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If one wants to eliminate the fraction, express it all in thirds, which can then be crossed out: [21 · 3 =] [211/3 · 3 =] 63 3

:

64 3

= Leipzig comma

If 15:16 is recognised as half-tone, then 30:31:32 are two quarter-tones; if 31:32 is a quarter-tone, then 62:63:64 are two eighth-tones and 63:64 one eighth-tone. The ultra-dominant 7th

is C, musically speaking, where the primary tonic is taken as

: the seventh of the 2nd fifth, mathematically: 3 x 3 x 7 = 63; corresponding octave 64!

however is at the

That these comma differences are easily demonstrated in practice will be shown in the following: In the Eroica (Trio from the Scherzo), Nikisch always left the 7th ‘plain’ in the 2nd horn, i.e., blown as a natural tone; this tone, notated B (when an E horn sounds D ) is slightly but noticeably flattened in relation to the octave or fifth. This chord on the 7th merges quite obviously into a tonal unity (‘concordance’). But if to such a 7th-chord the pure octaves or fifths of other brass instruments are added, likewise the equally tempered tones of woodwinds, then the ‘natural-7th’ becomes unacceptable: it appears impure and must be replaced by the upper octave of the normal, large under-2nd (i.e., the normal double under5th), i.e., for brass instruments, the appearance of octaves on the first valve = B (but for the natural-7th, on the other hand, B ‘plain’). The 7th is pure! It is only ‘impure’ if it must appear instead of the second under-5th. And it is impure in relation to tempered values. Many brass players with sensitive hearing know this quite well, for they pitch the 7th in unaccompanied ‘horn passages’ differently from when they play in combination with tempered instruments, or when the nominally identical tone changes its harmonic function. o plain, flattened B (rel. 63) 1st valve, pure 5th B (rel. 64)

4 Solo horns

In double-tone (bi-concordance)

4

horns in D

4

horns in C

= 63 instead of 64

which with both its pure 3rds and 7ths merges into a wonderful, tender, serene harmony {lit. over-harmony}, the upper C (1st horn) is definitely not an octave repetition of the fundamental C (8th horn).

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The tones are evaluated as

In the form intended by Nature, this chord of eight merges into a complete unity (absolute purity of the and is required!).

String instruments

Double bass C string

The excellent effects of the sounds

are immediately explained and made understandable by means of acoustic analysis. (Didymean + Pythagorean!)

i.e.

is in the upper triad sense

, or contracted to

it is the lower triad: actually it is consonant in both cases. As upper triad it correlates with an omitted root C whose 3rd and 5th are exactly E + G; the B correlates as a 3rd with its relative root G. The sound is thus in a functional sense a bissonance (C major + G major with omitted outer tones).

upper triad

Thus its bass notes are often C or G.

(See later: The “Substitutes”)

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As lower triad it is simply a consonant basis as a strengthening of the bass.

and develops in chordal sense its consonant

Bass note (yet not ‘fundamental’) It does not substitute for another sound; on the contrary, it perhaps lets itself be substituted twice. (See later.) A large number of dissonances which are completely unintelligible in the upper triad system, lose their mysterious character in lower triad evaluation:

or laid out otherwise

i.e.

Just as

or

(atonal sounds)

or even more simply interpreted: D upper 5-note chord + C lower 4-note chord (7th relationship)

is understood as the contemporary complex of the dominant-tonic succession,

the complex shown above at is thus only to be understood as the vertical tonality of the horizontal succession: minor mediant—minor bicordance {see p. 77}.

(in A minor)

Whether in sequence or in combination, the tonal values of the chords remain the same. In the succession each individual sound is of course Against this, in the sense of tonality, all individual notes must refer back to the

.

primary tonic (the sound root of the functional centre). If all 3 sounds appear together, they keep their chordal values:

i.e., in the order of presentation:

Thus again quite simple!

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VIII Erroneous Conception of the Minor Consonance, as well as the Major and Minor Concordance in the Sense of Dissonant Forms Although the concept of the minor chord as ‘natural consonance’ is already 350 years old (Zarlino’s fundamental work was published in 1558!), it is still not generally accepted. Even Rameau failed to secure this position in the Theory for all Times for the minor chord, with which he already had practical experience: as a natural sound coordinated by the major consonance. Culprit for the misjudgement of the character of the lower sound was obviously the ‘Precepts of Thoroughbass’, which was not in the slightest a theory but rather a wellestablished practical concern, serving regardless as a basis for the ‘Precepts of Harmony’ for hundreds of years. Thoroughbass (figured bass) notation is shorthand for “chords”, which has nothing to do with the character of harmony and sound relations, but comes from the bass part and evaluates from that, by intervals, the chords above it, without regard to tonal lineage and the functional significance of the elements. Third and fifth are denoted by 3, 5 or ‘not at all’, as absence of either figure was understood to mean ‘fundamental triad’. (This is not developed further here.) In this list of chords, a triad is a triad, regardless of whether it is diminished or pure, whether major or minor chord! The key signature automatically caters for differentiation:

diminished

major

minor

The diminished triad is represented as a major chord with sharpened root note or as minor chord with flattened fifth, and the minor chord as major chord with “flattened” third! Well yes . . But that still today so many theorists regard the minor chord as a major chord(!) with altered(!!) third should make you cry rather than laugh! A craftsman knows more about his materials than these people, who in the end still regard themselves as “Music Scholars” . . . . The major chord, in the sense of frequencies, is the complex of , or in narrower root position

Now

The minor chord is, in the sense of frequencies, either

or

In the latter case, if it had no fundamental, to which the 12 (i.e., 6 or 3) could refer. is the same as , i.e., reduced to 10 : 12 : 15, i.e., taken farther onto

the common numerator 60 as: when cancelled ! Thus it remains: also in the monistic conception {as opposed to plurality, that only one being exists}, the minor chord is accepted as a lower sound

and as polar counterpart to the major triad!

I defend the chord of the dominant-seventh as a naturally intended integral unit which I call ‘concordance’. It develops in close harmony in the 3rd octave of partial tones as a natural form and includes the naturally intended first three proportions of the mathematical prime numbers, 1:3, 1:5, 1:7.

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The next uneven number (no. 9) is no prime but rather a power. % Here, thus, a natural void gapes. Nature will include here a matching group in order to develop new types (dissonances!) on the other side of this border. The naturally directed border runs between and 7 and the 11, but not between 5 and 7! As Oettingen in 1866 mentioned nothing of a natural-seventh in his dual harmony, and his heir Riemann held the opinion that the seventh is a 2nd under-fifth knocked up two octaves, then of course their unoriginal and uncritical descendants likewise denied the existence of a ‘naturally-intended’ seventh in the sense of usable harmony. Prof. Krehl indicated, to a budding student of harmony, the natural-seventh 7 as ‘harmonically just as unnatural as a “simply hideous” sounding element’. Quite soon {lit. two or three days} after that the very conscientious candidate could be convinced by two organ stops, Septime 11/7' and Echo Cornet (8' 22/3' 13/5' + 11/7') and by a horn quartet, what the state of affairs is with the ‘unnatural’ naturalness of the harmonic seventh . . . Organ builders determine pitch — from established custom — according to the pipe length measured in feet. If C is 8 ft long then the ‘11/7’ ft is the 7th part. Thus the seventh is not tempered and no Riemannesque second under-fifth in octave transposition. Organ builders, completely as a matter of course, include the natural-seventh (of the Pedal 32' = 44/7', of the 16' = 22/7', of the 8' = 11/7') with the consonances, ‘because: tuned pure, it blends with the fundamental!’ They are right! The chord of the seventh is described by Riemann as brought into a common octave range , or in close harmony

, i.e.,

, i.e., if

. The naturally given form, however, is .

The difference in pitch between 7 and 71/9 = ‘Leipzig comma’; it amounts to about 1/8 of a tone.

and

, which I am denoting as

IX The Comma-Difference and its Functional Significance in Equal Temperament Tuning Equal temperament tuning levels out all comma-differences and creates an artificial range of twelve semitones. Nothing is more obvious than to ask, “Is then the minor chord still a naturally given lower sound?” “Is the seventh still a 7?” “What sense is there in the mathematical formulae when the measurements are still not correct?” . . . To these the answer: the essence of the naturally given form is in no way altered by artificial stylisation nor by a necessary compromise on practical grounds. If a painter excludes assistance by mechanical aids (ruler, protractor, compasses) and because of this his straight lines, angles, curves and circles do not match precisely the exact geometric (thus naturally given) forms, the observer experiences nevertheless the shapes portrayed in their natural, pure form. He satisfies himself with approximate values ‘in the sense of the ideational archetype’.

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But when it comes to understanding the archetypes in their proportions, one would not choose “stylised” circles, ornaments, etc. of a painting as geometric pattern. Thus: in one case, the “exactness” of the form is insignificant, actually nothing at all worth striving for, but in another case it is a sine qua non {an indispensable condition}. So then the countless deviations from the exactly pure form — as they are inevitable in the tempered 12-semitone system — are also by no means questionable in the ‘artistic’ sense, as we still experience the levelled values in a naturally differentiated way, by virtue of our conscious ability to perceive and our musical logic. (Tonal psychology, selective hearing, functional presentation.) The comma-difference tones and sounds are of course levelled in equal temperament, but they are still “experienced” in different values; because of this, they receive diverging grammalogues in my indication of functions. with reference to

:

fifth types

Principal chords E major (perfect fifths) replacement

third types

Mediants E major (comma lowered)

double third types

seventh types

seventh types

third-seventh types

seventh types

Mediants F major (comma lowered)

Principal chords F major (perfect fifths)

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Against 1st

taken as the norm (fifth tone), is 2nd

lowered a tenth, 3rd

lowered a

raised a tenth, 5th raised a fortieth, 6th lowered a fortieth, and 7th fifth, 4th syntonic comma (= 1/10 tone); Leipzig comma (= 1/8 lowered an eighth. [. . . . = pure; tone).] (Details in Acoustic Regulation of Sound and Determination of Function by S. Karg-Elert; C. Rothe Publishing House, Leipzig.)

X The Cosmic Physics of Tones and Sounds a) General Comments Chains of tones and sounds point to infinity; there lies their pole, to which they tend. pole; energetic maximum

pole; gravitational maximum

Each point on the horizontal can become centre for vertical forms, each point on the vertical can become centre for horizontal forms, each point on the vertical can become centre for new vertical forms (harmonic products or powers).

The expansions vertically and horizontally are so tremendous (hypothetically infinite) that our spherically limited tone-sense very soon faces insurmountable inhibitions.

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The linear sequence (Pythagorean leading notes:

is in the purely melodic sense a limma-chain ). The limma is, however, never ever a naturally

narrow to

given interval of ‘primary value’, just as little as is the whole tone step. The fifth-powers will only become neighbouring tones of through projection of themselves in the line; as and they are definitely in their original positions as indicated in the following spheres: Higher octaves #

%

This is a distance of nearly 16 octaves:

Deeper octaves

Each further addition of a flat or sharp tone shifts the borderline downwards or upwards by a twelfth, so that the bi-enharmonic interval covers a range of somewhat over 36 octaves in its (2/9 of a tone difference)

original form. Octave reduction will result in chains of fifths instead of chains of twelfths, i.e., instead 1

3

2

3

of the original form 3 : 1 : 1 we get 3 : 1 : 2 . In this case, the specific character type remains (the relationships of tension are admittedly no longer the same):

But even in this compacted form, the scale of fifths still exceeds the limits of our range of perception (‘D ’ would end up in the sub contra-octave range and ‘C ’ in the 6 stroke{ } range!). If also the double fifth (ninth) is reduced by an octave, so that it falls within the range of one octave, then the result will be the ‘whole tone’, the basic element of linear form.

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The further development of continuous scales is to be understood in the same way. (See also the table on page 4.)

Nature’s desire for the scale is immediately obvious. It is truly deplorable how superficially and worthlessly these fundamental issues are dealt with in most music teachings: e.g., a.) The melodic range is extracted from the 3 main triads, resulting of course in large and small whole tone steps, narrow major thirds and broad minor thirds, wide leading notes, comma-shortened (thus imperfect) fifths (D A), deep sharps, and high flats, in which no musical melodist believes . . . b.) The main harmonic realm is formed by the cooperation of 1st 3rd 5th || 4th 6th 8th | 5th 7th I

IV

V

and 2nd steps, where each chord would then appear with the unharmonic (because it is Pythagorean) third, something which again no musical harmonist believes . . . c.) But by far the worst is the derivation of the scale from a uniform overtone series . . . . !

(!!)

(!!)

(8, 9, 10, 11, 12, 13, —, 15, 16). Something more amateurish can not be imagined. A formation or a compass will be ‘harmonic’ merely by the addition of the natural primary third to the previously given complex of fifths. The ‘spatial’ distance of the interval of the third is in the original sense considerably larger than the relationship of a tone to its fifth. Fifths Thirds

We are again facing incredible spatial expansion: if triad suspended from

is set as central tone, then the

and that mounted upon it yield the following form:

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Consequently, the thirds in their original form are drawn centrifugally about 21/3 octaves from their fundamental notes. Even larger is the interval of the seventh in its original form, for it comprises about 25/6 octaves.

fundamental

Thus, and not otherwise, is how one is to envisage the original forms of natural harmony! % The sevenths have a tendency to reflect in the dissemination of sound. A very important discovery, which can surely be proven everywhere in living music, but which the Theory of Music — as far as I know — has hitherto had to overlook completely, as it did not consider the seventh to be harmonically valid on its own!

in the harmonic sense

in the melodic sense

Pythagorean leading notes (melodic) “Limmas” (uniform) - normal -

The third moves forwards (in major , in minor

), the seventh, however, backwards.

The case rests with the harmonious: each sound develops in its ‘dominant’ the natural fifth of the same kind in the forwards direction (centrifugal). Each sound (regardless of function) develops in its ‘mediant’ the natural third of the same kind in the forwards direction (centrifugal). But each sound develops in its “seventh counter-sound” the natural tetrad of the opposite kind on the seventh in reverse direction (centripetal forwards!).

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The about-turn depends on the naturally-given cleft between the harmonic and the ultra-harmonic fundamental values:

upper triad

lower triad

Just as the

can each be shortened by one octave, without forfeiting their specific 5th-

character, so also can the values.

and

each be reduced by two octaves, without losing their type-

limit for functional representation

the upper-7th lower-3rd is identical to the lower-3rd (B ) upper-3rd (A ). the lower-7th upper-3rd is identical to the upper-3rd (F ) lower-7th (G ). (7th counter-sound)

hybrid form:

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All functional representatives of a sound respect the gap of the ninth: parallel- and leading-note changed chords, variants, parallel-variants and variants of parallel sounds, same-third chords, counter-sevenths, mediants and counter-mediants; the concordant form of the mediant alone seems to be an exception, as its seventh “appears” to step into the cleft: . But in reality this is not at all the case; more precisely, it comes in only when there is tempered tuning, where all fine distinctions are put aside! The horror of the hiatus remains, and this very example is classical evidence for the naturally intended slight differences in value: the mediant based on the syntonic third is thus narrowed by a tenth to the unison (as measured at the 4th fifth), its seventh is narrowed by an eighth to the primary mediant (as measured at the 2nd counter-fifth): consequently the negative gap at the ninth amounts to one syntonic plus one Leipzig comma (1/10 + 1/8 tone, i.e., nearly a quarter tone). My grammalogues make this contraction obvious straight away.

b) The Hypothetical Centre of Sound Each consonance has its own sound root, which functions as unison in the sense of energy. This unison is ‘absolute’ in its sound, but in the sense of tonality it is a relative value when that sound is not a tonic. The primary tonic is ‘absolute’ for the valid range of tonality, but on the other hand the centres of tonality are relative values, measured at the general centre of all centres.

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It is obvious that C major and A minor can be assumed to be central keys , and this question was never controversial. But then we must assume logically two tonal centres: ‘C’ as prime major and ‘E’ as prime minor. Oettingen and his following (Riemann and a few others), however, do not accept

and

primarily as double primes, but refer these two keys to a central one , from which the C and E come! An assessment incomprehensible to me. Incomprehensible in all kinds of respects: 1. The allegedly ‘absolute’ unisons are degraded to relative ninths. 2. D major - G minor are regarded as central keys . . . . 3. C major climbs upwards (!) to A minor (via G major), or A minor climbs downwards to C major (via D minor). 4. The polar dominants are striving towards each other (!), instead of mutually repelling each other, as “strong” types!

I start a priori from a symbol in harmony (natural third!) and put it in the middle of our tonal range that serves practical music:

S. Karg-Elert

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This central third is, in my opinion, the germ cell from which our entire tonal range is developed:

{ EC

is given as 4 : 5 i.e., the original form 1 | | 5 , which by that very fact requires 5 : 4

3

the 3 : 1 : 3 : 5 alias 5 : 3 : 1

5 | | 1 4 : 5 : 6

6 : 5 : 4

Upper fifth

Central tones Simultaneous 3rd

Lower fifth

From here on, from this position, which from now on in this book I shall call the ‘principal original position’, all horizontal and vertical forms will be developed:

B major

X is the mathematical central axis, which maintains the same distance to central to central

. From

to D is a normal whole tone step 8 : 9, from

as it does

to D likewise; but

to

is not in a Pythagorean double whole-tone relationship 64 : 81, forming instead the natural-third, 4 : 5, which encloses two whole tones less one syntonic comma cut from each.

Two normal whole-tones minus one syntonic comma = one normal whole-tone + one narrow whole-tone = a natural-third 64 : 80. The ideational X-axis is at a distance of half a syntonic comma to the real tones D D.

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The X-axis can never be a real tone as the displacement of a note upwards or downwards by a twentieth would indeed likewise dislocate the central-third. The absolute pitch would be changed but, on the other hand, the proportions would of course remain the same! Only this comprehension of the X-axis as merely a spiritual concept, which can never be experienced by the senses, is fully covered by the laws of physics. By complete contrast with Oettingen, Riemann and Krehl I am obtaining original positions which correspond much more to naïve feelings than those noted on page 27. (Examples 48 & 49 )

Original positions:

a) X-axis (ideational). Central double-prime and at the same time, central double primary third and central interval of a third. b) Original positions of the tonics of the central keys. c) Second degree of distance (+ and -) from the centre. d) Dominants striving to diverge. Ninth-chords: not matching original positions not matching chord types of the same name. E major above e) Mediants (according to Oettingen, the other way around). F minor below

{

As seen from this original position of the C major and A minor chords, all other original positions become easily understood. But as now, for example, generates the note ‘E’ as 4th fifth and as 1st third, or the note ‘B ’ as 2nd counter-fifth and as 1st seventh, thus does it follow also for E and E, as well as for B and B in their divergent original positions.

narrow 2 octave interval

wide 2 octave interval

From this example, the following rule can be put forward: Each tone returns at a distance of 2 octaves, identical ‘in name’ but different ‘in type’:

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(tonic)

That each original position can be transposed at will by an octave is obvious. If notes of different types are brought into the same octave, this always results in minimal differences (up to a 108th of a tone! = difference between Pythagorean and tempered fifths) of a quite definite interval size, which always stays the same.

This makes 11 different notes with the same name! If one adopts 25 different note-names:

then we obtain, by means of centripetal octave changes, 275 (25 x 11) effectively different pitches within the compass of an octave! Two hundred and seventy-five notes in each octave compass . . . . . Two questions are now obvious: 1. How far does our ability to differentiate sounds extend?

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2. What value have such compilations of, for example, the 275-fold octave subdivision? re 1.) In a tonal succession, half a syntonic comma (= 1/20 tone) should be the ultimate that can be sensed as a tonal shift. In the simultaneous sounding of two notes of different pitch, the absolute tonal position will determine the limit of our ability to discern. The difference manifests itself through the speed of the beats. If for example a note Xa produces 1296 and Xb 1280 vibrations per second, the result will be 1296 minus 1280 = 16 beats per second, i.e., the same frequency as CCCC! , therefore the notes Xa and Xb differ by a 10th tone (syntonic comma). If both those notes in the next lower octave are compared 648 : 640, then they produce only 8 beats. ; the difference for this position is proportionally the same as for the first mentioned! Or a note gives 1260 vibrations and another 1280, then the superimposed waves , therefore generate 20 compression waves (1280 minus 1260). these notes differ by one Leipzig comma (one 8th tone). If, on the other hand, two low notes produce only 4 beats — and their effective frequencies were 32 : 36, then those notes would be a whole tone apart: . % One understands accordingly that very narrow, low-pitched intervals cannot be distinguished at all! re 2.) The value of a theoretical compilation of a 275-note octave compass is definitely not chimerical {imaginary, fanciful}. Admittedly, exploiting the entire tonal range within a short piece is not feasible, but the stepwise pacing out of the total range of differing partial centres may well be possible without further ado: ‘E ’ as limma to “D ”, ‘C ’ as C major

limma to “D ”, generally referring to A minor . Even if, in practice, ‘E ’ hardly develops its under-third 7th and likewise ‘C ’ its upper-third 7th, the instance of mediant-3rd development for principal tones not as remote from the tonal centre still happens often enough. The principle of this type of consequence remains, even if not every type establishes the entire 25 tone scale. relatively high overtones ____________________ (. . . hypothetical range . . .) low principal notes ______________________________ high principal notes (. . . hypothetical range . . .) ____________________ relatively low undertones That nature has endowed us with the ability to sense harmony logically, i.e., to comprehend intuitively beyond a real stylisation those incredibly finely graded pure values, is repeatedly to be stressed emphatically here.

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(lowered )

(raised )

C

30-note compass in pure values

c) Central Detail from the Tone- and Sound-Cosmos

(lowered )

(raised )

Treatise on Polarity

x—————————————————————————————— x – – – – –x– – – – –

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lowered

– – – – – – – – – – – – – – – – – – – – – – – – – – – –– – – – – – – – – – – – – – –x

raised

The 30 pure vales brought into a common octave range and compared with the 12 tempered semitones:

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d) Analysis The table above, which is most eagerly recommended for as careful a study as possible, is an excerpt from the present polarity teachings, and it shows all essential characteristics of tonal and sound relationships, comma differences and dissonance valuations, as they are put forward for the first time in this work. The first impression is confusing. Admittedly! It results from the multi-meshed overlay and overlap of semicircles in three different sizes. But this very form of representation is uncomplicated and analytically clear straight away, symbolising the naturally given tonal and sound relationships. % The upper system refers to C major tonic, the lower to A minor tonic. The separation achieves a fundamentally clearer, graphic display of polarity.

contra-

As a result of the combination of both types, the three principal sounds arise, in which it is very significant that the major-sounding subdominant in the major, and the minor-sounding upper-dominant in the minor are problematic forms.

-

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The inconsistency of the contra-dominant is “evened out” if, to the contra-fifth, the contra-third is also added: minor triad ( type) in the major, major triad (D) in the minor: “tempered” (balanced) contra-dominants.

In the typically qualitative sense (not the spatial sense) of undercutting and surpassing representative forms: parallel sounds (reductions) and leading-note changed chords (reinforcements) become known as incomplete fifth-related(!) double-sounds: (reinforcement of the

form, major-sounding lower triad)

(reinforcement of the

form, minor-sounding upper triad)

Here the narrow ‘tonal circle’ closes. and are the first to cross the border (= thirds of the ultradominant). The Pythagorean comma (O: octave-reduced 12th fifth) is of course not represented in 81

this tiny extract from the enormous tonal compass. But the Pythagorean third 64 exists twice in the linear series of fifths B : D and D : F If these tones are compared with the reciprocal thirds then syntonic comma differences result. Likewise (= a 10th-tone expanded double octave). The Leipzig comma is represented several times:

(= an 8th tone reduced double octave).

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Furthermore, complicated Leipzig comma differences result from:

(9/40 tone = 35th : 36th partial tone) The small diesis is recognised as 3rd third

_________ _ _________ All thirds can develop their perfect triad forms from both sides. (The fifth arcs of the thirds are omitted, so that the typographic picture is not excessive, causing confusion.) (dominant-mediant with 7th)

(tonic-mediant)

(

"

"

"

"

, or

(Contradominant-mediant)

, or

(

"

"

)

)

In the same way, the counter-mediants of all principal triads are to be understood: Tonic counter-mediant likewise

The connection D major : B major, or G minor : B minor becomes clear straight away:

(strongly +)

(strongly -)

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Beyond the dominants are the “ultradominants” ; they are not represented in the table (B major—D major | G minor —B minor), but B major and B minor are available as and . If ultradominants developed their outer mediants (G major—F major | E minor—D minor), then the tonal chain E minor—F major would be closed by the combination of the second (series no.) I with the first (series no.) II relationship . But even without the ultraforms this closing is also possible: through the (C major—E minor)

7th-relationship (7th-counter-sounds): (A minor—F major).

The same applies also to every other principal triad. The table shows a partial twofold to fourfold filling, similar in name, but typical, i.e., functionally different tones (compare particularly with the table, fig. 18, on pages 32/33 also). This range of relationships of a central sound yields the following tonal chain:

(Principal basis)

First relationship of fifths, 2nd relationship of thirds of the principal triads, 3rd rebounding relationship of the seventh upon itself. Through a further adjustment of the relationship of double thirds, the chain would be extended on both sides: G A

minor, minor,

B C

major, major,

D E

minor, minor,

F G

major, major,

A B

minor, minor,

C D

major_ major_

But despite ever continued widening, a uniform chain is by no means given, because the relationship fails to connect to the 1st, just as little as the 3rd does to the 2nd. Nominal continuations exist, of course ····G B D or · · · · B D F B D F···· D F A · · · · etc., but they result in comma 2nd

differences! B : B etc. The continuity is created merely by the Pythagorean chain run of similar types (I or II or III).

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One will now readily understand that the number of 12 nominally different tones is by no means sufficient in order to set up all kinds of ‘tonal keys’ (as the pianist only too easily assumes); rather, it requires a corresponding shift of the centre on a Pythagorean central basis! If, for example, the G major chord becomes fundamental, its dominant cannot be the D major chord of the 2nd relationship (i.e., the major mediant {as defined by Riemann} of the principal B major chord), but instead — the D major chord connects to that of G major in the same way as it does to the C major triad: 2nd

type

1st type

The same applies also for all other double and triple sounds {or triads}. The main significance of this table lies in the disclosure of naturally given dissonances. The trite explanation of chromatic dissonances as “altered forms” says not the slightest, of course!, as the question of the ‘alteration’, i.e., artificial semitone flattening and sharpening, still remains unresolved in its essence. The table gives a clear analysis of all diatonic and chromatic dissonances. Riemann speaks of “characteristic additions to the dominant and subdominant”. (His successors made “characteristic dissonances” out of this!); intended are the forms:

The fact that in this designation the corresponding natural-forms are completely blurred strikes every impartial observer, because it corresponds reciprocally to the forms: i.e., in my evaluation of polar function: D D

Riemann’s tremendously disturbing inconsistency is caused by the hermaphroditic nature of his values {meaning here: combining two opposite qualities}: Chord tone count:

in major upwards in minor downwards

polar

Dominant count D upwards D upwards

S downwards S downwards

parallel

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Equally unnatural is his explanation of the characteristic additions as ‘hybrid forms’ of D + S! Only in the tempered system is this true, but in the tempered system there are also no upper sounds nor lower sounds! The table shows the natural form of the dominant seventh-chord clearly:

Lower tetrad

Upper tetrad

The addition 6 or VI is recognised as octave-shifted third on the 2nd counter-fifth, which probably in name, but by no means in substance, resembles the tonic counter-seventh:

Parallel-sound Parallel-twin

The upper sixth of the F major chord is lowered from as third of the lower-fifth: D =

2nd

Parallel-sound Parallel-twin

The lower sixth of the E minor chord is

raised from as third of the 2nd upper-fifth: D = ________

but in the F minor chord, ‘D’ becomes a strongly elevated lower-seventh: D = .

but in the E major chord, ‘D’ becomes a strongly lowered upper-seventh: D = .

As always, third and seventh run contrary to each other: Consonance Consonance

He who does not feel the stress relationships of the tones of the same name ‘D D’ is just dulled with regard to the temperature of the differences willed by nature. For him, of course, there is also no difference between ‘G and A ’, as he hears comma-levelled values.

The counterpart also becomes clear:

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From a study of the table, a dual evaluation of chromaticism comes forward unambiguously: B — B or F — F . wide alteration) a) a fifth can become a third — (chord 24 : 25 b) a third can become a seventh — (chord 15 : 14

narrow alteration)

If the chromaticism concerns the third, so that out of a minor chord a major chord of the same name develops — or vice versa — then I talk about “variance”; instead of

The chromaticism of the fundamental, in the sense of a narrowing of the fifth, results in the ‘problematic triad’ (popularly = diminished triad), peculiar to different functions. In all cases it is a potential natural tetrad (concordance) or a double-fifth related dual sound:

1.

2. Or:

3. The simplest evaluation of function as ◦D in D major, or

◦

in G minor, is emphatically

the most unnatural form, as the syntonic and Leipzig comma difference comes in.

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corresponding, of course, in the minor!

If such comma differences are eliminated, i.e., if one type replaces another, then ‘Metharmonic forms’ result (technical terminology of von Oettingen).

Where the wide alteration of the fifth and ninth is a consequence of the third relationship, then the narrow alteration of the fifth and ninth mostly goes back to the seventh-relationship. The table analyses these forms in the clearest possible way:

A)

But the table shows still more: the narrow alteration of the fifth can also come about through a crossing of tritone {3 whole-tone} thirds: B)

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A) are closed, B) fragmented forms! One now understands, however, quite easily: each chordal tone is capable of octave shifts, i.e., the form of each chord is mobile without restriction, each naturally close position can be dispersed — the harmonic types remain unchanged. (A third remains of third-tone type, even if it is crossed to the sixth . . .) A and B, however, definitely do not stand in the relationship of position-transferred consonances, but they represent different original types. Acoustically, substantial differences remain, therefore, they are for me functionally clearly differentiated according to their nature: the original positions tell it all:

Even if A) and B) are shifted into the same position, the differences in type remain unchanged:

bissonant

concordant Whole tone!

That these natural differences effectively remain for the inner ear, I had already noticed 30 years ago, without obtaining an explanation from my “Piano-keys Theory-master” Jadassohn:

Contra-dominant type!

Dominant type!

Furthermore: the table lets the ambiguous nature of the diminished seventh-chord be perceived as the original so-called “small” ninth-chord. And as it is a narrowly altered form, then its existence results from the (secondary)

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seventh-relationship of two tetrads in counter-tonality (antonym):

in C major

in A minor

in C major [in C minor

in A minor [in A major

Comma-differences remain between sounds of the same name; they are therefore also designated differently by me in function. As can be effective as

(smooth)

or

(dispersed) ,

as more distant can be effective as

(smooth)

or

(dispersed) ,

then the diminished seventh-chord can also appear in fifth-related counter-form:

But still the table shows much more! If it takes on thirty tones related to

, then it also establishes the compass of a 30-note

chord! Undoubtedly. Thereby we have said only woefully little musically . . . . Just as little, as from the theory of certain preachers on Atonality, who spread their ideas in the following way: “The chromatic scale is a chain of 12 consecutive semitones. They can work together to produce the 12-note sound with over 479 million different possibilities of grouping.”

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Only! No no, for if it’s to attain a record, then the bagatelle of only 479,000,000 forms does not impress me at all! One should clearly understand: this number results from the permutation of 12 values. If instead of 275 original values I take about 30 only — values which any normal ear is easily able to differentiate, making up only a tiny part of the harmonologically conceivable elements, then I obtain by permutation such gigantic numbers, against which the 43/4 hundred millions appear a bagatelle! And once again: one understands me correctly if I — seemingly paradoxically — speak of a ‘bagatelle’: 12 tones are the ultimate that these theorists can employ as intrinsic values; the 13th tone is of course, in the tempered system, the 1st note again (displaced by an octave). But in the pure natural system there are no limits at all and 275 tones, or 53 (Oettingen’s harmonium) or 30 or 12 or 7 (diatonic system) or 5 (pentatonic system) are in the same way only excerpts from an infinite range that nature reveals in pure form.

I consider the 12 semitones, for practical reasons, a partially corrupted or even stylised concise form and amalgamation of innumerable natural elements, whose own will however still triumphs over all masking . . . My demand on pure natural means is an extremely modest one, as I call for only 3 basic elements: . If I still wanted to place the partial tones and into my toneanalytic studies, then octave-parcelling would go into the 1,000th part! I quite believe in the nature-intended unit forms (smooth!) approximately in the following formations: polar: deep high

deep high

in the sense of :

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Much more complicated is the form without adoption of the

or still differently: What riches in the forms of approximation {or convergence}!

XI New and Atonal Forms respectively A conversation between one of the radical leaders of the musical left-wing party and me (in 1914) should be imagined as follows, since it has significance regarding its principles: Aforementioned artist was of the opinion that the so-called “new sounds” are wilful formations which stand in blatant opposition to ‘surviving natural forms’(!). “The consonance is dethroned; we have put in its place the sharpest dissonance, the split whole tone, and chosen this as starting point for the further development of new sounds.” (This sounds like an edict {lit. ukase} or a senatorial decree!) To that my objection: “Do you — earthborn — really believe then you can escape the meshes of natural involvement? Your ‘inventing’ can always be only a ‘finding’ of naturally given possibilities.” The radical scornfully strikes and says, “There!, where is your theory of natural sound?” My reply: Sound is not primitive, but who then obligates nature to the primitive state? If I assume that (some) tones are called D D D (whereby it is completely inconsequential whether they are valued in the sense of D D D or D D D or D D D , as indeed their proportions remain the same), then they are representatives of the natural consonances:

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The dissonant three-semitone sound thus calls for still two more three-semitone sounds:

If however the 3 semitones are called C D E , then their analysis is much simpler: or

Here is the original form:

The narrow semitone-position is in this case a crossing form. That these formations are a starting point for new constructions I foresaw in 1902. Instead of the formula: fundamental + fifth + third comes fundamental + fifth x third

Finally, the formation C D E is still referable to the dominant seventh-countersound relationship, often invoked here:

important atonal five-note sound

diminished ninths

These are definitely not artistic constructions but apparently natural forms of the finest differentiation. Given the 3 varieties indicated, the possibilities of interpretation are of course not yet exhausted. Further types:

or

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These five forms compared with one another:

5 different semitone-sizes in pure values against 1 tempered semitone in the equally tempered system! The table finally shows also in a clear analytical fashion the naturally intended line of development to unlimited multisound (polysonance, which by kaleidoscopic transfers and crossings that destroy the natural organism, becomes the atonal sound). diatonic full sound Natural form destroyed organism, unintelligible

Syntonic comma:

illusory comma-differences

Leipzig comma:

Natural forms

or

or

All diatonic complexes are given. Of the chromatic dissonances, these are given: augmented triad and diminished seventh-chord, this (latter) however in dispersed form. The Leipzig comma is still doubly given: or

Natural forms:

or or

or

wide

narrow

Extreme diatonic substitutes also occur. Doubled chromatic intervals 14:15 | 24:25 | The so-called augmented 6/4/3 chord appears in natural sequence:

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Natural forms: extreme mediants

Nine-note chords

Nine-note chords

Ten-note chords

Leading-note and chromatic tensions grow considerably. They have an effect, which is all the greater, the more closely the chord is massed, e.g., when omitting the core:

in the most recent stylistic epoch very common If the central twin forms its dominant-mediant pair, the following results

Natural form (2nd type)

a twelve-note chord with a nominally identical tone-pair (syntonic and Leipzig comma). But if on the other hand (see table) the central twin develops its dominant counter-seventh pair, the B major and G minor sounds appear with their positions interchanged compared with that last example:

Natural form (3rd type) Ten-note chord The B major triad is lowered in type II about 1/10 tone, but in type III raised about 1/8 tone; the B minor triad is raised in type II about 1/10 tone, but in type III lowered about 1/8 tone. These are close to quarter-tone differences, which at a time of commencing quarter-tone propaganda cannot be ignored as hypothetical values.

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The much-praised semitone-complexes are also, of course, demonstrable in these natural forms: 9th-forms,

Type III

Type II

5th-forms,

Type III

7th-forms, 4th-forms

Type II

The newer literature is completely interspersed with such sound forms which popular theory, with its mechanical steps and cheap alterations, simply helplessly opposes. The ten-note chord cited last is the basis for the following examples: As quickly as possible

Quietly moving and shimmering

wild and aggressive

changing:

These extraordinarily expressive sounds are undoubtedly so-called “new forms”, which first appeared towards the end of the first decade in this century, but then remarkably rapidly won general recognition in all countries. (Their naturally given existence was already known to me before their practical introduction.) Sound forms alone still do not create art and so the emphasis of these forms as natureintended means should not straight away be identified as a sanctioning of works which introduce them. However, the stale subjective criticism which labels all hitherto unfamiliar as ‘unnatural’ must then be countered, so that limitless nature does not reveal herself to everyone to the same extent. Nature’s realm expands in subjective evaluation with the degree of gifted intuition. The cosmos has for everybody other illusory boundaries - - - - - - -

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XII Harmony as the Essence, the Chord as Manifestation Harmony is to be understood as completely abstract: devoid of space and weightless. Its form of representation in the concrete world of manifestation is the chord. If one strips the chord of all physical forms (‘manifestation’), then only the concept of harmony (‘essence’) remains. A trivial example: four brothers form a close circle of relatives. The weights of the 4 brothers are, e.g., 40, 50, 60 and 70 kg . . . . The brothers part; each goes his own way - - The brothers (concrete) are the bearers of the relationship. The bearers together weigh 220 kg; the bearers scatter to the four winds, but the relationship does not weigh 220 kg, and the relationship does not scatter to the four winds! Translated into musical terms: the chords

are the ‘symbols’ of

dominant harmony (i.e., the concept of the unified, inseparable affiliation of four single elements to a naturally given, easily understood relationship).

= the pitch of the chord has changed. The harmony has remained untouched. = the chord mass seems increased.

The harmony has remained

untouched. Inversion forms and positions can also be understood in a similar way: basic triad; third-bass and fifth-bass chords, or six-three and six-four chords respectively {= 1st and 2nd inversions}; narrow, wide and dispersed positions of a chord (but not ‘close and distant harmony’!). The ‘chord’ at a.) and b.) is the same (“six-four chord”), but the harmonic evaluation is fundamentally different: a.) = it is a C major harmony, b.) = it stands in for G major harmony. Hence I differentiate the sounds by means of functional grammalogues a.) tonic (“fifth-bass!”), b.) dominant (“six-four chord”, i.e., mobile sound

).

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= x same chord (E minor triad), but a.) stands in for the C major harmony and b.) for the G major harmony. Hence I also differentiate both chords in the sense of harmony: a.)

i.e.

b.)

i.e.

The chord is concrete tonal form; its individual elements (‘tones’) are basically oscillating substances. Each substance is subject to the law of gravity. Heavier matter predominates over lighter matter. Longer waves predominate over shorter waves. Thus also do the deeper tones (heavier substances, longer waves) overwhelm the higher ones (lighter matter, shorter waves). The deeper tone of a rudiment, or the deepest tone in a consonance, predominates and forces the higher tones into subordination — regardless of what harmonic value these have. This note becomes the root of the chord, which normally lies most naturally in the bass and is the preferred note to double {repeat at the octave}. Thus the ‘gravitational’ predominance of the tone A in the minor triad hanging from E is conceded straight away. 1/2

Minor triad hanging from E syntonic comma Sound tendency:

shortest wave lightest matter (string, pipe, reed, bell) same kind

always:

heaviest matter 1/2

Fundamental notes

syntonic comma Major triad based on C

(string, pipe, reed, bell) longest wave

But the relationships of energy lie the other way around. The higher notes show a stronger tension in relation to the deeper ones, and an increased activity. The major triad is thus a symbol of harmony, increasing in energy, decreasing in weight, and the minor triad is a symbol of harmony, increasing in weight, decreasing in energy. The fundamental note remains the deepest tone of the chord in root position. It is the “name-giver” of the major, as well as the minor chord. C major—C minor, A major—A minor. (Same rudiment.) It is in this sense of manifestation a ‘chordal prime’. ‘Harmonic prime’, remains of course the sound root in the sense of essence.

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XIII General Survey of (Consonant) Relationships in Sound based on tone commonness All sounds are related if at least one tone is common. Depending on the chord tones, unison, fifth, third and seventh relations arise. The relationship can be of the same gender or the opposite. A. Unison relationship a.) same gender (same sounds) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - (same sounds) b.) opposite gender ‘Counter-sound’ or “tempered*) contra(domina)nt” *) “tempered” = evened out (see later)

B. Fifth relationship a.) same gender ‘Principal or dominant chords’

b.) opposite gender primary: ‘Variants’

secondary: ‘Church-dominants’ ‘same-fifth’ Mixolydian || Dorian C. Third relationship a.) same gender primary ‘Mediants’ (major third-related) secondary ‘Auxiliary mediants’ (minor third-related) “Parallel variants” “Variant parallels”

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b) opposite gender primary: ‘Parallel chords’ secondary: ‘Leading note-changed chords’ {Riemann: substitute leading note for tonic, e.g., in upper triad CEG, change tonic (C) to its leading note (B), giving BEG; in lower triad ACE, change to ACF}

tertiary: ‘same-third’

Mediant parallels

D. Seventh relationship a) same gender primary: ‘Concordances’

Usually in complex form. But in simple formations ousted by means of ultradominants

E major C major A major

secondary: Likewise simpler to replace by auxiliary mediants

C minor A minor F minor

tertiary: ‘Tritone chords’ considerably simpler formation as ultradominant mediants and ultra contra-counter mediants

b) opposite gender primary: ‘seventh counter-sounds’

(The most important 7th relationship!)

secondary: ‘Dominant seventh counter-sounds’

simpler than tempered ultra-contradominants simpler than the dominant-variant parallel variants very common intermediate sound between

D lower triad C upper triad

tertiary: as same-fifth simpler

E lower triad D upper triad

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quaternary: ‘same-seventh’ last and purest consequence of the seventh relationship

— ultimate limit of the simple (direct) relationship — corresponds to normal linear perception:

or

B is higher than C | F is lower than E

The seventh relations can of course be ‘replaced’ by third relations, as can these by fifth relations. But then instead of a ‘simple with increased means’ comes an ‘extremely complicated with primitive means’. And that is very considerable! One thinks, for example, of the connection C major—E flat minor! Established through related 1st order: what a path!

The relations, 2nd order, shorten the path: T = t=

C major C minor

E major = DM (or tP) E minor = Dm (or tp)

Wagner, however, felt differently as he wrote, no — had to write — in the following passage in “Tristan” (last act): (he dies) Tristan: The lights are extinguished! . . . .

What perception! The flame inverts, symbol of death. This E minor sound can not be anything other than a seventh counter-sound. That is no “chord sequence” nor “harmonic connection” in the popular sense; it is a symbol! The same passage returns a few bars later: “Isolde came, with Tristan

(Life and Death) faithful to die”

Conspicuously, many examples from the literature of all times and styles can be quoted, in which the harmony and harmonic occurrences challenge hermeneutics {the science of interpretation, especially of the Scriptures}. Symbols everywhere — but one must have the means to interpret them objectively!

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XIV A) Mobile Sounds in general Chord tones, in the simplified harmonic sense, are to be understood as flexible values, i.e., they are to be seen as replaceable, without their original function seeming abolished. Only in this sense is the free-sounding melody and its simultaneously acting multiplicity: polyphony, comprehensible harmonically and functionally in the sense of wider distance.

At a) an A minor chord is produced, at b) an E minor chord, at c) a diminished chord, at d) one in F major, at e) one in G major and at f) an incomplete B minor chord — however, they are not to be understood throughout in this sense, but rather as “exit forms” of the C major chord. These can — if necessary — be identified by grammalogues, which show accurately their chord tone relations. I am choosing for the leading note exit (minor 2nd) the signs ” ” whole tone exit (major 2nd) ” ” ” ” chromatic interval ” ” ” ” hiatus (augmented 2nd) ” ”

and ” ” ”

(upper and lower movement). ( ” ” ” ” ) ( ” ” ” ” ) ( ” ” ” ” )

i.e., whole tone step together with a chromatic interval +

)

and am putting them in place of the 3-note or 4-note grammalogues, where they are recognised as unison, third or fifth exit {resolution}.

Inversion forms are treated in the sense of the root position:

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‘Splittings’ occur when a chord lingers and at the same time makes its exit or resolves on two sides. Splittings can occur as diatonic, leading note or chromatic passages, or form a hiatus.

(to F major)

In these examples, most mobile sounds occur in isolation; consequently, the grammalogues denoting movement must be linked to those for function. For a more detailed explanation {of 115}: a) C means contradominant*) = in C major the chord (‘Viennese sixth’), then I write C, i.e.,

; if the fifth moves to the sixth

, but if the fifth combines with the sixth (splitting),

then I mark it as C, i.e., b) D means dominant = in C major the chord and chromatically downwards

. Note ‘D’ exits upwards by a whole tone

i.e., together

The stroke above D is always the

natural seventh. The dot indicates the bass note:

#

together

.

c)

. Simpler to show as

d) Triad with sixth (split fifth), natural seventh and ninth as leading note making its exit downwards {NB: the following differs somewhat from the musical example in 115d}:

e) As a), but the sixth is altered in width D D (thus hiatus from C). f) Dominant ninth-chord with chromatic, markedly split fifth ‘D D D ’ and double fifth {i.e., ninth} ‘A A A ’. These examples show spectacularly how easily strong dissonances can be understood as mobile dominant chords, provided they come to a cadence. Their indication by steps (or degrees) must thoroughly fail here. Thoroughly! Whether the dissonant elements are ‘prepared’ or not, whether they act as changers, transitions, introductions, anticipations or suspensions, is a special distinction which has a part to play in the metrical and at the same time linear-melodic field.

*)

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It is obvious that these mobile sounds can be graphically indicated only by broad strokes and with a distinct effect. All melismatic forms remain disregarded in functional notation. Even the rapid cadence-like fragments with rebounds are understood as subordinate mobile sounds, and are ignored in the printed grammalogues.

Prestissimo

(Chopin: Scherzo in B minor)

F D B

to be notated only as J.S. Bach Well-tempered Clavier (I), No. 10

(in brief:)

b

C

b F

b C

F b

(i.e., B minor

as mobile forms in the sense of

e F

e b F

e

chords

{minor chords in lower case; major chords in upper case}

as mobile forms in the sense of

(in brief:)

F

b

(B minor

F major chords)

F C F

b

(D)

E

d E d a d

E major

a chords

A minor chords)

R. Wagner: “Tristan”

specified: in general: possibly! small cadency

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B) Mobile Sounds and Harmonic Polarity The parallel imitation of a voice-part results in contradictions between major and minor function.

The step widths occurring in the parts are copied exactly. But then there can be no question of harmonic analogy. If one does not distinguish between semitones and whole tones, then usable similarities of mobile forms appear ‘here and there’ — but ‘here and there’ also removes the opportunity of taking on approximate forms of similarity.

If one interprets mobile forms in a chordal sense, then some chords will result which match the stepwise assessment between major and minor:

But that is only a quite superficial similarity! In chromatic cases also a “distant” similarity is not possible. If parallel imitation results in the sequence major-minor I:VI, the sequence minor-major I:VI and so on, then even Nature discloses her will for gender opposition! Despite artificially enforced similarity! But Nature’s will for polarity becomes even more strikingly discernible when one establishes the harmonic grade of relationship in the forms above:

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If in a) a major chord is replaced by its parallel sound, then also in harmonic imitation a minor chord must be precisely replaced by its parallel sound. C major - A minor - C major A minor - C major - A minor Or b): if a major chord exits to its leading note-changed chord {as defined by Riemann: see ex. 102, p. 53}, then correspondingly the minor chord must aim for a similar relationship.

% C major - E minor - C major

F major - A minor - F major

E minor - C major - E minor

i.e. % A minor - F major - A minor

Or g): if the major chord varies, i.e., it becomes a minor chord of the same name, then logically the same applies for the minor chord and it becomes an identically named major chord. C major - C minor - C major

A major - A minor - A major

C minor - C major - C minor

i.e. % A minor - A major - A minor

%

polar

The reciprocity of these relationships is not questioned by any musician! But from this realisation, consequences arise which are in total contradiction to the understanding of minor dominants up to now. Consequences, which put all my precepts into a radically opposing position to all ‘Theories of Harmony’ known so far (including Riemann’s hybrid forms). The polarity between the major and minor consonance has further results, it also embraces the relatives — and not just the diatonic “representatives” but constantly also dominants (which are of course co-producers of the tonic-“representatives”) — and their fardistant relations, — further: the mediants, the dominant-mediants, the ultra-mediants; also the concordants — and finally the legion army of dissonances. (See separate chapters.)

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A. A melodic sequence in the major has the greatest similarity in the minor, when it is imitated in parallel:

But through melodic parallelism, the naturally intended harmonic polarity is prevented from having its effect. The virtually artificial enforcement of harmonic parallelism reacts promptly by requiring accidentals ( or ):

Nonconformance of leading note guides

B. An harmonic tonal sequence in the major finds its totally corresponding counterpart in the minor, when it is imitated in polar fashion:

Conformance of leading note guides

But this harmonic polar form allows no melodic polarity in the nature of things, without submitting to conflicts. All contradictions are eliminated, when the melodic sequence appears ‘in contrary manner’. C major

A minor

G major

C major

D minor

A minor

I have some concern that I could be completely misunderstood . . . it should not be said by any means that these melodic inversions have the benefit of ‘greater naturalness’ over parallel imitations! Not at all! Inversion and parallel sequence are in the same way artificial imitations, their values have to take place from the aesthetic side.

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Of course, one does not exclude the other! (compare the countless polar examples of the principal parts!). Anyway, the imitation forms are mimicry formations, which only outwardly pretend to be pure types. They will — necessarily by nature — always lead to some form of internal contradiction. He who asserts that an indication of polar function is placed in opposition to practice, ‘which even takes on as “dominant” the E major triad (5th degree) in A minor, and not the D minor chord (4th degree)’, is simply judged shallow and really illogical: If — to pick out the simplest but still most fundamental — I do not denote the E major triad in A minor as ‘D’ but as ‘ ’, then I do not make understanding of the harmonic fact more difficult, but rather facilitate it. The indication “tempered contra-dominant” does not, after all, exclude the familiarity of the sound in practice! But that marks it as an “art form”, which so far also proves itself to be without functional notation, as it contradicts the key signature (accidentals)! And in so far as this “apparent dominant” in the diatonality cannot develop relatives. If the harmonic element dominates, then it favours the naturally intended polar forms and subordinates the melodic! (“melodised harmony”). If the melodic element dominates, then it favours the linear analogous parallel forms and subordinates the harmonic! (“harmonised melody”).

XV Indication of Function in Figured Bass, Sound and Degrees Figured bass has nothing in common with ‘music theory’. It is merely a practice serving as shorthand for the accompanist, and actually a quite primitive and — it must be said — a decidely unmusical one! Figures replace notes in mechanically counting upwards from the bass note (not even from the root tone, because that would really be too musical!). Somewhat better is the notation by Gottfried Weber {1779—1839}, using chord-letters, still in practical use nowadays: C G e B e D G: corresponding major and minor chords. {But there is little ‘correspondence’; upper case = major, lower = minor.} Seventh chords with the 7 added (minor seventh) and 7 (major seventh, thus leading note!), diminished triads with an o added, augmented triad with ’ or + (Ernst Friedrich Richter {1808—79}). And so on.

Footnote for page 62: August Halm, 1869—1929, German Rudolph Louis, 1870—1914, German Ludwig Thuille, 1861—1907, Austrian Arnold Schmitz, 1893—1980, German Heinrich Schenker, 1868—1935, Austrian Schenker was Bruckner’s pupil in Vienna, 1887-89; Halm studied from Rheinberger in Munich, 1893-95, Thuille 1879-82. All were prominent musicologists. The work hinted at above as Ludwig-Thuille was a joint text, pub. in 1907. Schenker’s texts came out from 1906. Karg-Elert was evidently well-read in the contemporary literature.

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An inner relationship of the tones to each other or a hint of the harmonic event is understandably not recognisable. This shorthand represents at best harmonic facts. But it must fail in complicated cases, . . . . as the graphic representation is applied to only a tiny selection of dissonant types. The designation of degree proceeds from the scale and develops for each tone therein a triad. It proceeds thus from a melodic element and denies thereby the naturally intended harmony developing from the sound! It merges the major chords, likewise the minor, in tone and semitone sequence into a chain, by equating the 8th step with the 1st. Major and minor appear as quite dissimilar counterparts. 3 basic major triads = 3 secondary minor triads of a diminished triad 3 basic minor triads = 3 secondary minor triads

This system becomes contradictory when the minor takes on its artificially sharpened leading note 7th as a mandatory value. Then there are still only two basic minor triads (= I, IV), a secondary major triad (VI), but a basic major triad (V), an augmented III+ (which, however, in practice must be perpetually restored to a regular major triad!) and two diminished triads (IIo, VIIo). The confusion is obvious. But despite all contradictions, the designation by degree indicates a very great advance compared with that of figured bass! And outstanding treatises on harmony, e.g., by Halm, Louis & Thuille, Schmitz, Schenker, avail themselves of degrees! But whoever is capable of reading, recognises on each page the ignorance of naturally intended analogies. And from this a double form in the analysis of major and minor phenomena inevitably results. For example, one of these outstanding theoreticians says: (authentic) is more natural for the major, but IV I (plagal) is more natural ‘The cadence for the minor’, so here the natural will is clearly recognised; but analysis even avails itself of a double formula : (major)||IV:I(minor)||, which veils the naturally intended polar conformity! A second one says: ‘Just as in the minor key the fifth degree frequently seems to be major (as ), the fourth degree in the major will often appear to be minor (as IV),’ so, likewise, the polar nature is clearly recognisable here, but through the double formula :I(minor)||IV:I(major) completely veiled. A third one says: ‘The dominant ninth-chord is in the same way a major or minor cadence

G B D F A

in parallel tonalities; as G B D in C major is the counterpart of

D F A in A minor’, then that is already the recognition of radical polarity; but inconsequentially this is blurred by the reactionary indication of degree V IV. in the major in the minor

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Now a fourth one says: ‘Each triad develops its parallel on the lower 3rd degree’, so that radical parallelism is therefore complete nonsense! Accordingly, the C major chord develops as parallel the A minor chord, but the A minor chord the — F major chord(!) as parallel. And the A minor tonality receives the C minor chord(!!) as dominant parallel! Starting out from Gottfried Weber and stimulated by van Oettingen’s dual theory, H. Riemann introduced a ‘tonal key indication’, which quite obviously proved to be a compromise. The triads are described according to their ‘harmonic foundation’, whereby the upper sound receives a + and the lower sound a o. + c = C major chord || oc = F minor chord, alias c lower sound || All other chord tones are indicated as supplements by adding figures, the upper sounds by Arabic numerals and the lower sounds by Roman numerals (+ and o can then be omitted): g7 (= G B D F) || aVII (= B D F A) || 4 = fourth above, IV = fourth below the minor root. < = sharpening || > = flattening, e.g., + c6< = C E G A , etc. In 1893 Riemann published his Simplified Harmony, in which he relinquished the tonal key and — as the first — introduced a ‘description of function’. This demonstrates the essential meaning and the familial relationship between sounds and sound sequences, and attempts to trace back the whole harmonic event to the fundamental forms S, T and D. The teachings on function are a ‘logic of harmony’ and the most outstanding form of tonal analysis. Riemann is to be highly respected for having created a work on function, even though it still clings to a lot of contradictions and gaps. To have set up the fundamental principle of function remains a great feat in the history of the teaching of harmony. Even in 1902/1903 I had intuitively recognised polarity, without at that time even knowing any syllable of Zarlino, Rameau or the masters of the most recent past and present: without having heard or read Hauptmann, Oettingen, Riemann (although I had studied for 5 years with a celebrated ‘theoretical’ leading light, and on top of that at Hauptmann’s former domain!). The correspondence between the E major chord in A minor and the F minor chord in C major, the E minor chord in A major and the D major chord in A minor (alias the G major chord in D Dorian and the D minor chord in G Mixolydian), actually all the contrasting correspondence between Dorian and Mixolydian, as well as between Lydian and Phrygian (exaggerated bright major — exaggerated dark minor) forced itself upon me categorically, without my initial possession of the means to render this “felt” analogy tangible. I felt only instinctively that a ‘something’ veils the secret relations between the major V and the minor IV . . .

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The veiling of the correspondence of the “degrees” concerned was caused by the degrees themselves. I then placed the tonic basic triads in a table over each other and likewise the diminished triads, which in vertical comparison characteristically have no counterpart (see table p. 62), and I found immediately that with further counter-movements of the degrees, all parallel sounds also correspond to each other. And the so-called V and IV! And the E major chord in A minor is the same as the F minor chord in C major. And that the Dorian cadence corresponds to the Mixolydian cadence, just as the Phrygian corresponds to the Lydian. But analogous to the relative equivalence of the degrees concerned, I also gave them corresponding degree numerals.

compare this with example

I etc. Degree counting in major key etc. Degree counting in minor key

. .

Large figures: basic triads, small figures: secondary triads. Figures with dash = major chord, without dash = minor chord. This system of degrees (1902/1903) is the embryo of my precepts of polarity, which first proved its value beyond the primitive School of Harmony, with progressive development ever more proof, letting the common incomprehensibilities of the harmony of Chopin, Liszt, Wagner, Bruckner, Reger, Strauss, Debussy, Scriabin and Schönberg appear as relatively simple, natural forms, without having to grasp frantically for an exclusive system of overtones! But I doubt very much whether I would have found this path of development if I had not learnt Riemann’s teachings on function in about 1906. I confess this because of a feeling of thankfulness. It is painful for me, nevertheless, to admit that I must oppose Riemann’s theory in very many points. I felt that from the first reading, which so clearly revealed the nature of function to me, even if it nevertheless had to be opposed in so many instances.

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The harvest of Riemann’s seed consisted for me in the realisation of the eminent musical value of the description of function, which can express the inner nature of tonal phenomena and of linear events which take place in harmony, as long as it does not become sterile because of ‘simplification at any price’. I took over the term ‘leading note-changed triad’ directly from Riemann, as the designation is more striking and more factual than my earlier designation ‘counter-parallel-triad’. That should be all that applies to these points as ‘fertilisation’. Everything else ensues automatically from the following chapters. Only before an incomplete comparison between some Riemann function signs and the one carried out here. in A minor

Leading note-change

isolated

XVI Transition to the Second Part

Grammalogues for Polaristic Functions in General I differentiate between upper and lower case letters, as well as characters slanting towards right or left. The slant should be understood in the sense of direction of the system of tonality.

Major system

Minor system

Principal triads: Contra(domina)nt, Tonic, Dominant triads

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Upper case letters: major chords in the major key Upper case letters: minor chords in the minor key Lower case letters: minor chords in the major key Lower case letters: major chords in the minor key

correspondence of the genders between sound and system. opposition of the genders between sound and system.

If the lower case letters stand alone, then they are “variants” (gender change of sounds of the same name: F major—F minor; E minor—E major).

Contrants—Tonic—Dominants—Variant (possibly special names . . . .)

If they are placed behind a principal letter, or “attached”, then they are its replacement. They always stand a third higher or lower than the principal.

p or l or

= parallel = leading note-changed

minor chords in major, major chords in minor

Upper case letters over or under one another (major chords in major, minor chords in minor) “Ultradominants”

Second Contradominant and 2nd Dominant respectively. The first is skipped (erased, hence struck through).

Lower case letters over and under the principal insignia:

varied ultradominant and ultracontrant

major chords in minor minor chords in major

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Upper case letters as additions to the principal insignia: M = mediants and Counter-mediants. Major 3rd-related substitutes of the principal triads. major chords in major, minor chords in minor minor 3rd-related substitutes of the principal triads. major chords in major, minor chords in minor

(these are parallel variants) Upper case letters as additions to principal variants: variant-parallels (major chords in major) (minor chords in minor)

Strokes over

and under

: natural 7th and 9th additions.

Upper triad in major, lower triad in minor, lower triad in major, upper triad in minor

Movement grammalogue:

= whole tone, = leading note, = chromatic, = gap {hiatus} = ” , = ” , = ” , = ” Substitutes for chord tones (exits upwards and downwards)

Zero character: a missing chord tone etc. Parentheses: interpolated introductory or terminal {lit. execution or performance} sounds which do not refer to the tonal centre but to a fifth or third relative. They appear in front of these as ‘introduction’ and as ‘execution’ after them. i.e., in the key of C major: the chords of D major, G major

| G major, | G minor, | B minor, D major D minor F major

in the minor key as well. Curved lines downwards: introductory sound placed in front. terminal sound placed after.

Curved lines upwards:

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Precepts on the Polarity of Sound and Tonality (The Logic of Harmony)

by

Sigfrid Karg-Elert (State Conservatorium in Leipzig) Copyright of the Publisher in all countries All rights reserved, including translations

Leipzig, Publishing House of F. E. C. Leuckart Founded 1782

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SECOND PART

The System of Harmony in Polaristic Perception A Developmental Precept of Tonality

First Section Fifth relationship (Diatonality)

1st chapter: 2nd chapter: 3rd chapter: 4th chapter: 5th chapter: 6th chapter: 7th chapter: 8th chapter:

The Principal Triads ....................................................................................................... 71 The Tempered Contrants ............................................................................................... 73 The Ninth-chord Forms of the Dominants and Contrants .......................................... 77 The Ultradominants ....................................................................................................... 82 The Diatonic Substitutes for Principal Triads .............................................................. 86 The Problematic Triad ................................................................................................... 99 The Neapolitan Sixth-chord ......................................................................................... 104 The Comma-pure Modulation on the Basis of Principal Triads ................................. 112 Fundamental Guidelines ......................................................................................... 119 General Remarks on Chromaticism (which belong to the Second Section) ........ 134 The Concordance as a Means for Modulation ....................................................... 137 Pseudovariant Modulation ..................................................................................... 140 Ultradominant Chains ............................................................................................ 142 Epilogue to the Chapter on Modulation ................................................................ 147

Supplement to the First Section 9th chapter: Twins and Triplets .......................................................................................................... 156

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Intentionally blank

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1st Chapter The Principal Triads open, strong forwards to the tonic

weak backwards to the tonic

backwards to the tonic, weak

tonic

dominant

contrant

3 major triads (5th related) come together in a close system of principal triads

forwards to the tonic open, strong

tonic

dominant

contrant

3 minor triads (5th related) come together in a close system of principal triads

The central tonality is the tonic triad % It goes ‘forwards’ to the dominant # The dominant is in essence and appearance the ascendancy opposite the tonic. Since it alone forms the open side of the systems, only it can “concord”, i.e., form the natural 7th without contradicting the remaining principal triads. % The tonic goes ‘backwards’ to the contradominant # It is the lower upright triad in the upper sound system and (Against!) the upper hanging triad in the lower sound system It forms the weak side of the system and inclines remarkably lightly on gender-opposite influence. If the dominant develops its natural seventh, then it becomes a tendency-sound: it strives back to the central tonality. The weak contradominant endeavours likewise to develop a typical, distinctive form, which distinguishes it as a tendency-sound. It is not able to do it from its own strength and so it calls for its parallel sound*, which strives still further away from the centre as substitutes. Frequently, it co-operates with it at the same time and forms the contrant twin**, which, depending on the chord form and position, accentuates more either the prototype or the derivative form.

*

See chapter 5. See chapter 9.

**

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C major

A minor

polar:

authentic

plagal

cadence-curves: (polar:)

(polar:)

(the same, polar) and above all harm protect me in your mercy

% polar:

etc.

This popular phrase is a melodised polar harmonic sequence, as it appears in practice countless times: Here, functional conformity rules:

Not so according to

Riemann:

(hidden designation of degree)

and in the designation of degree:

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Lord,

as you ev’ry

way,

ere

polar:

a

wind command and ev’ry

plea

ere

a

wave order its

plea

we make.

etc. (the same polarity) and whose misdemeanour thus afflicted the fathers

% polar

Corresponding harmonic function with an entirely independently led melody.

2nd Chapter The Tempered Contra(domina)nts In

the functions are not determinable as the tonic appears rudimentary. The closest interpretation is the lower hanging triad as and the upper upright triad as D. Then the F minor triad correlates with the tonic of C minor ” G major ” ” ” ” ” ” C major ” D minor ” ” ” ” ” ” A minor ” E major ” ” ” ” ” ” A major, and thus all coincidental would be illusory. But in C major the F minor chord stands instead of the F major triad (C ) ” C minor ” G major ” ” ” ” ” G minor ” ( ) ” A major ” D minor ” ” ” ” ” D major ” (C ) and ” A minor ” E major ” ” ” ” ” E minor ” ( )

{ {

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They are really variant representatives of the natural contradominants in these cases. They represent mimicry forms: dominants, which they cannot actually be in their ‘tonality’. They are thus mutually interchangeable sounds of the strong sides of the variant keys. Figuratively expressed: C major receives the dominant (F minor chord) from C minor and shifts it as a strong negative-sound in place of its weak positive sound (F major chord), as it now acts as its “variant”. Reciprocally, C minor demands from C major the dominant (G major) and shifts it as a strong positive-sound in place of its weak negative sound (G minor), as it now acts as its variant. Because of this compensation, I am calling it the contradominant-variant ‘tempered contrant’. It is most remarkable that the strong side displaces the weak side of the opposite gender. The case is analogous to substituting the contradominant by the dominant of the parallel-key, whereby it becomes the contrant-parallel: open

open

C major

open

C minor open

(natural) D becomes

open

C major

open

A minor

open

open

(artificial) (principal:) D becomes

(artificial) c is =

(natural) (substitute:) Cp is =

Tempered contradominants are counter-sounds of the tonics. designated from their root!) c lower triad = C upper triad

(substitute)

||

(principal)

(Minor triads are to be

E upper triad = e lower triad

i.e.

Since they are strong types with open sides, then they concord (i.e., they are able to . develop their natural sevenths in the direction of sound) c

Parallel imitation: non-conformance of function. artificial natural

natural artificial

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polar:

polar:

(suspension)

parallel

(suspension)

(chain of suspensions)

(hybrid)

Joh. Schop Should I

not

sing

to

my

God?

Allegro Beethoven: Sonata, Op. 2, No. 1

Rather quick

R. Schumann I am not filled with wrath, even if it breaks my heart.

Allegro

J. Brahms: Rhapsody, Op. 79, No. 2

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Andante

J. Brahms: 4th Symphony

(see next example)

(G minor)

Hai - de

(D major)

Hai - de

or

Joh. Brahms: “Betrayal” (Ballade)

i.e., the reversed sequence of the previous example

or (Introduction)

(Closure)

reciprocal

The mobile triad in both examples forms, together with its related sound, a lower 9th-chord.

see next chapter

Peacefully moving

A. Bruckner: 4th Symphony

(Horn)

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3rd Chapter Ninth-chord Forms (Bicordances) of the Dominants and Tempered Contra(domina)nts The open sides: upper triads above and lower triads below, apart from the natural sevenths and their natural ninths (as perfect fifth of the fifth), form:

In the levelled equal temperament system the upper ninth-chord cannot be distinguished from the lower ninth-chord. Ideational comma differences admittedly exist: lowered B (1/10 tone ), lowered F (1/8 tone ) raised B (1/8 tone ), raised F (1/10 tone ) All ninth-chords appear frequently in two divergent forms: a) with omitted root (‘problematic tetrad’) b) with narrowed ninth (‘Italian ninth-chord’)

Similarity with in A minor in C major

in E major

in F minor

The possibility for modulations of these forms is obvious (the and require of course in a pure conception metharmonic re-interpretation, as the natural dominant-ninth chord in the major lies opposed to the corresponding form in the minor): (!) false position

normal position normal position (!) false position

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The “pure” ninth (perfect fifth of the perfect fifth) of the tempered contradominant is in the sense of harmony “naturally”: but in the sense of tonality a mixed form:

in C major (Mixolydian)

in A minor (Dorian),

on the other hand the narrowed ninth is artificial in the sense of sound: but natural in the sense of tonality: leading-note in C major leading-note in A minor. Both forms are common, but much more frequent are the tempered contradominants with narrowed ninth. Dominant- and tempered contradominant ninth-chords with narrowed ninth and omitted root form the ‘diminished seventh-chord’.

comma-differentiated similarity

comma-differentiated similarity

Regarding the Italian ninth- and diminished seventh-chord, see later on in much detail.

Natural ninth-chord: the situation (position) of the ninth and its tension {a feeling of suspense} with respect to the root is of most decisive importance for the character of the sound:

In free tonality an adjusted chain of ninth-chords of changing transposition forms. The ninths are marked as . The forms with the major fifth in the bass have the most favourable prospect of being held as minor ninth-chord, since their basses represent at the same time the root of the minor dominant:

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The tempered forms belong to the third-relations (see later bissonances of the countermediants). But at the same time the 2nd counter-fifth relationship and the simple seventh-relationship create comma-differing resemblance-forms: Original positions

comma-differentiated similarity comma-differentiated similarity

That these differentiations in the levelled 12 semitone system have become untenable, is self-evident, but their different inclinations are potentially still provable:

Also

7th counter-sounds

In the minor correspondingly polar. — This is only as pre-orientation; everything else remains reserved for later chapters. — The ninth-chords, which are characterised by a softer, more pleasant, more mellifluous and sensuous character (see the chain of chords No. 152), appear only with the forceful entry of the Romantic, and form with the ‘twins’, to which they have strong connections anyway, the basic elements of a hybridised impressionistic style.

Bicordance

Twin

mixed: G major + D minor, C major + A minor

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Andante con moto

Andante

C. M. von Weber: Cavatina from “Freischütz”

‘dolce’

(Woodwinds) ‘dolce’

(rootless “seventh-chord of the VIIth degree in the major”)

(9-8 suspension) Mendelssohn: “Elijah”, Chorus No. 5

Mer-ci - -

-

- - ful - - ness, Merci - -

- - ful-ness

‘espressivo’ Mer-ci - -

- - ful-ness

Chopin: A major Prelude, Op. 28

Andantino

Wagner: Parsifal — Flower girl

‘tender’

Wagner: Tristan

swelling

tremendously exciting, isolated ninth-chord

Allegretto

(Violin)

Contradominant twin

César Franck: Violin Sonata

Beginning! (Tonic twin, as suspension)

Waltz

Johann Strauss: On the Beautiful, Blue Danube

D -ninth becomes tonic-sixth (“Viennese Sixth”) ‘Twin’

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That m - y

sou -

- -

Liszt: “Oh come into the Dream”

- l in

bliss

a - wakes_______

Andante

Grieg: Nocturne, Op. 54

quasi D minor chord

quasi F minor chord

_

_

_

_

_

_

_

_

_

_

_

_

_

_

_

_

_

_

_

_

_

_

G major chord

B major chord

The hybridisation (major? minor? or major + minor) is in this example particularly characteristic! W. Niemann: Orchid Garden, Op. 76 No. 2.

Fred. Delius: “Violet” as you, oh vio - - - let

Root

D min + G maj

A min + C maj

problematic lower sound

Moussorgsky: “Songs of the Dead”

Moved to E minor:

(natural)

or E major (artificial)

Waltz Grieg: Op. 12, No. 2

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S. Liapounow: Op. 11, Carillon.

Pesante

(to G major) a minor + D major

Hybrid forms: C minor + F major very light

A minor:

C major:

velvety smooth

in A minor C major D minor F major

C major _ _ _

How complicated it looks at first sight, but how basically simple it is in the conception of polar 9th-chords!

4th Chapter The Ultradominants (ninth-related) If the tonic dominates in double form, or if both dominants tend towards the tonic {lit. tonicise}, then this results in an interlinked triple-system:

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The sounds that cross the boundaries of close tonality are called ultradominants and ultracontra(domina)nts. Provided that they appear before or after the simple fifth-relatives, they are evaluated as ‘introductory or closing sounds’. In these cases the simple functionletters are placed in parentheses ( ) and connected by a lower incoming slur or by an upper outgoing slur with the function-letter of the sound.

Introduction

Closure

In the sense of a ‘dominant of the contra(domina)nt’, the tonic triad is strong, opensided and therefore capable of being concordant! In the sense of a ‘contradominant of the dominant’, the tonic is weak, of closed side, developing thus the parallel twin as makeshift of a natural 4-note chord-form, or the parallel as makeshift of a variant — or else — it ‘tempers’ (varies) and becomes a stronger counter-sound capable of being concordant.

If the connection to the simple fifth-relatives is missing, then their ultraform actually becomes clear. They behave as exaggerated forms of the dominants. but:

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On the other hand, the dominants can make concordances and the contradominants can vary to make concordances. Finally, all double-strength triads can also make bicordances (i.e., behave as natural 5-note chords).

very interesting

The last two examples are stylised: 177 h) and i)

In the 13th to 15th century, the ultradominants very often replace the dominants. Whether the B major chord now functions as the D major chord as very many cases.

or dP (Mixolydian parallel) in C major, and

(Lydian dominant) or CP, admittedly can not be definitely decided in

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S’on me re – gar

-

de

Pre - nes

Dance-song from the 14th century (unaccompanied)

Palestrina “Stabat mater” (though also interpretable as:

(in D minor)

Herz- al- ler- liebstes

Orlando di Lasso Labourers’ Serenade Mä - del nun lausche Mädchen lausche, wie fein, wie fein, zahm und brav

(A major)

Zu nehmen Preis und Ehr -

- re und Kraft

or

die da Leid,

J. Brahms: Requiem. Leid tra - gen

(F major)

so

fern ins fremde

J. Brahms: Songs of the Virgin (unaccompanied) Land bis sie Gott, der Her – re fand. continuing in C minor

better:

(pure principal chord in G major, which however (C minor) is not intended, as the G major sound functions as '!' in C minor!) (Variant types, see later)

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5th Chapter The Diatonic Substitutes for Principal Triads (Bissonances) If two neighbouring principal triads combine into a double triad and get rid of their peripheral chord tones, an apparently gender-opposite triad results, which acts as replacement for both original principal triads:

The appearance of an E minor chord is thus half a C major chord and half a G major chord. The appearance of an F major chord is thus half an A minor chord and half a D minor chord. In this sense it is an incomplete double triad (bissonance) = third + fifth, which is simultaneously a new root + its third. This minor chord is likewise an upper triad, just as that major chord is a lower triad.

The tonal direction accordingly ‘begins’ the substitute with the “minor third”, which, as is generally known, is only a secondary appearance — difference between third and fifth of the same root. Hence I denote these substitutes by a lower case function-letter, inclined in the direction of the principal triads. p l (apparent minor chord in the major) (apparent major chord in the minor) Their relationship to the original principal triad is divergent: E minor chord to C major chord = in the forwards direction, E minor chord to G major chord = in the backwards direction, F major chord to A minor chord = in the forwards direction, F major chord to D minor chord = in the backwards direction,

{ {

thus strong! thus weak! thus strong! thus weak!

major third-related minor third-related major third-related minor-third-related

As strong replacement it is a leading note-changed sound = l . As weak replacement it is a parallel sound = . [If one compares a chord with its parallel sound, then a correspondence with the potential key signature appears in the sense of tonality, whilst the key of the leading note-changed sound always differs by one accidental. C major—A minor = both without accidentals: C major—E minor (1 ) +strong (brighter) A minor—C major = both without accidentals: A minor—F major (1 )

- strong (darker)]

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+ stronger + weaker than than

polar

- stronger - weaker than than

In the major, the parallel triads lie underneath F d the principal triads

C a

G = major chord e = minor chord (weak!)

and the leading note-changed triads (lie) above the principal triads.

a F

e C

b = minor chord G = major chord (strong!)

In the minor, on the other hand, the parallel triads lie above the principal triads

F d

C a

G = major chord e = minor chord (weak!)

and the leading note-changed triads (lie) underneath the principal triads.

d B

a F

e = minor chord C = major chord (strong!)

The strong substitutes for the strong principal triads exceed the limits of firmly confined tonality! open

open

a) Tonic-parallel is tonally the same as the contra leading note-change: b) Tonic leading note-change is tonally the same as the dominant-parallel: c) Between major and minor triads, reciprocity exists without exception!

etc.

In Riemann’s teachings, confusion over functional notation has already entered an acute phase. Nothing tallies any more: re a)

but

re b)

but

re c) (half polar)

(pure parallelism!)

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a.) b.) C major D minor

a.) C major

b.) E minor

a.) C major

b.) E minor Complete reciprocity

Riemann No correspondence at all

The practical value of the recognised reciprocity of tonal functions impinges in an unforeseen way on modulations. (See next chapter.)

The substitutes appear a) as changed sounds between their principal triads: and all other functions. The same in the minor. b) as cadence replacements: Instead of comes: Instead of

comes:

Tonally equal, but functionally different: Tonally a double fifth-relationship, but functionally a simple fifth-relationship:

Conversely: a fifth-relationship tonally, but functionally a double fifth-relationship:

A new tonal form results from the sequences: weak replacement of the weak side, strong replacement of the strong side, functional double fifth-relationship by the tonal triple fifthrelationship: (comma difference in pitch!

)

All forms also in reversed order! Likewise all forms in the minor!

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—Changer—

schematic

)

)

)

)

Mediation between principal triads:

strong substitution

weak substitution

strong substitution

weak substitution

Parallel imitation (sequential minor cadences in the major, major cadences in the minor) major

(minor)

major

(minor)

major (minor) major

(minor)

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minor

(major)

minor

(major)

or introductory sound as parallel: minor

(major)

minor

(major)

Isolated substitutes in replacement of the tempered contradominant auxiliary mediant

mediant

leading note triad

tritonant

or auxiliary mediant

leading note triad

mediant

or

tritonant

or Neapolitan cadence (open). See chapter 7.

If two substitutes are forming a cadence with one another, then the introductory sound may be understood as “parenthesis”:

or

or

also understandable as

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The sudden entrance of the 3rd fifth-sound is always misunderstood because of commadifference: the triple dominant becomes valued as auxiliary mediant, whereby comma errors would result. ‘Metharmonic change’ thus occurs in these cases; this is the triggering of a sound of comma-differing similarity (but) of opposite function. Original positions: thus

D F A

D F A artificial welding (metharmonic) tonal closure of the circle

(Metharmonic)

A blurring between major and minor continues . . . The tempered contradominant makes bridges between the variant keys but the diatonic substitutes, on the other hand, mix the parallel keys and their respective nearest relatives.

c) Tendency towards the Tonic by Substitutes If the D minor chord is , then the G minor, A minor and A major chords are its 5threlatives ( , , ); if the same chord is some sort of substitute, then the G minor, A minor and A major triads from then on remain its , and ; they are not however diatonic functions any more, but single connecting sounds to those substitutes. They are therefore shown or as introductory or closing values. within parentheses and marked by

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In the same way, the following parenthetical sounds result for C major and A minor: ( ) ( )

A maj. A min.

E maj. E min.

B maj. B min.

F maj. chord F min. chord

Substitutes: Cp (D min.) Tp or Cl (A min.) Tl or Dp (E min.) Dl (B min.) chord ( ) G min. D min. A min. E min. chord (D) Substitutes: (C) (c)

D maj. (G maj.) C maj. C min.

G maj. or

(C maj.) F maj. F min.

or

C maj.

F maj.

(F maj.) B maj. B min.

(B maj.) chord E maj. chord E min. chord

chord

in C major apparent closing of the circle in the 12 semitone compass

in A minor

likewise in the minor (polar)

can not make a concordance, as sound direction and origin The substitutes Cp Tp Dp go against this, but they can become dominant seventh-chords through a connection to fifthrelated substitutes:

but

or

applies to any substitute whatsoever

but

or

The making of a concordance is an unequivocal tendency, which gains particular worth for modulation. Very attractive also are the ‘problematic bicordances’ as parentheses:

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polar:

Appearing in this accumulation, the introduced substitutes mask noticeably the gender of the key. The major form is felt as if it were the minor form, and vice versa. The introductory sounds are interpreted as auxiliary mediants. Enharmonic similarities enter, and so on. These are significant symptoms: Tonality widens, the key is destroyed.

d) Double and Chain-Introduction of Substitutes Each substitute can place before or after its two-sided cadence:

valid likewise for the other substitutes.

In these cases, a short close-modulation has already taken place. The fact that it has not been completely effected has metrical reasons: even if the D minor chord falls on a heavy beat, this sound appears relatively light in the midst of powerful tonic chords.

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But different here:

new tonic

Details in chapter ‘Modulation’. Very common is the case where the introductory sound approximates itself to the tonic, i.e., lets itself be introduced: e.g., the A minor chord as Tp sends out a proclaimer (E major) and this in turn sends in advance a pacesetter (B major). The B major chord is D of the E major triad, but this is of the A minor triad, the parallel of the C major tonic triad.

The further consequences arise from the previous ones by themselves. Examples from the literature.

suffers affliction

A. Gabrieli (1510): Ricercare

Heinrich Schütz (1585): St. Matthew Passion on the stem of the cross, on the stem of the

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A major cross for us

bitter

bitter

death.

eternity and reigns there

death

A major on the stem of the cross for us

And reigns there

with the father in

with the

father in

eternity continuing G minor

At the same time a comma-pure example of modulation: G minor — D minor — C minor ). — G minor (Curve Compare the variants with the diatonic substitutes! A, D, G major chords

(By the way: what marvellously

C, F, B , E , A major chords

expressive music!) Leo Hassler 1564: from “Pleasance” (Villanella)

g minor

degrees proceeding downwards: (introductions)

J. Haydn: Piano Sonata in E minor, Hob. XVI:34

E minor

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% % For the Viennese Classical typical (stereotype) form: contradominant parallel as modulation-switch to the parallel key. According to Oettingen, Riemann and their following this would always mean a comma-error, as these assume equality between +D9 and ◦SIX: comma-difference

distinct striving away from the dominant

striving towards

or

Original positions

Original positions according to Oettingen etc.

or

in Riemann’s functional notation =

, or

resulting in:

comma-error

Mozart: “The Marriage of Figaro”

Oboes

Mozart: “The Marriage of Figaro”

For the Viennese Classical typical (stereotype) Cadence-form %: sixth-chord of the contradominant in the major (thus an apparent contradominant-parallel) before the 6-4 chord of the dominant.

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R. Schumann: Op. 21 Novellette, No. 1

(Cadences)

(weak) (strong)

(weak)

(tendency towards tonic)

cadence R. Schumann Arabeske, Op. 18

Rising curves. Modulation.

or

Curve shape very clear, exemplary, if also a somewhat primitive modulation (comma-pure!): D maj. chord B min. (B maj.)”

E minor range F major range

or

D maj. chord B min. (B maj.) ” G maj. ” E min. ”

G maj. E min.

” ”

C maj. A min. F maj. D min. B maj. G min.

” ” ” ”

C maj. A min. F maj. D min.

” ” ” ”

” ”

B maj. G min.

” ”

(double sided cadence)

Frédéric Chopin: Prelude in B major, Op. 28, No. 11 harmonic sequence

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(end)

R. Wagner: Parsifal

Solenne.

Trumpet solo

(tenors)

allegory,

the inadequate

All that is

transitory

here is an

is only an

event

Franz Liszt: “A Faust Symphony” (potential functional standstill) (leading note-changed triad)

(tonic seventh!) isolated

All further tonal changes of the diatonic substitutes are distributed over the following chapters.

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6th Chapter The Problematic Triad Chords of the VIIth major and IInd minor degree. Diminished triad. A ‘touch-me-not’ {i.e., don’t meddle!} of harmony-teaching. In my opinion, the difficulty of interpretation lay in modern theory’s attempt to handle the sound in question by a formula that is too simple. in the minor. Riemann: in the major, In my functional notation, that would be: But that is not sufficient, as then D is the same as tonally, and is the same as Cp, so this interpretation must also apply.

The conception in the minor lies closer in the chordal sense: the formation introduces the root tone of the concordance (D in D minor). Thus, for the major, Cp also lies closer than D because of the root tonality.

‘F’ is not at all a seventh! But the principal unison (C_) can, indeed should, be doubled in the or simpler . first line. In the sense of the minor, however, ‘D’ is principal root.

But if ‘F’ is to be understood as narrowed upper-fifth to ‘B’, then logically ‘B’ must also be understood as narrowed lower-fifth to F. Now that is indeed not really possible, as ‘F’ in the sense of the normal upper-fifth ‘F ’ requires the B upper-sound, {i.e., B major triad} and ‘B’ in the sense of the normal lower-fifth ‘B ’

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requires the F lower-sound (alias B minor). Merely reversed, this yields a functional sense: ‘F’ is the upper-fifth, ‘D’ the upper-third and ‘B’ is a unison, altered inwards; likewise, ‘B’ is the lower-fifth, ‘D’ the lower-third and ‘F’ a unison altered inwards. Riemann actually disputes the possibility of altering the unison. He says that it would become a 3rd. B D F : B D F = ( G ) B D F. That can be true, but it does not need to be the only interpretation. Apart from the Pythagorean and Didymean chromatic intervals, there exists another, previously unnoticed until now (Leipzig chromatic interval 14 : 15), which is demonstrable in all narrowings.

Hence the chord in question stands instead of the B major and B minor triads (the unisons are narrowed).

in the major: (instead of D major)

(instead of B major)

(instead of B minor)

(instead of B major)

(instead of B minor)

in the minor (instead of G minor)

(major) (minor)

uniform fall of the functions

fall, rise and stand still!

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One recognises clearly that the irregular cadence sounds tend to the diatonic ultraforms

(major)

(minor)

and seems to be a removal of the chromatic interval, thus a The removal of the simplification. But — it turns out, in fact, to be a chromatisation and a complication. Again and again a proof: Nature does not want a closed circle of tonality. Nevertheless, if it is artificially forced, Nature then reacts with the introduction of mimicry forms and problematic formations. Yes — the indication of degree! It is in such cases a ‘refreshment’ of frugal ones. It indicates where these problematic forms stand, designating them by a zero: i.e., ‘there is something wrong here’. Amen. Incidentally, this does not burden the logical conscience. Admittedly: the “rules”. . . . ‘a diminished interval may not be doubled’— “the leadingnote may not be doubled” — however, in practice the ‘F’ is nevertheless sometimes doubled, and sometimes ‘B’. So now there is a rule: “no rule without exception”. Nothing then! What is to be doubled? The functions tell it without exception: the real or substituting root tone of the initialled tonality: or or

S. Fornsete, 1240: “Summercanon” thus: st

IInd

st

“degree” obstinately maintained through the whole piece, exit of the chordal fifth therefore: from the triad of the IInd degree

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Fr. Landino (1350): Ballata

(A minor)

always sixth instead of fifth or H. de Lantins (1400): Canon

Gilles Binchois (1400): Ballade

or Guillaume Dufay (1400): Mass Soprano in G minor: Phrygian Alto in C, Ionian (major)

Tenor in E , Ionian (major) Bass in C minor (Dorian)

Guillaume Dufay (1400): Mass

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rings around the saviour sound

which is the same as: Fr. Liszt “Christ” ‘Thou art Peter’

2 flutes

Edv. Grieg: Op. 74, No. 1 Psalms (unaccompanied)

lovely to regard.

O you my Sulamit wondrously fine

G. Sgambati “To hope”

Valse lente.

Lopez Almagro: Andalusian Suite

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Clarinet

Gustav Mahler: 3rd Symphony, 3rd movement (“What the animals in the forest tell me”)

roh!

Cuckoo

3 Trumpets, solo

. Bassoon

.

V’cello, pizz.

instead of seventh-countersound

7th Chapter The Neapolitan Sixth-chord and its Derivations The original mobile sound of the minor dominant gradually grew stronger into an isolated form: (Phrygian)

Suspension resolved

(isolated)

or

or

The B major chord in A minor is known as

, its root tone is of course ‘B ’. The B

major chord in A minor is always a bissonance: and substitutes the D minor triad (stronger replacement of the strong fifth-relative). The Neapolitan chord does not yet dare the distinction of its own bass-note, it remains “fundamentally” a dominant, which introduces the ‘minor (leading-note) sixth’ instead of the chordal ‘fifth’, thus the designation ‘sixth-chord’. (The case is similar to the ‘problematic sixth-chord’, ‘classical contra-sixthchord’ and to the ‘Viennese sixth’).

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(isolated mobile sounds that have become fixed) The use of the small -sixth results in a Phrygian element (lowered 2nd degree of the fundamental scale in the minor) and stamps the sound (conceptual dissonance) a pseudoconsonance. {Karg-Elert borrows two terms here. “Conceptual dissonance”, coined by Riemann, applies to the first and second inversion of a triad, usually based on the dominant, which moves next to the triad upon the lowest note of that inversion. “Pseudo-consonance”, coined by Schenker, is similar.} Surprisingly, the sound has its effect only later: if immediately after it the tempered

contradominant occurs. Thus the characteristic charm lies not at all in the sound itself but in its functional relationship to its resolution {lit. removal}. (‘Tritonant sound’: double-fifth + third = B —F—C : E.) Both sounds form cadential complements to each other: As in the 16th century, to the beginning of the 18th century, the variant finalisation for movements in the minor key was in the modal tradition {viz., end on the major triad}, so then the Neapolitan cadential sequence appeared before the minor chord (in the middle of the piece) as well as before the major chord (finalisation). The later adoption of the Neapolitan cadence in the major key happened as a consequence of that given previously. It is however obvious that for an identical tonal sequence in the major, functional values other than the minor must result, and the followers of the degree theory and of the functional theory of Riemann, firmly connected to it, have absolutely no reason to register an agreement between the major and minor cadence in preference to my non-agreement.

logical difference in value natural, artificial

artificial, natural

Degrees: Riemann:

(My functional designation reacts automatically to mimicry forms by indicating contrary grammalogues.)

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If one wants to put forward a rigid formula for alleged modulations between Neapolitan sixth-chords, then they could be: Any major triad can appear in sixth-chord form; the bass takes over the third of the root chord, rises a full tone and forms the basis of a new major chord which will cadence authentically to the subsequent minor or major sound: After this tried and tested recipe, you can steer automatically, Reger-like, towards any target. Thus: or

a)

B E A | A maj min maj b)

G C maj

or

F | F min maj c)

F

B maj

or

E |E min maj

or

or

NB 1)

NB 2)

Thousands of modulation schemes could be made in this way as stereotypes. But 95% turn out wretched. . . They work like a conjurer under the deceit of variants. Now, how suspicious is this minor or major! The major chord can of course appear as variant for the minor chord — but not a major key for a minor key of the same name, unless not merely a ‘variance’ is intended. The path from C to G major is exactly opposite the path from C major to G minor, as G major functions as an intensification, whilst G minor functions as a relaxation towards the C major key. I know that the general public think considerably more liberally than I do over original positions and their comma-differences; — but I also know that the general public cease to be liberal exactly where, logically, they should be liberal: free tonality compared with atonality, thus on the other side of tonality, where I would not let original positions and commadifferences apply any more, as here the 12-semitone system enforces its character, but not earlier! How superficial is the opinion that the Neapolitan sixth-chord is an original miracle cure for modulation, made evident by the fact that each tonal connection which seems to or c, come from it, in the sense of its original function or respectively, and is carried out so precisely, only that these functions offer far richer opportunities for nuance (root positions, seventh and possibly ninth additions) than that rigid formula.

(

)

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see No. 240 a)

or

or

or

or

or equally well also the other examples

ditto

Mà

Claudio Monteverdi: Orfeo, 1607 (4th Act) chi me’l vi-eta ohimè sogn’ o va - neggio

(Historically the oldest form of the Neapolitan sequence) G. Frescobaldi: Fugue (G minor)

a)

b)

J. S. Bach: “Mass in B minor” Kyrie

Ky – ri - e

e - lei -

- son

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Treatise on Polarity Handel: Judas Maccabæus (1st Chorus) lamented the sons of Judas

Grave

lamented

(Italian) J. Haydn: Variations in F minor

Andante

(diminished seventh-chord)

free substitute

W. A. Mozart: Rondo in A minor K.V. 511 1787

potential

Beethoven: Sonata No. 8 (Pathétique) Opus 13, 1798

Neapolitan sixth-chord largely reinterpreted

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Weber: “The Enchanted Huntsman” is it not deception, is it not delusion?

then B major chord

G. Rossini: Overture “William Tell”

Andante 4 cellos solo

Franz Schubert: Impromptu in E , Opus 90, No. 2, 1827(?)

Frédéric Chopin: Scherzo No. 3 in C minor, Op. 39, 1839 Introduction

Prestissimo

C major ending

C sharp minor chordal passage

quasi

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Treatise on Polarity Robert Schumann: Sonata in F minor, Opus 14

R. Schumann: David’s League Dances, Opus 6

Presto

Johannes Brahms: May Night, Opus 43, No. 2

Tears run passionately down my cheek ———————

J. Brahms: Requiem, 2nd movement Introduction

B minor B major Andante

Franz Liszt: Nocturne (3rd Dream of Love)

pointless enharmonic change

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Treatise on Polarity Liszt: 1st Piano Concerto pointless enharmonic change

or Sergei Rachmaninoff: Prelude, Opus 3, No. 2, 1892

Lento

allargando

Charles-Marie Widor: 8th Symphony (ending), 1887(?)

Edward Elgar: “The Apostles” (Oratorio)

E

-

li,

la - ma

sa-bach-tha

-

ni

Neapolitan — twin — hermaphrodite B major E major G minor E minor

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8th Chapter Comma-pure Modulations on the Basis of Principal Triads In no area of the practical operation of the teachings of harmony do the facts of natural polarity become so tangible in appearance as in modulation. Modulation means tonal change. Change excludes abrupt juxtaposition of distantly related keys. These are better designated ‘shifts’. The clearest picture of “comma-pure” modulations is obtained by imagining the commencing and final keys in their archetype positions.

Centre:

From signature to

signature, ascending by 5ths,

from

signature, descending by 5ths.

signature to

On the keyboard (up to a4), B major (5 ) is the ultimate limit of the sphere of excitation {or tension}, (down to AAAA), B minor (5 ) is the ultimate limit of the sphere of relaxation.

Further expansion results accordingly. The pathway between starting and finishing keys is counted in steps of a 5th.

For example, C major to C minor = 31/2 (upwards) C major to C minor = 31/2 (downwards)

to

The half-steps are always substitutes: C major , A major = 3 , thus 3 steps + 1 substituting 1/2 step striving away = 31/2 steps upwards. C major , E major = 3 , thus 3 steps + 1 substituting 1/2 step striving away = 31/2 steps downwards. The modulations C major —A minor—E major—G major— C minor or C major —F minor—G major— C minor contain a comma-error in both cases: instead of C minor, C minor is reached; instead of C minor,

minor is reached, something that

clearly manifests itself in the original-position of the final tonic triad.

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Instead of:

instead of

Parallel of the 3rd contra-dominant

Variant, same third as the starting key

Leading note-changed chord of the 3rd dominant

One has thus not overcome any path distance, but has returned to the (partially displaced) starting point. To C major and C major, the path is twice as long as to C and C minor. Twice as much effort must be undertaken. But now one can quickly reach the C and C major chords: C major A minor G major C major || C major E major, G major C major, but this C and C major has its origin in the C major zone:

raised by 2 commas!

lowered by 2 commas!

With C major however we understand a distant region of unusual relaxation (7 ) [extremely dark!], and inverted C major is understood as a distant region of unusual tension (7 ) [extremely bright!].

Standard basis (principal triads) Mediant basis (lowered by 1 syntonic comma) Ultramediant basis (lowered by 2 syntonic commas)

One should not object that these differentiations have only a purely theoretical significance, as in practice there is ‘no demand for comma-values’. Yes, if one refers to M. Reger’s lopsided “Contributions to the Teachings of Modulation”, in which the “Neapolitan sixth-chord” holds orgies fit for prosecution, then it is really pointless to exert oneself so much about an ideational purity. . . . But it appears to me that often there merely exist mistakes over the concept of ‘modulation’. Modulation presupposes tonality, or more precisely: key {under a key signature}. In by far the most cases, however, ‘sequences of harmony’ merely masquerade as ‘modulations between two keys’.

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It is obvious that no modulation without harmonic movement is possible, yet every transition or harmonic movement is not necessarily a modulation. The examples of modulation by Reger in question are one and all practically occurring harmonic events, but only to a small part ‘tonal changes’ in the purest sense. C major—F major—C major is possible as a tonal sequence, likewise C major— A minor—E major or what is the same: C major—A minor—E major — but this sound chain is not a tonal transformation! Then it becomes either a) the last chord in the key of the first or b) the first chord in the key of the last or c) both chords referring to an ideational tonal centre (which, by the way, the Riemann functional notation, from which Reger in all his life never broke free, cannot carry through at all in many cases) — then in all three cases the talk is about a widened tonality, but not a ‘tonal change’.

C major

C minor

C major

C minor

C major

C major

C major

C major

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Or even: from C major to C minor (i.e., transposed from C major to C

C major

enharmonically reinterpreted: D major; G,

minor):

C minor

D,

B,

B minor

These harmonic sequences are without exception understandable. Their functional course will later be clearly recognised in the following chapters (only the first two degrees of relationship will come into effect; by means of the seventh-relationship, the line of approach in C major—C minor could still be much shortened!). Without question these sequences are of practical use, and the literature demonstrates similar examples in abundance (by Reger and his epigones {i.e., inferior imitators}). In peaceful effect certainly of characteristic expression — granted — but ‘modulations’ they are not, probably rather shifts, tonality breaks {or jerks, abrupt angulations} or diagonal harmonic movements, shuntings {or rankings} (last example) in the limitless realm of elevated tonality. Even as wildly as the examples behave, they remain a more or less strongly chromaticised fiddle of the cyclical sequence

which again is a partial substitution of the original cadence.

True modulation is a comma-pure key change. Movement at a distance from the tonal cadence of the original key. Linear development on the basis of the principal triad. But all cited examples are turning round and round and spiralling up or down to comma-differentiated derived sounds (mediants, ultramediants. . .).

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Example (C major—C minor) becomes a comma-pure ‘modulation’ (linear course on the basis of the principal triad) through the following correction:

The difference becomes clear, without a long explanation, from the following table: Example C major—C minor ; comma-pure, start and finish on the same line (perfect-fifth distant C minor).

Principal triad basis

Example C major—C minor ; comma-difference. Finish on another line (\ lowered centralC minor). (= Nos. 267 and 276)

Mediant basis \ Principal triad basis

The first example shows a tendency to move {lit. strive} away. The end of the path signifies a distant target. C minor is with regard to the initial C major under really considerable tension. The second example shows a swinging tendency. The end of the path appears as a chromatically disguised start. This C minor is lowered by one syntonic comma compared with that perfect-fifth distant C minor. The characteristic tension is diminished by 1/10 of a tone. In the same way, all the following schematic examples, which do not have to pass for pure modulations, should also be understood.

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Diatonic original form: C major—C major

(= No. 271)

Principal triad basis

mediant type (= No. 267)

C major—C minor Mediant basis \ Principal triad basis

C major—C major

triple double

Triple mediant

mediant type (= No. 269)

Ultramediant Mediants Principal triads

C major—C minor Ultramediant

(= No. 266)

Mediants Principal triads Mediants

C major—C major Principal triads Mediants Ultramediant

(= No. 268) Mediant simpler: principal triad

Mediant

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C major—C minor lowered by 4 commas

7-fold comma-difference with respect to fifth-types

Principal

raised by 3 commas

(= No. 270)

nearly everything replaceable through seventh-relationship (simpler!)

The explanation of the logical connections in these sequences and technical instruction in free-tonal treatment is not described in full until later chapters. The purpose of citing such tonal-sequences was as follows: the principal difference between linear and quasi-diagonal tonal development should be demonstrated. I can not emphasise strongly enough how important — in a paedagogic sense — the exact mastery of the pure principal triad modulation seems to me. It alone gives a clear picture of the stress relationships between the beginning and end of a modulation phase. The original C major does not really ‘lie’ a semitone higher, and the original C major is not a semitone lower than the original C major, rather do both ‘lie’ in tension and relaxation regions extremely distant from each other. The musician should gain an idea of the distance and learn to appreciate that distant zones can not be reached by quasi ‘vertical twisting’. To reach C minor from C major must be considered the surmounting of a distance and not equated to a turning on the spot (variant); for the ‘C minor chord’ is still not quite a key representative. When teaching, I have on countless times had the practical experience that after this ‘preliminary instruction’ the students, within the hour-limit, have so clearly grasped the nature of modulation, that already in the second lesson they have, almost without exception, obtained excellent results, i.e., provided solutions in part to some really complicated tasks. If absolute security in comma-pure principal triad modulation is achieved, then nothing stands in the way of permitting all styles in the sense of free tonality. But also not earlier!

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Fundamental Guidelines Principle of Modulation: The starting key reaches out towards the final key even if the final key is contrary to the starting key. Their tonally identical but functionally different chords (harmonies) form the switch.

A

Switch

Z

§ 1. The tonic triads of the starting and target keys and the switch should, if possible, fall on accented notes, as only in this case does their significance appear distinct. § 2. The target tonic should be surpassed {or overstepped} as much as possible: i.e., if one goes up to a target key, then the tonic should be introduced as much as possible through its higher fifth-relative D or ; if one goes down to the target key, then the final tonic should be prepared through or C (perhaps c). In other words: the tonic should rebound, or expressed differently again: after ascending, the perfect cadence is the one preferred, and after descending, the plagal cadence. The reason is simple nature: should the final tonic be brought in reversibly cadencing, then at a rising trend the lower fifth-relative already lies on the path, in reality or potential. And the other way around. + surpassing E major chord C minor ” A major F minor D major B minor

” ” (major) ” ”

G major E minor C major A minor F major D minor

” ” start ” ” ” ”

finish

start

finish

minor

- surpassing

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§ 3. For remote modulations, the switch should be evened out as much as possible half way along the path. § 4. Tempered contradominants should not — at first — function as switches (comma errors); in that way, the process can then pass off in the same way for natural contradominants also, so that the tempered form is not determinant. The tempered contradominant can function as switch, if it is reinterpreted as tempered ultradominant. § 5. More attention should be paid to natural part writing (melodic soprano, clear bass cadencing or linear bass!).

For purposes of faster communication, tonal distances are given in steps of a fifth. substitute striving away, 1 : fifth sound {according to Riemann: in major keys, the major 5th degree (e.g., C E G — G B D); in minor keys, the minor 4th degree (e.g., A C E — D F A)}, 11/2 : fifth and substitute striving away, 2 : double fifth . . . . . figures over the stroke = upward direction, under the stroke = downward direction. major, minor tonic triads of the starting key. and = variants on the way.

1/ : 2

1st Group: Modulations to the keys of tonic substitutes, dominants and their substitutes. These modulations present nothing new; they come about through changes towards the tonic of the dominants and their replacements.

or

From C major to A minor i.e.,

1/ 2

(harmonic half-step downwards):

1/ 2

. . . . Cadence . . . .

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E minor The twins in the second bar already loosen the strength of the major key C major C major ; A minor the Cp can not form an under-7th [it is still an upper sound bissonance: (B ) D F | F A (C)]; by adding the ‘B’ it becomes a minor key concordance:

and brings about the cadence in A

minor.

= mediant triad of the tonic, see later

Not the mediant triad of the tonic but the differentiates the modulation, even if that already refers to A minor. The B major sound is a feature of A minor accentuation (overstepping), but is still the strong substitute of the strong side. The same example with introductions.

One should clearly understand: the introductions ornament and elevate the introductory sounds to greater significance, but they do not bring about any decisions in modulation and D, similarly in the parallel key, their themselves. As Cp C Tp T reinterpretation is not even so convincing, as long as metrical markings have no influence. minor

minor

minor

heavy

heavy

heavy

heavy

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All modulations can be shifted in terms of polarity: From:

from:

becomes:

becomes:

polar form of the last example:

In example 284, because of the introduction of and (chromatic mediant and principal triads), the A minor key became more sharply differentiated from the C major key. The result of a purely polar transposition into C major is the introduction of the F minor and B minor chords, through which the C major key is recognisably moved away from A minor. The decisions fall on the accent.

polar to No. 285

The movement of the tonic to its parallel sound — or to the key of the same — is a reversal; no matter whether it is in the major or minor key.

6 switches:

(polar)

The harmonic half-step movement which is directed forwards leads to the tonic leading note-changed sound, or to the key of the same.

6 switches: or

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Here the much vaunted Neapolitan cadence can be deployed.

But it is, as everywhere, more a characteristic nuance in already fixed tonality than a suitable means of modulation! And it is nearly incomprehensible that such a pre-eminent master as Reger has gone to the trouble of writing a booklet, in which the Neapolitan sixth-chord is praised as helper in all needs of quick modulation. (Foreword!) Suspicious enough, that he finds a matching use in the minor as well as its variant key: A C F = Neapolitan sixth-chord for E minor or (!) E major. The “or” is characteristic of the variant nonsense. The Neapolitan sixth-chord is a mobile sound, a leading-note minor-dominant concerning this, and its suspension which suppresses its resolution. (See my designation mimicry-form in the major respectively.) What distinguishes it as a cadence sound is nothing else than its original function: (or c ). Thus all the much vaunted little tricks of modulation of the Neapolitan sixth-chord are to be registered on the account of the potential always requires the outbalancing (retaliating) , on the other hand the of the dominant: or alone suffices for introduction of the tonic triad.

a)

insufficient

b) suffices

Neapolitan sixth-chord (insufficient!) natural dominant triad (suffices!)

Here an important principle of function becomes recognisable: if the seventh is added to the substitutes, then they cease to be that; they become dominants and bring about the conclusion of the modulation. Upper-sevenths (of minor key substitutes) drop the sound a fifth, under-sevenths (of major key substitutes) force a rise of a fifth.

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The introduction of cadencing counterparts of the opposite side is of course exempted.

seventh

the A minor chord rises (through the seventh) to the E minor chord

The accentuation of the counter-side in the starting key shifts this further away from the final key, as

is too easily interpreted as E minor. A minor—F major polar as

or

(*) problematic triad) the C major chord drops (through the seventh) to the F major chord

e.g., C major to G major

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The last example shows the characteristic harmonic half-step chain, which later on also plays an important role in close modulation or as a middle member in longer distance modulation:

There is nothing against widening it towards both sides, whereby the C major key as well as the G major key receives tonal strengthening. The finishing key appears overstepped.

(Dorian) without previous or The 21/2-jump (D Cp) is, through T, easily interpreted as immediately subsequent explanation. Therefore, provided there is no preceding C major episode, it can act as T instead of D. Where the D minor chord is interpreted as , which, by means of a preceding or occurs in the most precise way (see next example), and the A major chord is interpreted as T, which happens in a most precise way through a cadence being put back, then a comma-pure modulation from D minor to A major results. The principal A major triad is reached through a ramble on a normal basis, and has nothing to do with the variant A major chord from the beginning; likewise, the A major key can put in the D minor chord as c, which has nothing to do with the principal D minor chord from the beginning. But the A major chord can also function as or C or , whereby the final key would be C minor, E major or G minor.

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% artificial A major chord

% natural A major chord

(NB: this example is put into this place for reasons of principle. It really belongs to a group to be discussed later.)

The engaged means are of perfectly simple type (substitutes and dominants), but they are very well suited to distant relationships. At this point that should do as an approximation . . . .

a)

C major—G major

to

overstepped +

Back to modulation (Examples 294—298). If that example is suitable — and it is — then its polar form (from A minor to D minor) is also usable. The employment of the tempered minor cadence is not imperative, if it happens for conventional reasons, then, through these, the polarity of the modulatory function is not annulled in any way. The cadence has of course nothing more to do with modulation.

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the same, polar A minor—D minor to 1

is reinterpreted as

or Dp as

overstepping -

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.

C major—B minor

A minor—B major (E major)

(E minor)

polar:

overstepping overstepping

to

to

The distant modulations indicated in most of the treatises on harmony are of the type shown in the last example, e.g., from E major to E minor , i.e., equally C major to C minor . But beginning and end are only tempered contradominants, thus “chords” and not “pure keys”. C major—F minor, D major, G major—C minor chords are quite well accommodated in one single key (e.g., D major):

C major chord

C minor chord

i.e., C major chord to C minor chord within D major. C major and C minor are, however, not ‘keys’.

In order to modulate from pure C major to pure C minor, considerably greater actions are needed!

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From

, or from

to

to

= C major—D minor , or A minor—G major

4 switches:

false:

that is: cadence cadence

cadence

NB: the major key ending in minor-key pieces was usual in earlier centuries. But despite the D major chord, it remains a modulation from C major to D minor (with variant ending) .

Saint

Ga-bri- elle, ____

I

love ____

you.

problematic triad reinforcement of the target key (widened cadence)

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2nd Group: Modulations to the keys of the 2nd and 3rd fifth-relationship The extreme substitutes of the dominant-pairs are each 11/2 steps away from the centre.

Consequently, the tonic triads of the more distant keys are no longer interpretable as direct tonal functions of the starting key. They must be interpreted as ultradominants and their further chains.

But there still remain bridges between double to triple-fifth related keys.

As previously, the targets can be reached through harmonic half and whole-step movements.

C—A major

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Ble - ss - ed Vir - gin

thou heav - en - ly maid

Tenor

A min—C minor

(belonging to C minor)

alto

open (the itself does not occur here; nevertheless, all functions are referred to it.)

This episode no doubt finishes with a G major chord, but not in the key of G major. The path to G major would actually be opposed. (See example 304.)

The shorter paths are just so much more obvious . . . . The exclusive use of fifth-relatives seems a little too dry, unless melodic or polyphonic means are not in the fore. The sequence

is completely unnecessary, as — standing on its own — it would not be intrepreted as C T D in the first place. An overstepping on both sides (plagal: recommended.

Model:

and polar:

— perfect:

) would at least be

i.e., contrary:

and polar contrary:

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The examples are primitive but matter-of-fact. Their expression is gained — as already stated — through melodic and harmonic means. Functionally, nothing will be changed by these. But they show something new: apart from the polarity, there is also a contrariness: i.e., opposing development in the same gender: ‘up’ becomes ‘down’, strong sides are interchanged with weak ones, ‘D’ becomes ‘C’, ‘p’ becomes ‘l’. The whole and half-steps remain as such; indeed: from the minor third semitone-step (principal triad to its parallelsound) comes the major third (principal triad to its leading note-changed sound). The contrariness results consequently in opposing ‘similarities’ only, but not in naturally intended exact correspondences. The contrary forms represent same-gender polar mimicry

, but the polar-

contrary forms on the other hand represent opposite-gender parallel-mimicry.

Excellent effects result through curved paths, through rebounds to jumps or also through periodical half-step changes to whole and one-and-a-half steps:

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The path-distance from C major to C minor must be overcome. The goal is reached most speedily by means of jumps (there are cases where chain jumps become necessary) — backward steps, it is true, retard the sequence of events, but they act as a balancing mechanism. In the example above, all stages from the B minor to G minor (variant of G major) chords are touched; however, instead of the scanned half-step movement: B minor—D major—F minor—A major—C minor chords, strong protrusions and mild halfsteps backwards occur. These curved shapes are stylised imitations of the cadence: or

If this sequence sounds acceptable — and that is the case — then its contrary, polar and contrary-polar forms are likewise useful. In the bar in two sections with two switches: (last example) norm: i.e., C major to C minor polar: i.e.,

A minor to A major polar function:

contrary:

Sarabande

i.e., C major to C minor

polar contrary:

i.e., A minor to A major

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, i.e.,

The function above can be strung together like a chain: e.g., from C major via C minor to C major , or

, i.e.,

,

,

e.g., from C major via C minor to C major . Or the minor-key groups starting with A minor via A major to A minor . A major A minor Two examples in addition: from C major to G major and from A minor to D minor .

Sarabande

contrary polar

[contrary polar]

the same, polar:

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If that is still not enough for the insatiable chromaticist, then the groups can also be rearranged, and this results in a modulation from D minor to G major or from [8 ] [8 ] G major to D minor . That would thus have started with C major and A minor from C major to G

minor (= 16 !)

or from A minor to D major (= 16 !) These modulations are completely free of chromaticism, both in the tonal as well as the functional sense. They are purely diatonic in nature, developments of the principal triadic basis, simple results of the primitive 5th-relationship. The single introduced chromatic form is the tempered contradominant, but it has no influence whatsoever on the modulatory event, as it stands only as a ‘balancing mechanism’ in cadences — thus already carried out, only casually confirmed. It is unnecessary to say that the sequences are completely comma-pure.

General Remarks on Chromaticism Ever since chromaticism absolutely and thoroughly undermined tonality, the feeling for comma-pure modulation — particularly among young people — has decreased considerably or completely died out. An exercise to modulate from C major to G major does not impress most people any more; it is solved under the rules in a 3-fold manner:

1) by born ‘Mediantists’:

2) by born ‘Dissonantists’:

3) by born ‘Enharmonists’:

a jumble from alterations, after that

= G major exactly

re 3) The last smart alecs are — to a certain extent — the most famous people. They are abrupt and draw, from the standpoint of 12 semitone music, radical consequences out of enharmonic studies. One should also allow them to notate the B major chord as and the B minor chord as

; since, finally, the ‘E ’ “today” (!) is

indeed the same as ‘D ’ and ‘C ’ is the same as ‘D ’. (See the famous Orthography of Whole-tone Harmony.) Admittedly, admittedly, the piano . . . . . etc.

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But the sound human mind should tell them, that — when G major means a sharpening in relation to C major, and D major the same towards G major, etc. — G major must display a quite conspicuous tension towards C major . . . re 1) We talk over and over again about the difference between identically named sounds and keys of principal and mediant evaluation, and it will be dealt with further still in the relevant chapters . . . . re 2) The dissonance! I really believe that it is not treated as peripheral in this work, and its unique significance is truly not underestimated — but it has nothing to do with ‘modulation’. I assert this, in full consciousness, bearing in mind the certainty by which many will label me as conservative (which, by the way, will certainly signify no disaster for me in any case). Dissonance carries the seed of atonality within itself, its solutions are always ambiguous, i.e., when it is not stretched in a narrow key, in which case its versatility suffers losses.

This case, understandably, does not come into

question for modulations, as it is to do with ‘coming out from the narrow key’. The almost universal opinion now is that the versatility of the dissonance is the typical element in modulation.

Not at all!

In the sense of tonality, the dissonance is

polysonance; if it emerges from a mixture of sounds x y z, then it also functions in this sense, and consequently, the consonances x or y or z can stand equally well instead of the dissonance. An example of this — by necessity anticipating later chapters —: If in the sequence C major, A major, D minor the A major chord is the mediator, and in the sequence C major, G minor, D minor the G minor chord is mediator, then the complex formation C E G B flat can also stand in their place. But — the A major and G minor chords are of course no means for modulation, but only introductions of the D minor chord, thus the hybrid form (the diminished seventh-chord) is also no modulatory element but an introductory element. So then in tonalistic music, all dissonances are introductory sounds, which either replace a dominant triad or come interceding between a tonal replacement, without which it is also understandable enough! A sequence of isolated dissonances automatically annuls tonality. Consequently, one can insert any arbitrary major or minor chord after a chain of dissonances;

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if a leading-note or semitone guidance exists between the last dissonance and the newly inserted consonance, then the connection turns out so much the better. A modulation is not in any case performed thereby, but merely the following process has been carried out: abrupt breaking-off of the key or tonality, atonal episode, unexpected new entry of a consonance, which will manifest its tonal function only in the course of further development.

C major

tonally completely indifferent can equally well and equally badly also be notated in a different way

G major

And despite all wild manoeuvres: if C major beforehand had asserted itself as a valid tonality, then it would also have proven its old dominion and binding power over the dissonance mush. What appears after the intentional smashing-up of tonality is an A major (!) chord, which, as a matter of course, can be referred to C major as a worthy tonic representative . The costs of the many and were also to no avail. G major can not be understood as such, because it does not result from a logical process of development. In the following example: The G major chord is understood as that, but it is too subdued, it takes the place of the G major chord, to which it indeed also functions as its collective colour. (E major chord = varied E minor sound in C major; G major chord = varied G minor sound in pure E major, but chromatically-shifted G major sound in pure E minor and C major respectively.)

or Even when the chord-changes occur more slowly, a mediant such as G major has no originality: it strikes back — if also not completely, yet still considerably — not, however, reaching G major on the Pythagorean basis:

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Example 319:

D major chord B minor ” G major E minor C major A minor F major D minor B major G minor E major C minor A major F minor D major B minor G major E minor C major

Example 318 b. G major G minor—G major E minor E major C major

This is for establishment in principle and defence of the principal modulation. +

The Concordance as a Means for Modulation The addition of the seventh stamps the sound concerned as dominant (or tempered contradominant respectively, which is of course a borrowed dominant of the variant key). As the dominants and ultradominants make concordances anyway and furthermore strive towards the tonic, then they are eliminated here as terminal sounds. The contradominants through seventh-introduction become dominants of the ultracontradominants. There’s nothing new in that.

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The ultra-contradominant making concordance works even stronger: [NB: Harmonic Sequence · · ·]

C major—E major B major—D major = C major—D major

polar:

A minor—F minor B minor—G minor

or

= A minor—G minor

More important compared with that are the concordances of the diatonic substitutes. C major to A minor E ” B ” ” F ” C

(important)

(likewise)

C major via A major to G major (C major C)

All seventh additions drive the sounds backwards: upper sounds strive downwards, lower sounds upwards to their successors.

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The most important form unquestionably is the switch

, it intercedes between

and 21/2. The last sound (21/2) can now be re-evaluated to each of the 7 tonal functions: ex. 322, 4

in G major

in D major

in A major

in B minor

in F minor

in C minor

in E major

As these tonic triads can again be re-interpreted to 6 new tonal functions, this presents a wealth of further modulation possibilities:

C major—D minor

Very strange, that this comma-pure, strong leading-note means for modulation has not long ago found its “special pamphlet”: actually the sound in question is the polar counterpart ). Like this, it also allows of the much-praised “Neapolitan” chord in the minor key ( : one of the most effective conjuring tricks, if it does not depend on one comma more or less:

i.e., = from A minor to B

minor

but with double-comma difference.

If this example should be valid as a ‘Neapolitan event’, then the major key counterpart does not need to take second place:

i.e., = from C major to B major but with double-comma difference.

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Pseudovariant Modulation The reason for the comma derailment lies in the fact that a variant or mediant stands at the switch and becomes re-interpreted as a principal or diatonic triad.

The B major chord here is merely a varied B minor chord. F major B major E major A major F minor D major B minor G major E minor C major A minor F major

perfect-fifth principal triads

variants (mediants)

Consequently, the switches faux pas!

or

, and

false original positions

or

respectively must also cause a comma

false original position

p false original positions

false original positions

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And nevertheless the tempered contradominant is able to intercede e.g., between C major and E major or C minor respectively, and between A minor and F minor or A major respectively, without yielding any comma faux pas {lit. derailments}:

a) b) c) d)

the F minor chord is not Cp, but varied double-dominant triad in E major, the F minor chord is not , but varied dominant triad of the contra-parallel in C minor, the E major chord is not , but varied double-dominant triad in F minor, the E major chord is not D, but varied dominant triad of the contra-parallel in A major.

The comma faux pas is averted by the fact that at the switch, two variants are standing, which annul each other. Neither determines the modulation, because a) the variation can be imagined just as well as b) the whole chord:

Cadence in E maj or C min

Cadence in F min or A maj

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Ultradominant Chains The fastest forms of change are presented by chains of 9th-relatives. The archetype is perhaps to be sought in Palestrina (Stabat mater: E, D and C major chords). Ultradominant chains are especially found in Liszt (Dante-Fantasia), occasionally also in Puccini. Set in a narrow tonality and then appearing sporadically, they show a marked tendency to repercussion:

But in a more liberal environment, their continuous sequence is certainly possible:

(excellent effect)

likewise in opposite direction

For modulation purposes, overstepping or return — which of course boils down to the same thing — is advised at the end. overstepping

overstepping

C to B major

A to B minor

As one can see, the bass notes stride along the Pythagorean whole-tone scale. If the ultradominants take their exit from the tonic, then only the even-numbered distant dominants (2, 4, 6, 8, 10, 12) can enter. The 5th retrograde step making a cadence lets an odd-numbered principal triad reach its tonic (see last double-example). If a double 5th retrograde step takes place — advisable with strong over-tension — then an even-numbered principal triad comes finally.

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contrary

contrary

If the chains begin with D or C ( or ), then of course the target shifts in or out a fifth. The substitutes bring a desirable change. They can complete the chains and lead forwards or backwards in half-steps — or they themselves can send out the chains.

or C maj—C min

C major!

C minor Phrygian A min—C maj

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mixed form

pure

C — F major

F maj—tempered F maj

tempered

If the ultradominants are introduced, overtaking, through their dominants, then the latter show themselves as triple dominants. They resemble — except for the comma-difference — the third-relatives of the counter-fifths (contradominant-mediant triad CM). But while these incline forward to the tonic (a consequence, which results from the potential C), the triple dominants pull back to the ultra-form. The comma-lowered CM will offset the relaxation of its original principal, whereas the triple dominant will reduces its over-tension:

the other way around in the minor key. As C form contrary, in the minor key contrary-polar:

contrary

also with tempered introduction

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polar:

contrary-polar:

Diatonic chromaticism and chromatic diatonicism respectively! (naturally with exclusion of the tempered contradominant). A very succinct example of this is given in the so-called ‘G minor’ Rhapsody by Brahms, which indeed declares itself in G minor only in later development. Brahms: Rhapsody, Op. 79, No. 2

Starting from E major: (starting from G minor, 1 /2 added throughout)

E major

G major

B major

No mediant triads! No comma-differences! Pure principal triad development! The triple-dominants admittedly do not lead more quickly to the target than the simple dominants, as they repeat of course; their specific character lies — as opposed to the straightline chains of fifths — in the curved passages. C

A

D

B

E major chords

C

G

D

A

E major chords

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Two examples follow, which mix the forms in question with other means, explained earlier:

return

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Epilogue to the Chapter on Modulation The whole of the chapter above is heavily schematised, and I can very well understand the justifiable objections, that practical music shows a much, much freer modulation, does not recognise any comma considerations and, above all, puts the dissonant element in the foreground in innumerable cases. All this is certainly true, and I would not be an informed, practical musician and would not understand anything about composition, if I wished to doubt the justification of these objections . . . One should note well: in this section, we were merely demonstrating the consequence of the relatively most primitive relationship (dominant type). The complete consequence, which is of course many times greater, is as generally assumed . . . . The examples indicated are unembellished without exception, but this does not stop the taking on of all imaginable polyphonic, rhythmic, motive and dissonant means: only one thing remains out of the question for the time being: ‘variant re-evaluation’. That it plays a large role in practice I know as well as those who cannot wait to combine the most distant sounds ‘on the very shortest route’, e.g., C—B—D major (which comes about only through a two-fold variation — or mediant modification respectively — of the original form C major— B minor—D major). Later chapters will deal with these tonal combinations in sufficient detail . . . I am exceptionally anxious to clarify the inner essence of the modulatory process, for didactic reasons. Assume a primitive modulation from C to A major. It proceeds directly, without curves.

Actually, it could not be more unembellished, but as this example expresses the inner essence as clearly as possible, from now on this elementary of the process of change type of bicordant, chromatic, or variant or mediant embellishment respectively, or substitution is quite safely usable: 9th-chords

7th-chords with altered fifths

e.g., the same

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diminished 7th-chords

somewhat lengthened

(ideational foundations)

ideational foundations:

with stronger dissonances NB1)

NB2)

NB 1 C G E A B or C Foundation:

5 9 7 3 5 F1

C

F

B

E

NB 2 E B G C D or E Foundation: A

5 9 7 3 5 1

A

The primitive 5th-case example still forms the recognisable basis. And further: Instead of F and B major, F and B minor can occur, as the straight line to the E major chord is not broken. Mind you: the F minor chord is in no way Tp in A major but rather a function which, however, does not become effective:

,

foundations to some extent variants thoughtless/harmless

Or instead of the F major chord comes its counter-mediant: D major, instead of the B major chord its 7th counter-sound D minor:

stylised:

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Or even further: instead of the F and E major chords entering their same-3rd (Mp): F and E minor chords, whereby instead of the F:B and E :A major chords making cadences, collective changed-sounds*) arise (F minor:B major and E minor:A major chords): (*) Collective changed-sounds, see later) Leading notes

% %

E D

B

D C

A A

G F

C

G (G )

F

E E

note the inner legitimacy!

This might be enough for the time being! These examples are not forced but are obvious as simple outcomes on inner legitimacy, as clear consequences of natural tonal conditions. The following chapters teach the means for similar combinations. This technique becomes more reliably backed up, more exact, well, I should like to say: the more “puritanically” the present chapter on ‘comma-pure modulation’ is worked through.

Below, some ‘stylised’ examples of simple modulation forms:

Semplice.

Molto lento.

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Quieto.

as if a V’cello

sonoro

Allegro.

risoluto

con umore

con molto espress.

Larghetto misterioso.

(3 violas muted) senza espr.

Sarabande.

as if a string quartet

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Ländler. (commodo)

Curved path (overstepping) Villagers’ dance, not too fast!

rough

(original positions) (fifth steps) (primitive)

Similarly. Half step! (primitive) Rigaudon. Oboes

stiff

bassoon (dry)

Agitato e dramatico.

F minor, for the purpose of repetition F major, metharmonic!

very common

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Some examples in the ‘a cappella’ style: Tempo giusto. For

glory.

thine

A -

(moderately moving )

is

the

- men,

venture

a

not

of E major

kingdom and the power and the

-

-

-

of this

I

this

I

men.

A -

must -

- men.

-

-

-

-

must - C minor

de –

clare,

very expressive

-

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That your peaceful

beauty

has

made

me

richly

endowed

with

great

Tenor marcato

intense

A minor D major sorrow

Oh

wonderfully

Oh Won-der! very expressive

Oh Won-der!

Oh

a minor created

woman,

might

I

yet

your

thanks _______________

(A major)

Mixolydian

____ receive;

wonderful

F major [Lydian]

hymn-like, broad they stand before

(E major)

your

lovely

body

driving forwards in my

(Tenor accentuated, most mellifluous) B major (Mixolydian element)

driving forwards

or

elevated song _________________

elevated song

(as from the distance) Saint

Saint

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8 solo parts

Saint Ga-bri - el - le,

I

estatico

love

with greatest ardour

F minor

you!

E major as at

the beginning

coloured and indulgent increased contraction of the steps

(elegant - -)

pressing (accel.)

as if horns

Basic modulation is quite primitive:

All charming, expressive dissonances are merely ‘mobile forms’, they have no influence at all on the modulatory event.

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Supplement to the First Section

9th Chapter Twins and Triplets As already mentioned in the 5th chapter — “The Substitutes” — the diatonic minor sounds in the major key and the major sounds in the minor key are incomplete complexes each of two 5th-related principal triads:

If one or other of the principal triads appears in full {combined} presentation, “twins” result.

Tonic-parallel twin. Contra(dominant) leading-note twin. Likewise the other principal triad twins. They definitely appear as concordances tending to the diatonic, of closely related dominant and mediant triads. original type concordants

twins

original type concordants

twins

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Historically, the twins appear first in diatonic circular step-sequence:

closed key-cycle (dissonant tetrads)

fifths

prime

thirds

modulating chain (concordant tetrads)

The functional designation is i.e., the remarkable: NB 1. dominant triad of the tonal dominant = D major7 = G major7. NB 2. i.e., the tonal dominant is introductory dominant triad of the subsequent sounds etc. G7—C7 etc. Example 374a: The artificial closure of the circle results in a narrow range of keys. The natural primitive concordants and likewise natural 5th-relationships are — for artisticaesthetic purposes — artificially sylised here. The result is — dissonance! What power, what tension, what austere, manly character lies in such an asymmetrical chain: Schütz, Handel, Bach. Example 374b: Nature poses infinite chains of all types of relationship. But the mechanical uncoiling of chains of many links without any setbacks or stylisations is not enough of a personal achievement, already to be declared as art. Also, the first example is a mechanical form of fall in degree, but here there appear continuously changing kinds of voice movements (whole tones and semitones, pure and diminished fifths) and of functions (main and auxiliary steps); the automatic mechanism is thus only approximate. The mechanical dominant-chain form is very common as a fooling about with form in the piano-music of Chopin and Liszt (fioritura {“flowering”, i.e., melodic embellishment, as in baroque times, when often improvised} and suchlike). Of all twins, the contradominant-parallel twin in the major has been the most preferred all along.

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Tonic and dominant parallel-twins still show the character of a suspension in the style or respectively had already attained independence as an of the classical period, whilst isolated form.

But from the middle of the 19th century, Viennese waltz-music also stamped the tonic triad as weak tonic parallel-twin (completed on the weak side!) and as sharp tonic leading-note twin (completed on the strong side!), without subsequent resolution!

With this “Viennese sixth” also, Arnold Schönberg introduces himself as a Viennese to the musical world in his gigantic choral work ‘Gurrelieder’. It begins and ends with long-held tonic-triad twins.

Mahler’s “Song of the Earth” finishes with the same Viennese sixth. A quite wonderful example of twin sequences is offered by Schumann’s ‘I bear no ill-will’ at the visionary passage: “I saw you truly in the dream”. How pale and conventional this sounds without the blurring colours, which arise only through the mixing of major and minor chords.

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I saw you truly in the dream and saw the night in your soul I saw the serpent

together: i.e.

An exclusively E major interpretation is certainly not given, as with the E major key, C minor tonality is established at the same time. Thus the C minor interpretation would also be possible, even more obvious, as C minor is introduced making a cadence, not E major, and the C minor lower-sound tetrad

is interpreted as

rather than

.

i.e., in C major: (original positions) (comma!!)

cadence-curve

better

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Liszt (“Catholic type”) and after him Grieg, Macdowell, Debussy and, above all, Delius have developed a formal twin-sound style. The accumulation or stereotypical use of twins creates an impression of an indefinite state without contours, of floating, swimming, of opalescent, coloured shimmering, of clear-obscure {the last is especially vague in its contradiction: the French word clair and not klar is used; the second word is spelt obsur and this is presumed to be a typographical error}. No wonder that it was particularly favoured by the

impressionists and exoticists {our attempt to match Karg’s word for ‘practitioners of exotic things’!}.

The cause of their tonal character is obvious. These formations have no explicit tonal tendency, no explicit gender, they are upper and lower sounds at the same time; the energetic opposition of passive and active sounds, and positive and negative forms respectively, seems fully compensated:

.

Change of position:

or Parallel-twin

Leading-note twin

The leading-note twin appears often as suspension of the parallel twin. Although this is itself a suspension (= double sound), it nevertheless constitutes a relative resolution regarding that weak dissonance). (strong Tempo di Valse.

Robert Schumann: Arabeske, Op. 18 (1839)

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NB: This pedantic notation for suspensions has of course been chosen only for instructive reasons.

In impressionistic music (Debussy, Ravel, Scott, Scrjabin, Delius, Karg-Elert, Schreker, — are fairly common. Niemann), blurred endings to the tonic twin — , occasionally also Endings:

minor:

minor:

or

or

Johann Strauss: “Fledermaus” (or “Bat”)

Georges Bizet: Carillon (The Girl from Arles)

(E Lydian: Dl)

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Georges Bizet: Berceuse (Lullaby)

A

swan, my si - lent

Edvard Grieg: “A Swan”

Edvard Grieg: Melody, Op. 47, No. 3 (1888)

E minor

twice

Edvard Grieg: Homesickness, Op. 57, No. 6 (1893)

(diatonic sequence)

W. Niemann: “The Magic Book”, Op. 76, No. 5 NB) precisely:

twice

also as 5thor 4th-tetrad

Claude Debussy: The Blessed Damsel (1887—88)

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C major solo violin

quartet, harp

quasi G Dorian

Claude Debussy, Suite bergamasque (Prelude—1890)

Claude Debussy: Ballade (1890)

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Modérément animé

twice l.H.

Claude Debussy: Pagodas (Engravings, No. 1—1903)

Gabriel Grovlez: Litanies of Jesus

Naïve, tender and fervent

Arnold Schönberg: Songs of Gurre (opening)

(Original: figured)

theme accentuated

suspension to:

etc. Arnold Schönberg: Songs of Gurre (conclusion)

opening motif inverted

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in the blue of the night

Josef Marx: “A Young Poet . . .”

moves

A major: Josef Marx: Album Leaf

Alla Sarabanda.

S. Karg-Elert: 17 Aphorisms, Op. 51, No. 13—1905

(*)

(*) The fragment is not quoted exactly from the printed version; changes have been made, presumably to suit the text.

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flexible

{Op. 77 is not the source; it has perhaps been written expressly for the text.}

quieto e semplice

S. Karg-Elert: Heathland Pictures, Op. 127 (1920); closing bars of No. 1: “A Wide Plain”.

Tranquil

lovely

and mild, lovely

S. Karg-Elert: “Evening Star”, Op. 98, No. 1 (1914), for voice and organ

and mild a picture of the moon comes up

3rd manual: Celestes 8’

2nd man.

(chain of twins) (actually 7thchanged chords)

S. Karg-Elert: Exotic Rhapsody, Op. 118 (1917?)

respectively

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(as if harmonics) (as if horns)

referred to F referred to C Alexander Scrjabin: Nuances, Op. 56, No. 3

Rudolf Schartel: Mandrake, Op. 20 (Art-harmonium)

overtone formations chain of twins

The Triplet If a principal triad appears with its parallel triad and its leading note-changed triad in combination, then a triplet results:

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The triplet usually appears as contradominant replacement in a cadence, in which case the dominant usually follows it. how lovely you

are

In this case, it appears as diatonic, thus a variety (d) of D. Between Cp and admittedly comma-differences: or

there are

as 7th (Leipzig comma to the primary )

difference to tonic third

(See No. 409)

As for all forms of dissonance, the triplet comes only as a mobile sound on

(

fifth—sixth fundamental note—leading note

transition

)

Suspension

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The most usual forms put the leading note or the third in the top part: Leading-note 7th

(pseudo-9th)

3rd

(pseudo-5th)

A minor F major

chords

D minor

in F major in C major In functional change: F = suspension (leading note) A = transition (9th)

mobile forms

Its strong 5th-sound character puts it close to the pentatonic and 4th-sounding forms, and that is why one can refer to the triplet as a remarkable bridge between old and new harmony.

or

fifths fourths

In the sense of colouristic retouching (colouristic additives), the bundles of fifths and fourths appear mostly at a distance from the original consonance:

detached

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If the major key triplet resembles the 5th-related double principal triad in the minor, then the minor key triplet resembles the 5th-related double principal triad in the major:

Major and minor forms float {merge} into one another. Tonal resolution proceeds noticeably.

in – clines to

a

pu – re heart Peter Cornelius: from “The Lord’s Prayer”

Franz Liszt: The Bells of Geneva

triplets as movement form (suspension on the Cp = third)

Edvard Grieg: Wedding Day at Troldhaugen, Op. 65 (1897)

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I

S. Karg-Elert: Christmas II (a cappella) from “The Christian Church-year”, W 11, 1909 I - ya, I want to be his

- ya,

{presumably No. 4: “Born in Bethlehem”, the last in the set; No. 3: “From Heaven above . .” is by implication ‘Christmas I’}

soloists sequence

own.______

I

-

ya,

I

-

ya,

choir

tranquilly take away my

solo 1st violin

heart

as

pledge

S. Karg-Elert, from the “Symphonic Mass” (manuscript)

F

4 solo violas 2 muted horns

(from the “Kyrie”) violin

(B)

*)

{probably W 64 (1923), left incomplete, now lost — see “Harmony of the Soul”, letter to Bellmont, August 1923, pp. 27—28 as well as the Gerlach catalogue, p. 127.}

Je - sus

*)

Chri - st

(Oboe, English Horn, Hecklephone { = baritone oboe})

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fastoso S. Karg-Elert: ‘Praise Zion’, Op. 106, No. 6 from “Cathedral Windows” (organ) — 1923 {both bars merely approximate bar 3 in the Elkins edition.}

Frederick Delius: Springtime (1897)

Conclusion

Cyril Scott: Cavatina (Elkins, without opus no.)

i.e.

in a perception of the major key:

with double substitutes,

in a perception of the minor key:

savage, warlike

Alexander Scrjabin: Prelude, Op. 59, No. 2 tritonic sequence

E maj A maj with double substitutes or

minor chords

(atonal)

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Intentionally blank

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Third Part The System of Harmony in Polar Perception A Developmental Precept of Tonality

Second Section Third and Seventh Relationship (Chromatic Tonality)

10th Chapter: 11th Chapter: 12th Chapter:

13th Chapter:

14th Chapter: 15th Chapter:

The Ecclesiastical Tone Systems (belonging in part to the First Section) ........ 175 Variants as Metharmonic Means of Modulation............................................... 188 General Comments on the Third Relationship................................................... 199 Tritonants ....................................................................................................... 199 Collective Changed Sounds ............................................................................ 201 Chromatic Chords ......................................................................................... 202 A) The Style of the Mediant ................................................................................ 203 B) Acoustic and Functional Individuality of Mediants ..................................... 207 C) Obvious Tonality Changes through Comma Differences ............................. 207 D) Metharmonic and Enharmonic Concepts..................................................... 209 E) Harmonological Tonal Interpretation............................................................ 211 F) Cadence Broadening........................................................................................ 213 G) Mediant Tonality-jumps, Shifts, Parentheses and Sequences...................... 221 The Seventh Counter-sounds (Counter-concordances).................................... 309 Abolished (Decentralised) Tonality .................................................................... 312 a) General Comments .......................................................................................... 312 b) Parallel Chord Shift ......................................................................................... 313 c) Atonal Complexes (see also XI in the Introduction) ..................................... 320 d) Consonant and Concordant Tonal Interpretation in the Freetonal Sphere and its Tendency towards Bi- and Poly-tonality respectively ..................... .322

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10th Chapter The Ecclesiastical (Church) Tone-Systems One can look up information about the original forms themselves, the “Greek scales”, in the appropriate specialist works (among others, K-E.: “The Fundamentals of Music Theory”, Book I, Speka Publishing House). A detailed discussion is unnecessary in a work on harmony. Church tones in the western world take over forms and names from the Greek ‘keys’, but unfortunately muddle the names for the given forms. If the chuch tones are notated without accidentals, then this results in 3 forms like major keys (C = Ionian, F = Lydian, G = Mixolydian) and 3 like minor keys (D = Dorian, E = Phrygian, A = Aeolian). This relates to C in the major key and A in the minor key: C (major) Ionian pure major type!

A (minor) Dorian

C (major) Lydian

A (minor) Phrygian

C (major) Mixolydian

A (minor) Aeolian pure minor type!

One speaks of a Lydian 4th (augmented ” ” Mixolydian 7th (minor ” ” and ” ”

Dorian 6th (major Phrygian 2nd (minor

compared to the pure major form)

compared to the pure minor form).

Dorian and Lydian appear as parallel keys Phrygian and Mixolydian

= raised 6 + 4, = lowered 2 + 7.

But this perception completely blurs the natural counterparts. As the relationship between major and minor does not result from parallelism, but from polarity, the church-tones should thus also be received merely as complements, which show polarity between themselves. But this applies only between Dorian and Mixolydian and between Phrygian and Lydian. The typical criteria of church-tones are, in harmonic perception, always 3rd-tones of a typically diverging principal chord. Accordingly, the Lydian 4th is ‘3rd of ’ ” Mixolydian 7th is ‘3rd of d’ ” Dorian 6th is ‘3rd of ’ and

”

Phrygian 2nd is

‘3rd of

’

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NB: Just as between major and minor keys of the same name, an exchange of the dominant = c, so then an exchange occasionally takes place also between often takes place: D = , major and minor keys of the same name as

and

:

in the minor =

,

in the major =

C major : D major chord = T :

A minor : G minor chord =

C minor : D major chord =

A major : G minor chord = T :

:

(Lydian 4th)

Norm

same sound direction

:

(Phrygian 2nd)

Mimicry

contrary direction

Norm

same sound direction

Mimicry

contrary direction

Comparative Survey of the pure and mixed Systems in polar perception positive strong open

tonic basis

tonic basis open strong negative

C original major (C Ionian) — Norm — bright, positive, active

A original minor (A Aeolian) — Norm — dark, negative, passive

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positive overly strong open

tonic basis

tonic basis

open negative overly strong

C major Lydian

A minor Phrygian

ultra-major

ultra-minor

overly bright, exaggerated major character

deeply dark, exaggerated minor character

open, positive strong

Leipzig comma

extreme, eccentric.

open, positive strong

tonic basis

open, negative strong

Leipzig comma

tonic basis

open, negative strong

tempered C major

tempered A minor

strong on two sides

strong on two sides

The weak sides are replaced by strong dominants of the variant key centrifugal

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negative weak, closed

negative weak, closed

T.B. T.B. positive weak, closed positive weak, closed

C major Mixolydian

A minor Dorian

weak on two sides

weak on two sides

The strong sides are replaced by weak contradominants of the variant keys. centripetal

Modulations, of course, take place also in the church-keys, and indeed between transposed similar types, as well as between similar step or transposed varieties of different types. The relevant literature proves that the individual types have never come through tonally pure. Above all, the minor key forms show as variant: Dorian in cadence (though also only here): instead of ; Phrygian as a rule, ending with instead of . The ultra-forms occur — if not always, then at least very often — as extreme opening sounds of the simple dominant, whereby they lose their main appeal. In the Mixolydian and Dorian, the dominant variant-parallels often appear in a sequence, which makes their interpretation dubious, and lets the perception of the sounds in (metharmonic).

comma difference

or

comma difference

question appear closer to

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In the works of Chopin (particularly Mazurkas), Liszt, Brahms, Dvorák, Grieg, MacDowell, Debussy, Sibelius, Sinigaglia, Respighi, the Russian innovators — in so far as they reveal a nationally emphasised character, and in all exotic and pseudo-exotic pieces in harmony — unmistakable influences of church-tones stand out markedly. The superficial music critic is quick to judge with the catchword ‘Grieg’s harmony’ . . . The specific church-tone harmony consists merely of the giving of precedence to the typical dominant before the tonic. The dominant variants are anhemitonic (without leading-note) {lit. without semitones}, they appear very often in the top part as a third that jumps off in the direction of its chromaticism, which is also usual in .

mimicry

artificial natural

‘Aeolian’

natural artificial

tempered

comma differences

comma difference comma difference

: d as well as (‘countersounds’). between

:

represents the relationship between the tempered contradominants

Against this there are comma differences between

:

as well as

: d. A new meaning is however certainly possible, if a later back-reinterpretation

follows. The episode lying between the two represents a ‘transitory modulation in the form of parenthesis’.

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Comma differences

ACE

CEG

Original positions

(bass clef)

(C major)

(A minor)

(comma-raised A minor)

(C major)

(comma-lowered C major)

It is considerably simpler to forego re-interpretations and take up only introductory or closing forms.

Dorian introduction

Lydian closure

Mixolydian introduction

Phrygian closure

Mixolydian introduction

Phrygian closure

Dorian introduction

Lydian closure

C major group

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A minor group

In shift and sequence, the specific church-tone cadence (alias the exotic character of the over-sharp, over-blunted and over-sharp-sign-notated form) stands out distinctly:

polar:

(comma-pure)

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C major Mixolydian The key

is

Mixolydian,

the position is

wide, now it

similar to

D minor Dorian

Tonic

chord

and

gets narrow again,

(comma-difference)

parallel

in

Parallel

Dorian

key,

in

variant.________

A - - men. Dorian

key,

sounds

same

as

major.____ (variant)

E minor Phrygian

Phrygian

ultradominant,

(in eunuch tone)

with the usual solemnity

C major Lydian typical

of

the

church style.

Lydian

ultradomi-

bright, merry

(Variant)

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-nant,

profanely

merry

harmony.

Free tonality, changing or modulating church-tones (characteristic of the a cappella style before Bach and Handel). Most

beautiful

of

all

maidens,

accept

this

teaching:________

G major

(tempered) (Dorian to)

C minor

(tempered)

if

G minor Phrygian:

you really love me,

tempered A minor

(variant)

you also really

give me . . .

(

D minor Dorian

(variant)

F major

(D major): )

I

quarrel

A minor

D minor

(Neapolitan)

if

C Mixolydian:

without

(Neapolitan)

cause.

(variant)

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Church-tone effects at stationary harmony

Chopin: Mazurka, Op. 68, No. 3 (1829; B.I. 34) “Tritone” leading note to the fifth NB1

B major Lydian

Chopin: Mazurka, Op. 67, No. 2 (1849; B.I. 167)

D major Phrygian (variant) referred to D major chord

“Phrygian 2nd” NB 2

Animez

Debussy: Ballade (1890)

D minor Dorian

“Dorian 6th” NB 3

The scale of D is in itself Mixolydian The scale of is in itself Dorian NB 3 The scale of C is in itself Lydian NB 1 (or ) is in itself Phrygian (Variant) NB 2 The scale of

Grieg: Violoncello Sonata, Op. 36

Dorian

polar:

Mixolydian

Grieg: Piano Concerto, Op. 16

(orch. tutti)

MacDowell: Andalusian Dance, Op. 37, No. 3

Dorian

polar: Ferdinand Pfohl:

“Beach Pictures”, Op. 8, for piano; 1892

Mixolydian

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(continuation)

(under)

(over)

(continuation)

Paul Juon: “Intimate Harmonies”, Op. 30, No. 7, ca. 1905 (Legend has it that . . .)

marc.

Paul Juon: Op. 30, No. 6 (Episode), ca. 1905

standing tonic chord NB: “tempered E Mixolydian” (minor/major)

similar to (comma-difference) ‘C minor’ = 3rd of c ‘D minor’ = 3rd of d

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valid tonic triad Draw

Felix Woyrsch, Dance of Death, Op. 51, ca. 1905

near, all that is great and small, join in to my roundelay.

Phrygian (pure)

Phrygian

as for previous number

(pure)

Liszt: 2nd Hungarian Rhapsody

Phrygian (variant)

better: 7th countersound (C minor:) (C major:) Wagner: Valkyrie (end)

Karg-Elert: Exotic Rhapsodie, Op. 118

in G, referred to the Lydian minor, occasionally to the variant tonic triad

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C major Lydian

C major Lydian

Mixed forms

Liszt: 13th Hungarian Rhapsody

tempered minor — Lydian tempered E — Phrygian Gipsy-major Gipsy-minor

tempered A minor — Lydian

Anton Rubinstein: Dance of the Lights of the Bride, from the Opera “Feramors”, 1861—62

(NB 1) Lydian 4th (NB 2) Phrygian 2nd (Neapolitan 6th-chord). 187 - English

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11th Chapter Variants as a Means for Modulation (Switches for Mediant Sidings) Finally, we are to push open the gates of transitional chromatic modulation (commadiffering modulation). Compare, by contrast with this, chapter 8 in the first section of the second part. and

and can become

.

and

and

and

and can become

and

. and

Likewise the other way around: each diatonic principal triad or diatonic substitute can become a variant. _____________ Most commonly, the tempered contradominant (strong variant) is used as a switch: It shows in the major key the downwards = directed path.

A) and in the minor key the upwards =

On the other hand, the tempered contradominants of the tonal substitutes show in the major key the upwards = B) directed path. and in the minor key the downwards = The following schematic examples place the tempered contradominant on the switch. If instead of the c or , the t or were entered, then the target key would move closer by a 5thstep, for d or

two 5th-steps closer and for

or

one 5th-step further away.

*) 3 means 3 steps of a fifth (in metharmonic re-interpretation) downwards, etc. 188 - English

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(closer) The same, polar:

(further away)

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(closer) (Neapolitan cadence)

(further away)

Just as the F minor chord functions as

in A minor and the E major chord

functions as in C major, all minor key examples are also transferable to the major, and all major key examples are transferable to the minor:

e.g.,

with major key start:

or

with minor key start:

Very common is the merging of two modulation groups into one larger line. Similar types can follow one another as sequences, or different forms can follow one another: e.g., twice the first major key example {i.e., ex. 465}

i.e.

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The same, polar:

Such chromatic shifts can of course be combined with diatonic modulations. These mixed forms are the most commonly used in practice, e.g., from chromatic

chromatic

diatonic setting

C major to E minor :

See the penultimate example (485), stylised:

If the third motif-group would once more put in the tempered cadence

, then

the target would be moved still further away: first C minor, which could perhaps follow a diatonic modulation with conclusion in B major.

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For reasons of readability, enharmonic re-interpretation is advisable:

When entering B major, the whole chain can be carried through orthographically quite comfortably: unessential

Original: enharmonic:

the same, stylised: (in Chopin’s tongue {i.e., manner}) Animato (F major)

(B major)

(D major)

(G minor)

(C major)

(E major)

(A major) chords

(E minor)

(F major)

(A major)

(B minor)

(F minor)

this from B major to A major with 3 commas

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= A minor : B minor; i.e., = B minor : C minor

The same, polar: Broad

For today the glorious day has appeared

The tempered contradominants of the diatonic substitutes show the way — as already mentioned — concerning the possibility for modulation in the forward direction

.

sequence

sequence

schematic:

sequence

whole tones

whole tones or

C major

B major

in plausible metrical evaluation:

G major

F major

Compare the progression from the weakest to the strongest type The tempered contradominant of Cp forms the auxiliary mediant (minor 3rd-relationship) a) of Tp ”

” mediant

of Tl or Dp

” semitonic

of Dl

” tritonic (double minor 3rd-relationship) d)

”

(major 3rd-relationship)

b)

(leading note sound)

c)

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Instead of a repetition of the above type (sequence), a changed type can of course also occur. e.g., Solemn

Violins, flutes

Woodwinds Horns

(Bruckner’s type of movement) Tuba

The polar counterparts for this purpose: (compare 493 with 491)

schematic:

whole tones

whole tones or

A minor

B

minor

D minor

E minor

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Type change a + b + c + d (analogous to No. 492) (Fl.)

to (Quartet pizz.)

The borders of tonality are drawn out to infinity (nullified tonality). In the present example, the functions of which form the polar counterpart of No. 492, the chances are favourable — just as there — for an orthographically correct execution. On the contrary, an A minor start would result in an F minor (!) final chord; enharmonic reinterpretation would be necessary from the viewpoint of readability, through which the continuity of the chain of relationship would indeed be broken at some point. _____________ Now this would be a suitable point to present the ‘fruit and flower piece’ {full development?} of a good many treatises on harmony: examples of modulation according to all keys. But a reasonable distance can be taken just from that, as the principle of mediant adaptation has been sufficiently elucidated in the above schematic examples: see No. 492. (Transient modulation in the keys of the minor 3rd, the major 3rd, the leading-note and the tritone.) Through a combination of these types, all imaginable possibilities arise. the the the the

c

in in ( ) of a (c) of a

the major leads through tonal re-interpretation the minor ” ” ” ” major principal triad ” ” ” minor principal triad ” ” ”

downwards upwards upwards downwards

identified with perfect-5th keys!

One just emphasises precisely:

It is however remarkable that, through enharmonic re-interpretation, the target chord appears on the “reverse” side, e.g.: G major

i.e., transposed G major A minor A

but enharmonically G minor minor - re-interpreted: G major

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Schubert: Impromptu, Op. 90, No. 3 (originally G major)

logical functional notation: But these abrupt jumping and substituting functions are really not intended at all; the enharmonic remedy here is merely an emergency aid, in order to rectify difficulty in reading, whereby admittedly a much larger inhibition appears: the harmonological breach.

the same in the now popular G major edition

ditto whole tones:

The path from G major to G minor, or from G major to G minor is an opposing one, not the tonal c, but the

of a tonal substitute shows the direction of the path: G major E minor G major E minor

B major D major G minor B major D major G minor

Similarly, in the case of Schubert’s 4th Impromptu. Here a sound chain appears from the A minor chord via the A major chord back to A major. That would be thus: A major A minor

A major

But actually there is an enharmonic re-evaluation and back-interpretation, which reinterprets the intended B major chord as the more easily read A major chord. Consequently the sound curve runs in the following way: (transposed: A minor)

(A major)

A minor

A major (B major) B major

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Original notation orthographically correct: transposed:

A minor G major

C major

G major

C minor

A minor

C major

G major

C minor

G major

function:

R. Wagner: Parsifal

orthographically correct: transposed:

B

maj

B maj

C min G maj

A min D min

A min E maj A maj chords

C min

A min

A min

G maj

D min

E maj

A maj chords

function:

The outstandingly euphonious sound-chain produces, in polar transfer, a quite common sequence:

(incredible)

F major

as: as:

E instead of F (plausible)

E maj

B maj

D min A min

E maj

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The entry in C major (instead of C major) confirms the accuracy of the last interpretation: the chain runs continuously to E major (i.e., = E major from a beginning in C major). ditto

metharmonic

metharmonic

referring to A minor

The same, polar: (A minor to F minor)

In the above section, the tonal pairs bridge C major : E major were to be understood on the common mediator A minor A minor : F minor

”

”

”

”

”

C major

C major : A major

”

”

”

”

”

F minor

A minor : C minor

”

”

”

”

”

E major

C major : E major

”

”

”

”

”

C minor

A minor : F minor

”

”

”

”

”

A major

C major : A major

”

”

”

”

”

A minor

A minor : C minor

”

”

”

”

”

C major

Without bridges these tonal pairs become mediants and auxiliary mediants. (Next chapter) 198 - English

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12th Chapter General Comments on Third-relatives Since the beginning of the previous century, the last-named tonal pairs appear more and more often in immediate sequence (i.e., without connecting “bridge”): e.g., C—E major chords — without connection to the A minor sound; C—A major chords — without connection to the F minor sound, etc. Consequently, these tonal pairs must be applied directly after one another: a) in the sense of chromatic substitutes, b) in the sense of a new kind of authentic or plagal, imperfectly cadencing connection sounds. All tonal pairs mentioned (see page 196, below) are 3rd-related, which I combine under the collective name “mediants”. They form in principle two different groups: a) those that appear “shifted” by a major 3rd upwards and downwards against their principal triads — and b) those that appear “shifted” by a minor 3rd upwards and downwards. In all cases, the tonal gender remains the same! The (simple/plain) “mediants” lie in the forward direction of their principal triads (i.e., a major 3rd above the upper sound or below the lower sound). The “counter-mediants” lie however — as the notation “counter” clearly shows — in the backward direction of their principal triads (i.e., a major 3rd below the upper sound or above the lower sound). The mediants are variants of the strong leading note-changed sounds and chromaticise each by one tone (mediant = third); the counter-mediants however are same-3rds of the parallel sounds and chromaticise each by two tones (members of the chord = root + fifth): The principal third becomes root of the equal-gender “mediant”. The principal root becomes third of the equal-gender “counter-mediant”.

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_ _ _ _ _Ultra

Ultra _ _ _ _

_ _ _Ultra _ _

_ Ultra _ _ _

The notation “auxiliary mediants” is to be understood as a general collective term; in special cases I interpret that: a) parallel variants (a minor 3rd, shifted backwards to the principal triad), b) variant parallels (a minor 3rd, shifted forwards to the principal triad). The parallel variants each chromaticise by one tone (3rd), however the variant parallels each chromaticise by two tones (framework of chord = root and fifth) and represent the same third as the leading-note changed triads.

Explanation of the verbal notation

Tonic - Parallel - Variant

Tonic - Variant - Parallel

Tonic - Parallel - Variant

Tonic - Variant - Parallel

(Mixolydian parallels)

(Dorian parallels)

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Just as tonal correspondence but functional difference exists in the diatonality between and , and between and , so also is there tonal similarity in the functionally P P divergent values: c = TM || t = DM || CM = Tp || TM = Dp ||. The same in the minor. Fundamentally of a different kind are the metharmonic differences between the dominant and auxiliary mediant types:

: Cp || and

: dP. Their comma differences

manifest themselves in the conflict between their original positions:

over the D, Cp under

the C,— corresponding polar form in the minor. _____________ All principal “variants” are valid as bissonance forms of the counter-mediants:

Variants of The mediants however yield bissonances of the principal “same third” (i.e., X = mediant parallels).

Same third in Same-3rd triads appear more remote when a parallel sound comes to stand beside a counter-mediant of a similar principal triad: e.g.,

|| or

etc.

Tritonants appear through mediant widening of the double dominant, e.g., D —(F:)—G major chords (CM : D) or F:—(G)—B major chords (C : DM). Correspondingly in the minor (very common in the Neapolitan cadence). They also come about when double dominants are put in against their mediants (i.e., striving towards one another): CM : DM e.g., (F)—A:—E —(G) major chords. Correspondingly in the minor. Finally, the two auxiliary mediants also form a tritonic relationship similar to the principal triads: A—(C)—E major chords (TP : tP). Correspondingly in the minor.

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Collective Changed Sounds Primary type: this arises when a harmony goes into resolution with the variant of its countermediant or its dominant-mediant parallel, C major : A minor or G minor chord. The same relationship arises in the sequence, counter-mediant : leading note-changed chord of a similar principal triad, A major : E minor chord (TM : Tl) or C minor : F major chord (

:

) etc. The form is easily remembered: the outer tones of a triad move in semitone steps outwards; the third strikes out in semitone steps to the side, so that from each major chord, from now on, a minor chord arises, and from each minor chord a major chord arises, respectively. or F major

D minor

C minor (D minor)

A minor

C minor (D minor)

F major

D minor

G major

D minor

F major

G major

D minor chords

Secondary type = tritonic variant: (fundamental tritone distance, major chord exchanges with the minor chord): C major chord (results in) F minor chord, or C minor chord (results in) F major chord. Or chromatic roots of which the lower one forms an upper sound and the higher one forms a lower sound.

Both these forms are tonally consonant conversions of the actual pseudo-consonant dissonances, which are formed by four-fold leading-note exits from natural triads and tetrads:

C major A or G minor chord with

A minor D or C major chord with

C major F minor chord with

A minor E major chord with

chords

Chromatic Chords Chromatic raising or lowering of the entire chord: e.g., A —A major = TM : CM or TM : TP or cP : Tp. Or rather, E : E major chord, DM : TM or DM : Dp or tP : TM etc.

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13th Chapter A) The Style of the Mediant The mediant (triad) is the romantic type in harmony, as opposed to the dominant (triad), which corresponds to the classical style. Fifth-like remote relations (which are inserted as such, but were not risked beforehand) are included in the narrowest cadential range as sounds of similarity through the mediants.

Chromaticism breaks into diatonicism like a flash flood. individual elements of the chord:

Variants

At first it loosens the

Same thirds

further on, new functional sounds are put up in chromatic shift: e.g.,

A major A major

C minor C minor

E major E major

in C major

F minor F minor

in A minor

which, to some extent, belong to the variant keys or the variants of the parallel keys. They open — by an introduction of the ultra-principal triads — the entire tonal range of the 12-step system, indeed, they even introduce artificial welds, even in double form: G major E minor

C major

F major

A minor

D minor

(see the table) they establish a scale, which already surpasses the boundaries of chromaticism and lets the first beginnings of enharmonics germinate. Enharmonic enh.

enh.

enh.

enh.

17-step scale! The 12-tone compass is exceeded by far, and a further development of the extended line seems in itself impossible.

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But more important than all these mighty territorial expansions are the new tonal formations, which result through welding of principal and mediant triads. Their number is legion! All chromatic dissonances are the results of mediant introduction:

etc. See later “Chromatic Dissonances” The unadorned diatonic cadence appears with the intrusion of the mediant triad strongly suppressed; in its place come veiled misapprehensions or chromaticised or varied introductions respectively. The fifth-steps appear mediated through inserted variant middle parts. C G =

C

E

G

or

C

E

G

Instead of modulatory stepping comes the “shift” springing in and out (tonality buckling/cracking/snapping). Colour predominates the portrayal. At first only a mild glaze and occasional colouristic retouching, the colour moment gradually becomes the primary means of expression (Impressionistic Style) until finally, at the beginning of this century, veritable orgiastic blossoms bloom (partly by Reger!) and through colour-inbreeding this brings about an absolute sterility and supersaturation.

The beginnings of mediant tonality are very hard to trace. The amazingly bold radical in harmony, Don Carlo Gesualdo (1580—1614), head of the Palestrina opposition, already writes in his “Madrigals” an astoundingly developed mediant style with all differentiated ramifications and chromatic-enharmonic close analyses. As he however did not create a school, and has only in the last decades become, as it were, newly discovered, he can not in the historic sense really count as an epoch-making creator of style. In J.S. Bach’s romantic works, not in the tonally resolute fugues, not in the baroquestyle suites, but in the incredibly bold fantasias and in many quasi freely improvised recitatives, mediant tonality appears quite distinctly, yet still however without putting on more than occasionally striking highlights. With the unromantic Mannheimers, it would be hard to find any mediant sounds.

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Haydn’s ingenious harmonic boldness in abruptly juxtaposing tonally remote movements and episodes (e.g., the great E major sonata, 2nd theme and likewise the 2nd movement in E major, and many others) are mediant tricks! Mozart’s sometimes quite intricate chromaticism, often misunderstood and thoroughly misinterpreted in its time, is a lavishly colourful bloom of mediant tonality. The “middle” Beethoven shows a remarkable preference for abrupt “shifts” and enharmonic re-interpretations (in particular of the diminished 7th-chord), which likewise result from 3rd-relationships. But by and large, Beethoven’s style — particularly in the last epoch — is by far more diatonic than chromatic; colour recedes behind monumental architecture. Schubert however develops mediant tonality quite powerfully; it becomes the specific feature of his harmony. The subsidiary themes modulate transiently out of preference by a major or minor third; the minor brightens up to the major, the major darkens down to the minor; the first chords with the same third go into resolution; counter-mediant variants [partly in enharmonic re-interpretation E major — B minor (C minor)] place the most powerful colour contrasts next to one another. The sounds are still indeed predominantly consonant and the shimmering colour play is the result of tonal sequences of distant relatives of comparatively short duration. In this sense, Schumann has unquestionably been stimulated by Schubert. The later master has hardly gone further than his great predecessor when it comes to mediant boldness. But with Schumann the ‘sound in itself’ is considerably more subtle and more attractive (motion and double sounds respectively). To a large extent, surprising effects which often touch upon the experimental, were achieved by Berlioz through complete mediant chains [e.g., D —F—C—E—A—D (C ) major], which nearly annul the unity of the tonality. A style more on edge is that of Chopin’s mediants; here the tritonic sound sequences (B—F—B —E—A—E —) and the semitone string-pulling and multiple veiled chromatic inflexions become quite striking and almost create an affectation. As well as that, the sound itself already becomes wonderfully colourful by means of dissonant mediants. Also strongly chromatic from mediant tones is the harmony of Spohr, yet it lags noticeably behind the fascinating power of Schumann, Chopin and Berlioz. Liszt makes distinct use of mediant tones to a large extent; nearly every harmonic cliché appears in ensuing repetition at the third, mostly alternating between mediant and auxiliary mediant. Diminished 7th-chords and augmented triads chase one another in long stereotyped chains. A much affected manner. But in sacred moments he manages true mediant miracles which were to be powerful stimulus for Wagner, Bruckner, Grieg, Debussy and Scrjabin.

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The point of culmination is unquestionably reached by Wagner. Already in the early works, in which only traces of Spontini, Meyerbeer, Spohr, Weber and Marschner can be recognised, the mediants appear very pointedly. But already in Tannhäuser and even more in Lohengrin are there complete chains of mediants, which, interrupted by rebounding dominants, have a very personal character, noticeably different from the style of Schumann, Chopin and Liszt. However, in the “Mastersingers”, the more narrowly understood “key” again predominates with its pithy principal triad relationship. The “Ring” is pronouncedly mediantic, both in the individual dissonance (altered triads to 5-note chords) as well as tonal resolutions. But the highest and most consistent development of mediant tonality is in “Tristan”! There is hardly a bar in this gigantic work that does not display mediant tonal interlacing (alterations) or linked mediant relations. Characteristic of the harmony in Tristan is the consistently implemented style of chromatic suspension, which is again predominantly mediant in nature (mediant thirds become leading-note suspensions). A true mediant-cult is pursued by Bruckner. Whilst Liszt prefers to shift smaller or larger contained episodes a third higher or lower, or — similar to Berlioz — places mediant imperfect cadences immediately next to one another — in Bruckner the phrase itself often occurs in multiple mediant transient modulations: The minor 3rd relationship predominates; the later Bruckner (8th and 9th Symphony) strongly prefers the counter-mediant variants known from Parsifal (Lamps of the Holy Grail) (e.g., C major—A minor) and mediant leading note-changed sounds (C major—G minor) respectively. With Reger we find everything that is possible in tonal resolution: sequences as per Liszt and Brahms, Wolf and Bruckner, Wagner and Bach, only just in excessive accumulation and unprecedented — a short-lived urging which hardly makes possible a full impact of specific appeal. Between remote mediant relations there stand stylistically — as a rule — in Reger the diminished 7th-chord and the Neapolitan 6th-chord (thus requisites for classical dissonance). But these do not at all cause the mediant transient modulations and excesses, as is commonly assumed, and as Reger in his “Contributions to the Teaching of Modulation” wants us to believe in the case of his favourite sound — the Neapolitan 6th-chord — but rather those dissonances enter as ambiguous parentheses between tonal pairs, whose relationship is understandable anyway (more details in several chapters of this book)! The dissonances in question indeed call in a whole army of possibilities for resolution of sounds; there is nothing to be said for just this and not that sound occurring, but everything to be said for the — mostly mediant — remote relations giving precedence to a leading notestriving introductory dissonance. That this at the same time functions as terminal sound of the preceding chord is understandable, but of secondary importance. The strung-out curves and the goal-striving linearity of Wagner’s mediant tonality and the mighty steplike, graded mediants of Bruckner stand sharply against the unruly, rampantly vagrant, tireless but still lame mediants of Reger. Important in Reger is the introduction of previously unused minor mediants.

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B) Acoustic and Functional Individuality of Mediants. Each note can become a root tone (harmonic prime). Thus, for example, starting from chord, likewise A as well as A .

, E as well as E can be the basis for an E major

completely inadequate step notation acoustic evaluation of the prime notes:

Pythagorean

octave multiples:

in my evaluation of function:

syntonic (Didymean)

In the first case, the E major chord is a more distant 5th-relation, which possesses nothing at all in common tonally with the C major chord, but in the second case the E major chord is a direct 3rd-relative! The original positions illustrate this difference very clearly:

or C major key progressing to E major

in C major the E major chord

the C major chord in E major

each one syntonic comma difference

C) Obvious Tonality Changes through Comma Differences*) The passage:

represents a progression of principal triads on a Pythagorean basis. E major is consequently the perfect-5th “key” of E G B *)Acoustic Regulation of Sound and Determination of Function, by S. Karg-Elert (C. Rothe Publishing House) {‘Distonation’ probably means ‘disordered tuning’ but ‘Detonation’ has the same meaning in German as it does in English, and that is clearly inappropriate here, viz., ‘an explosion with much noise’. The term comes from Riemann’s ‘detonieren’, itself perhaps from the French ‘détonner’ = to be out of tune. Riemann used it to describe pitch differences between natural and tempered sounds. Although itself manufactured and corrupt, I have changed ‘Detonation’ to ‘Destonation’ in the title, merely to distinguish it from any reference to an explosion. The original spelling appears once in the text (p. 208) and is left for readers to decide which version, if either, is more tolerable.}

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If this E major chord is followed by one in C major which is not tuned to the commencement but determined through a tie E–E, then a C major chord results which is raised by a syntonic comma.

The second time the passage ends with E G B, the entry of the new repetition begins with the rebounding counter-mediant of the closing E major chord, thus: C E G. Each new repetition moves up by a syntonic comma, so that the beginning of the 5th repetition yields a 5-fold comma elevation. Now as a syntonic comma (80 : 81) is approximately equal to a tenth of a tone, the resulting tonality change is amply accounted for as a semitone (!). The reverse:

lowered by one syntonic comma lowered by two syntonic commas

At the close of the 5th time (5-fold comma lowering), a changed tonality of a semitone is produced!

% But, mind you: never more will C or D major, nor C or B major respectively be attained, because where would such a modulation have taken place? In both cases, logically the finish would be with the C major chord, if only correspondingly “raised” or “lowered”. This proof can always be furnished; but the conditions for success are that a very slow tempo be taken and that at the repetitions the singers avoid any tonic correction (renewed adjustment towards the initial chord); thus progressions from one chord to the next are only in perfect fifths and thirds. The relative pulling up and down by musically exact singers in mediant-laden unaccompanied passages is often proof enough of their absolute, i.e., “perfect” intonation.

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Compare on the pure harmonium {tuned in mean temperament?}:

i.e., the mediant of the perfect-5th A major chord

lowered

normal

raised i.e., counter-mediant of the perfect-5th E major chord

In the tempered 12-semitone system, even though the mediant appears with the 4th dominant chord identified, harmony precepts — on the basis of logical evaluation of function — must still hold firmly to the irreplaceability of the mediant! The extraordinary upheavals in the field of harmony, which come as revolutionary by the forced entry of the Romantic era, by no means go back to the extension of the 5threlationship, but they are consequences, which arise from the original mediant harmony.

D) Metharmonic*) and Enharmonic Concepts In the ‘finite’ i.e., ringlike closed “equal temperament” (system of 12 semitones), all ‘infinite’ tone differences smaller than the semitone are levelled. In this way, the following appear to be equivalent: the third with the A the seventh ” ” th ” ” B the 6 upper fifth nd the 2 third ” ”

4th fifth 2nd lower fifth 6th lower fifth lower third, etc.

A)

E = E

| B

= B;

same names:

‘metharmonic’ equivalents.

B)

F

| G

= A;

different names: ‘enharmonic’ equivalents.

= G

Both re-evaluations are already known here from diatonic concepts: metharmonic

C major: A minor: The second D minor chord is there instead of the due D major chord, which however would lead to a C minor ending. (D minor : D major = variant, C major : C minor = same third! Variants and same-3rd triads are always opposite each other.) *) technical term of Oettingen. 209 - English

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normal run of fifths

(further procession)

metharmonic: (metharmonic rebound)

enharmonic

(passage in fifths)

The F major sound x comes instead of the due G major chord which, however, would lead to D major. (Pythagorean comma, ‘D ’ = 1/9 tone deeper than ‘C’.) In mediant harmony, metharmonic and enharmonic changes are particularly common: metharmonic

metharmonic

the same in original positions correct

wrong (comma faux pas) correct (tonal)

wrong

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% Such chains are always comma-free through metharmonic re-evaluations:

or

comma-pure ultimate goal. correct

etc.

in the sense of C major = enharmonic faux pas

comma-pure ending. The counter-mediant comes instead of the 2nd mediant.

E) The Harmonological Perception of Tonality ‘Absolute’ pitch is by no means a sign of musicality. The very first master composers were to a certain degree without such, but many indifferent musical craftsmen do possess it. The decisive factor is ‘relative’ pitch, the facility of apperception {i.e., conscious perception}, the ability to draw a conclusion from the material given, functional hearing; briefly: harmonology {the logic of harmony}, as I would like to call that intellectual imagination and critical employment of the inner ear. Absolute (= perfect) pitch is purely physically (acoustically) set. Relative pitch, on the other hand, physiologically and psychologically, is capable of fulfilling subjective divergency where objective analogies exist; it does not — as does perfect pitch — evaluate the single sound as a thing in itself but as a component of a process, an harmonic event in which ‘to make a condition and to make conditional’ are causally inseparable.

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But musically functional hearing reaches beyond the tonal pair; it can refer, according to an innate ability or after methodical training*), tonal groups and chains to a higher unity, and only thereafter (thus only after listening to the whole group, the individual members of which still remain unevaluated), evaluate those functionally ambiguous sounds as unequivocal. The ear perceiving absolute pitch hears the sequence:

as F—E major, “leading-note chords” (16:15)

but it will evaluate it in the transposed form:

as E major—E major (!) “chromatic chords” (25:24)

because — only because for the time being no cause exists to assume such a “rarely used” sound {viz., D major}. But also relative pitch can not differentiate between the E–D , E–E and F –E major chords; even a following sound might not suffice for a definite decision concerning E major– A minor or D major–G minor | E –C major or D –B major. Only from the tonal validity of a whole phrase could one draw an a posteriori conclusion {i.e., reasoning from effect to cause} by means of harmonological ability.

% In dubious cases, the inner ear will register simultaneously enharmonic double values and afterwards — when the tonal problem has yielded a tonally recognisable determination — cancel one of the two interpretations, which would not lead to the intended goal:

to be decided afterwards _ _ _ _ _ _ _ _ _ _ _ _ _ _

According to the enharmonic concept, the 2nd mediant replaces the 1st counter-mediant, and the mediant replaces the 2nd counter-mediant (small diesis, 125:128 = 3/16 tonedifference). *) NB: Here a blessed area opens for Conservatoria and Music Schools: “Training in Relative Pitch”. To complete the average practice of the course ‘Aural Instruction and Dictation’ seems to me really insufficient at present.

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F) Cadence Broadening Cadence broadenings through mediant modification of each principal triad:

extreme D type (important)

(original form)

C major

(A maj)

(B maj)

(A maj)

(B maj)

(F maj)

(C maj)

(G maj)

(D maj)

(E maj)

(D maj)

(E maj)

(C Maj)

(original form)

extreme C type (important) extreme

(original form)

A minor

(F min)

(G min)

(F min)

(G min)

(D min)

(A min)

(E min)

(B min)

(C min)

(B min)

(C min)

extreme

type (important)

(A min)

(original form)

type (important)

(G major)

(E minor)

(F major)

(E major)

(C minor)

(E major)

(C minor)

(A major)

(F minor)

(A major)

(F minor) (D minor)

C major

A minor

Cadence broadening through mediant modification of a principal triad:

C major:

A minor:

(A maj)

(E maj)

(B maj)

(A maj)

(E maj)

(B maj)

(D maj)

(A maj)

(E maj)

(D maj)

(A maj)

(E maj)

(F min)

(C min)

(G min)

(F min)

(C min)

(G min)

(B min)

(F min)

(C min)

(B min)

(F min)

(C min)

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re d) % The 5th-related mediants and auxiliary mediants of the same type (placed in a line, in the above example) would be better notated functionally as (C) | x (D) | x. One recognises straight away the enormous broadening of tonality, which takes place through mediant changes on both sides. Through the introduction of extreme parallel and leadingnote changed chords of the outer mediants, the tonal range grows by a fifth:

Admittedly, the unambiguity of functional perception already finds its limit in the tritonic chordal succession: ex abrupto {suddenly} in the 12 semitone system, C : F major is not distinguishable from C : G major. Beyond this limit, all further distant relationships in enharmonic re-interpretation seem to come closer to the related sound: Distant relationships Dominants

B

F

C

G

F

Dominants

C

G

D major chords

Mediants

D

A

E

A E B

Mediants

through

enharmonic re-interpretation

chromatic

can be understood

increasing distance through

% increasing proximity

leading-note triad

tonally as

in E major But if the replacements keep their connection to their original principal triads, then their functional meaning remains easily recognisable:

polar:

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The more the principal triads and fifth cadences recede, and the stronger the substitutionchange is accentuated between the - and + side, the more the tonality will work itself loose, and the tonal change becomes so much more colourful and exciting. also interpretable as C major

C major tonality: F

C maj

C min

E major tonality: like a sequence in form also interpretable as A minor

polar:

A minor tonality:

What a wonderful, expressive, unspent minor key chain! How many treasures lie undiscovered in the harmony of minor keys . . . The harmony of the past century (Romanticism) was predominantly set in the major, quite in opposition to the harmony of the 17th and early 18th centuries. The dissonant style (today fashionably entitled: Atonality) which broke out like a sudden flood has further consequences of thwarting the still unexploited consonant harmony of the minor. The greatest venture in remote, mediant minor key relationships was dared by Reger.

A further example of mediant accumulation: D

whole tone descent: E of the accentuated sounds:

auxiliary mediant

B

A

chromatic

countertritonant

tritonant

primary collective-

leading-note chord

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G

F

E

-changed sound

tempered ultra-contrant

countertritonant

countertritonant

(or C minor: G major)

tempered contra-dominant

The C major tonic — except at the beginning — is not represented any longer, but theoretically all sounds can be referred to it. But any other arbitrary sound can likewise form this chain functionally (whereby occasional enharmonic re-evaluation then becomes necessary), as of course mediant harmony includes all sounds in the 12-semitone compass, in which each sound can be the centre. An extremely important element in mediants is thus recognised: They broaden the key to a boundless tonality, but at the same time they also carry within themselves the seeds for destruction of tonality, as through intersection and enharmonic levelling of distant relatives, the tonic loses its unambiguous centralisation. In the example above: The effect of C major tonality becomes particularly problematical through the introduction of mediants on the accent. Particularly uncertain is the ending of a phrase. It can be an F major triad just as well as one in G major:

complements

The same, polar: whole tone ascent F

C

D

G

B

minor

or A major: D minor

E

F minor

or

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% It is important that within the tonality A), all contra-dominants, substitutes for the 6th (the 5th replacing or acting together with this) and B), all dominant-substitutes, can appear with the natural 7th, as long as these chords ‘potentially’ make 5th-cadences:

Original forms (diatonic)

i.e. with mediant-changed tonic or

or

or

with mediant-changed introduction: or

or

or

seldom! or

(similar to

= very often

or

)

a) and b) merely refer here to the tonic or its replacement. Instead of this, another principal triad or its substitute can of course also function. In these cases the introductory sounds are put in parentheses: etc. A considerably increased tendency is experienced by the contradominant and its replacement through wide alteration of the characteristic sixth:

Graphic grammalogue

5th

6th 5th

6th

widely altered 6th 5th

widely altered 6th

or all very common

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or

or

in C major i.e., in A major

in E major

transcribed into C major T :

or

Each major chord can be freely introduced through these forms. Notable are the transposition forms:

referred to C major: A

C

F

A

G

B

C

E

mediant-cadences

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A

G

D

C

G

F

F

E

leading-note cadences The polar counterparts — unfortunately — are often stripped of their essence in musical notation through incorrect enharmonic re-interpretation of the lower-6th as lower-7th:

(dominant type) (correct)

(false)

(dominant type) and

respectively (correct)

(false)

The last chain-example in the minor: ( ED ) B

or mediant cadences leading-note parallel {= consecutive} 5ths commonly

leading-note cadences

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B) The dominant types, however — analogous with the diatonic forms — produce their natural 7ths. weakest type

strongest type

weakest type

strongest type

If, apart from the 7th, the dP also forms the 9th, then this 5-note chord equals in metharmonic re-evaluation the c ( In the sense of cadence:

or

with 7th and 9th should be functionally unbelievable!). the latter interpretation is the simpler compared with

the first. syntonic comma (polar corresponding form in the minor)

If the dominant mediant adds to its natural 7th the narrowly altered (so-called “Italian”) 9th and suppresses the root, then the result is the diminished 7th-chord of the 2nd order, which is typical of romantic (thus also mediant) harmony in the works of Liszt, Wagner and Bruckner, and plays an important rôle in the Italian verists, strongly repressing the diatonic diminished 7th-chord of the 1st order

.

{‘Verismo’: Italian “realism”, a movement starting in the 1870s which spread to neighbouring countries and died out in the early 1900s. Originating in literature, its philosophy was that of the rigid representation of truth and reality in art, including the vulgar and the ugly. ‘Cavalleria Rusticana’ was the first veristic text to be made into opera (by Mascagni) and first staged in 1890. Leoncavallo’s ‘I Pagliacci’ (1892) was prompted by the success of Cavalleria and the two became the most celebrated veristic operas.}

Italian 9thchord

diminished 7th-chord, 2nd type

Characteristic of the 2nd type is the tie (metharmonic!) between the “Italian 9th” (i.e., the diminished 7th of the real tetrad) and the root of the chord of resolution.

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The following tonal chain shows a mediant relationship in both the points of support and the lead-ins: D

F

(C)

G (E)

D

(G)

B

C

(F )

F

(B)

A

A

G

B major (A)

G

B

(F )

C

(A )

G

B

G) Mediant Tonality-jumps, Shifts, Parentheses and Sequences The juxtaposition of tonally different movements or parts of movements in the classical epoch takes place in 5th-related fashion and as variants [e.g., 1st and 2nd movement, chief and alternative section (Trio), or 1st and 2nd theme, or rounded-off episodes and phrases]. For example, C major—G major || C major—Trio: F major || C major—C minor | C major—A minor || C major—E minor ||. (That the keys of the parallel and leading-note changed chords are formed through potential 5th-relationships should be sufficiently known.) The Romantic loves to avoid robustly precise 5th-steps and prefers, in the juxtaposition of tonal groups, the more colourful but gentler mediant harmonies. Where the suites, partitas, sonatas and concertos of Corelli, Bach, Handel and Telemann still show mostly a tonal correspondence in their movements and at most a majorminor change between variant or parallel keys, then already with Haydn and Beethoven a marked key change is beginnning between the individual movements, which occasionally makes excursions into mediant (chromatic) harmony. An exceedingly bold thrust is taken by Haydn in the great E major sonata for piano; 1st and 3rd movement: E major; 2nd movement: E major!! First theme of the 1st movement: E major (transition: C minor—G major), 2nd theme: abruptly in E major:

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Beethoven: C major Sonata, Op. 2, No. 3, 1st movement: C major; 2nd movement: E major. Waldstein Sonata, {Op. 53}, 1st theme: C major; 2nd theme: E major. Pathétique {Sonata, Op. 13}, 1st theme: C minor; 2nd theme, E minor (auxiliary mediant). The progression is C minor—F major—B major—E minor; thus Hammerklavier Sonata {Op. 106}, 1st movement: B major; 3rd movement: F minor = (!). 1st theme {of 1st movement}, B major; 2nd theme, G major, later B major, i.e., and nd (!!). E major Piano Concerto {Op. 73}, 2 movement: B major (C major) = . This abrupt mediant juxtaposition becomes particularly characteristic of Schubert: great B major Sonata {D 960}, 2nd movement: C minor = or understood as D minor: (!). G major Impromptu {does K-E mean the “simpler” version of the G major Impromptu, from D 899?}, subsidiary theme: E {E in original G version} major = ; E major Impromptu {from D 899}, Trio: B minor as C minor = (!), etc. Schumann: F major Novellette {Op. 21}: the constituents ABC are in the keys of F, D , A major = T, , . Chopin: A major Polonaise {Op. 53}, Trio: E major (F major) = . The base-keys of the E major Prelude {Op. 28, No. 9} = E—C—A major, etc. In the course of development of mediant forms, the key-fixed groups show a gradually smaller expanse, until they comprise only a few bars and finally only one bar. F major

A major shift F – F –

schematically:

C major

E major phrases

shift A– A– F – A– F – A–

C– C– C– C– E–

shift E– E– E–

mediant tonal chain:

In the above instances, the groups move on in mediant fashion in the same direction and thus raise so much more surely the original key (tonality). Far more often, mediant parentheses are in the following form where, furthermore, the original key reaches its value on the other side of the connected mediant episode: E major or E major e.g.

inserted episode

C major

A major or A major shift out

C major return shift in

The principle of such a mediant phrase parenthesis also remains when the rebound follows, not in the original key but — as is often — in the 5th-related key: e.g. C major

E major or A major shift out (mediant)

G major shift in (mediant)

later return to C major diatonic

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The shift in and out can take place abruptly or it can occur more calmly through an interceding {or mediating} chord, in which case the character, more or less, of a chromatic modulation is revealed.

C major part . . . . . . . E major part . . . A min, D maj chords

(abrupt)

G major part . . . . . .

(interceding)

C major part F min, E maj chords A maj part . . E maj chord

(interceding)

G major part

(abrupt)

As shortest mediators, diminished 7th-chords and altered triads and 4-note chords function here in preference:

Bridge

The original tonal chain

Bridge

C– E– G– B –

Bridge

is understandable enough, it needs for

modulatory reasons no switches and “modulation dissonances”. The dissonant introductory chords will of course alleviate tonality kinks, but they are inherently far more colouristic and expressive elements, as a functionally “necessary” means of modulation.

If a melodic-motivic copying of the preceding cliché enters simultaneously with the mediant shift, then a free-tonal (chromatic) sequence results. The groups are, as a rule, interlinked through chromatic linear construction {or voice-leading}:

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Schematic presentation of shifts (dominant sequences act as cadences, not as shifts) with bridges:

type of shift

tritone

mediant

auxiliary mediant

whole tone

whole tone

auxiliary mediant

mediant

tritone

In closed sequence and free interlinkage (typical of the so-called “New German” style) oboe

violin II

violin I

bass clarinet bassoon

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clarinet trumpet violoncello

cor anglais 3rd and 4th horns

Mediant path (motif entries) in the examples above:

B major chord " B G " E " " E C " A " " A

or

also referable to G as centre

A similar example (shift or sequences) with another mediant progression: oboe

violin II

clarinet

violin I

flute

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oboe

violoncello bassoon horn

Curved path of the example above (15 tone compass;

= dominants,

= mediants)

D maj B maj B maj G maj E maj E maj C maj

C maj

C maj

A maj A maj F maj D maj D maj B maj G maj

The forms of shift are easily obtained from the table above.

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Popular theatrical semitone shifts upwards (already amply used), at the same time, mostly in increasing rhythmic shortening (escalation gesture): Liszt — everywhere; Rubinstein — D minor Concerto; Tchaikovsky — Pathétique {Symphony}; Grieg — Ballade, Op. 24; Italian verists — everywhere. (triumphant)

etc. . . . . mostly leading to a gaudy apotheosis (“grandiose”) - -

The shift or sequence character is completely lost when rhythm and melody appear independently through-composed {meaning here perhaps ‘composed in continuity’}:

In the style of a waltz elegantly

blaring out

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marcato

The potential sequence is completely blurred. Free tonality takes over.

(enharmonic)

The same, polar: A E G&D

F

C

E&B

min. maj. minor min. maj. minor

G D F&C

E

B

D &A

min. maj. minor min. maj. minor chords

The potential sequence is blurred. Free tonality takes over. A large number of examples of mediant type from the literature follows in approximate historical sequence. The way to obvious mediant harmony leads first of all via the tempered contradominant, thus e.g. is the strikingly acting same third sequence later on (Schubert, Bruckner, Wolf): G major—G minor = T : or T : first of all a movement form out of

G minor chord linear: suspension of the E major chord isolated : leading-note change of the E major chord Everything else is listed in the remarks.

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io ti se

-

guo!

Mà,

Claudio Monteverdi (1567—1643) 2nd Song of Orfeo chi me’l vie - ta chimè,sog’n o va - neggio

(mediant)

Neapolitan tritonant

In the example above: through the omission of bridges, the following strong mediant effects result:

1)

2)

1)

2)

1) = counter-mediant 2) = counter-mediant variant (so-called collective changed chord, 1st order)

Claudio Monteverdi, 3rd Madrigal (5-voice) M’è più dolce il pe - nar -

same 3rd or Girolamo Frescobaldi Toccata cromatica (organ)

In the example above: instead of the B major chord, the C major chord can be used equally well; both can replace the expected G minor triad ( ): B maj

counter-mediant variant

developed from: G min G maj B min

C maj

B maj chords

developed from: G min E maj E min C maj chords mediant leading-note change

same 3rd

primary collectivechanged chord.

same 7th but:

C min:

C maj

C min: B maj

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A major

F major

Lentement

(See also the next example)

In the example above: the functions are actually very simple tonally; nevertheless, the tonal sequences (B major) G minor — E major, A major — F major must be described as mediant effects: E maj — A maj B maj

G min

F maj

C maj

in the

(variants) B major: E major = tritonant (Neapolitan formula) G minor: E major = 7th countersounds (metharmonic)

1 E

3 G

5 B

7 D

recte {forward}: tritonic parallel

E

G

B

D

Heinrich Schütz, 1585—1672 Easter Dialogue G min E maj

A maj F maj (as in the preceding example)

(Women’s voices) (Men’s voices)

What marvellous expression! What pictorial effect, which points directly to Berlioz! (Requiem) G minor: E major = 7th countersounds (metharmonic).

A F

D

B

G

E maj chords Tonal relationship through a tonal group

= fundamental tones: lower tetrad

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Handel: Messiah (27th recitative)

RecitativeThy re – buke hath broken His heart;

He is full

of

heav-i-ness,

He is full

of

auxiliary mediant

or

chromatic E major chord heav-i-ness;

Thy re – buke

hath broken His heart.

He looked for some to

have

harmony E major chord pity

on Him,

looked for some

but there was no

to

have

man,

pi - ty

nei-ther found He any

to

com-fort Him;

He

on Him,

etc. (C maj C maj)

Certainly one of the most shattering passages in the entire musical literature of the world! The harmonic means of expression are exceptional for the (ordinarily) diatonic progressions of Handel! The complete 12-tone horizon is encircled: A major chord up to C major chord. in brief

tempered contradominants

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Remarkable are the tonal sequences bridged by diminished 7th-chords: G minor — E minor chords auxiliary mediants D minor — B minor chords : quasi: The diminished 7th-chords on the words ‘rebuke’ and ‘comfort’ can be interpreted as potential F and E major triads; they refer in dominant fashion to the resolutions of the 6/4 chords: B and A triads. {Unfortunately, in translating the English text into German, whilst ‘rebuke’ falls on the same chord as the matching ‘Schmach’, ‘comfort’ (for ‘Trost’) lands on the ‘wrong’ chord! The harmony Karg intends is sung to ‘any’ which perhaps defeats his purpose.} But they can even more easily be rated as

potential E and D minor triads; in these cases, their reverse connection is more free of jumps and is functionally simpler: diminished 7th-chords as upper sounds lower sounds (jumping!) (proceeding/striding!)

Curved path of the preceding piece:

upper sounds lower sounds (jumping!) (proceeding/striding!)

(cadence, B minor)

Careful study of the preceding table is recommended, as it presents important functional metharmonic changes

which will be frequently referred to in later chapters. The

degrees of relationship result from the original positions and ties themselves.

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J.S. Bach: B minor Mass Crucifixus

B maj

D min

Original function:

apparent: C maj A min

Movement forms:

At NB {ex. 557} the

E major chords appear together, A major represented by means of D minor

The sound can also be understood as

and

G B C F , the latter moving into

the root position of the A minor chord:

i.e.,

primary collective changed chord (See above)

variant

B maj: G maj chords, potential B maj: B min. The minor chord (counter-mediant)

enters with the leading-note change: The counter-mediant is thus recognised as the “leading note-changed chord of the variant”.

Counter-tritonant

parallel B maj F maj twin. D min A min

countertritonant parallel

B major: D minor can also be interpreted as metharmonic 7th-change

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(see above)

(as above)

J.S. Bach: High Mass, No. 19 Confiteor

[To the G minor chord]

at which place the mediant enters:

this sound acting meanwhile as

to

longer cadence run:

D major: E minor = metharmonic same-7th: D F

A

C

E

G

B

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Variant parenthesis (structural formation), very much more common tonal quality Beethoven, Op. 2, No. 3

potentially

metharmonic

ditto 4 times

Recapitulation In the example above: a) C major to G major

b) C minor with deviations to G major B minor — A major

c) C major to the end

but

Counter-mediant parenthesis, greatly expanded

Beethoven: Op. 7, 2nd movement

C major

C major

auxiliary mediant false cadence, tonic representative!

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counter-mediant entry 4 bars A maj, 4 bars F min via D maj, E min, F min to G maj, B maj, C min, G maj C major recapitulation

in itself a diatonic group of counter-mediant sidings

Abrupt substitutional jumps in counter leading-note parenthesis on the (D) of the enharmonic re-interpreted notation.

in

Beethoven: Op. 7, Finale

dominant of the

F major chord _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _

tonality proceeds

rebound

etc.

The shift (63/4 bars) is sharply marked dynamically: E :D

D in E major

F major parenthesis (In the orchestra such shifts are more especially marked by means of contrasting colours.)

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Mediant and auxiliary mediant stations within the range of tonality. (Typical developmental character.) Tempered contradominants (variants) as switches. Beethoven: Op. 10, No. 2, 1st movement

Allegro

D min

Group function referred to the general key

to B maj F maj B min C maj—F min—

D maj

etc. is correct (but usually D min notated as ‘A ’)

bridge D maj 1st theme (Recapitulation) in the parallel-variant-key D maj

proceeding in F maj to the end

D maj

The formal parenthesis (development) is closed by a however, only at least 15 bars later (bridge).

, the harmonic component,

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Auxiliary mediant fluctuation (tritonic range) within a parenthesis. F major

Beethoven: Op. 10, No. 2, 3rd movement

etc. A maj

D maj

(apparent)

F maj

The development b) is extremely interesting, as it contains an immensely fine comma deviation: in reality, the D minor chord

is not a Tp at all here, but a

(!), i.e., varied

triple dominant; equally, the D major chord is no but a triple dominant and equally the A major chord is not a direct 3rd-relation but a distant 5th-relation (chord of the Pythagorean third = 4th dominant): original positions

diatonic retransition

This quotation is really a classic example of the functional evaluation of original represent mediant or variant forms, as they are 5th-relatives (or their positions! All bissonant substitutes) of a mediant type (jumping to A major in F major). Fundamentally different is the case of Beethoven’s Funeral March from the Sonata Op. 26: Maestoso

bar 1

ditto ends with the C major chord 8th bar

auxiliary mediant shift up bar 9

ends with the D major chord

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It should be superfluous to stress that after the C major (chord), C minor and not B minor follows (transposed A minor—C major | C minor—E major). Thus the 2nd phrase ends in E major. After the D major ending a bridge follows:

which “leads

back” to the A minor section (recapitulation), i.e., in relation to the preceding D major (section): A minor

This reversal from D major is simply and intelligibly: B min A maj

D maj

B maj E maj

A min

But instead of a D major ending, one in E major must be taken and instead of an A minor final key, one in B minor should then ensue. An ‘enharmonic’ faux pas would then result! This however does not happen, rather the case is analogous to the minor-third shift in Schubert’s “Wegweiser” (see ex. 586): start and finish are absolutely the same tonally; a mediant-shifted group is placed after the start. It breaks off abruptly without further development and just as abruptly enters the (chain-) introduction to the end-group: transposed: [A minor]

1

to

shift

from one

recapitulation

to

the same functions apply, of course, for A minor also (the original position of which is assumed to be 4 octaves lower).

E : B major chords and E

: B major chords respectively are ultra-mediants, they

equal the enharmonically transformed simple counter-mediants E : C major and E

:C

major respectively. In the Beethoven example above, the adoption of an enharmonic faux pas must be rejected, as a logical chaining does not exist between bars 16 and 17. But if it is still to be assumed, then

as

[E major : A minor

is counter-tritonic variant =

is also intelligible enough

].

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Beethoven’s “Pathétique”, where, in the first movement the two main themes in C and , was already mentioned earlier (see E minor stand in auxiliary mediant relationship = p. 222). The Waldstein Sonata, Op. 53, offers a number of interesting mediant and apparentmediant problems: in the first movement, C major is constantly pushed aside and only at the beginning and end, as well as the introductory episodes, is presented in perfect tonic form. Bar 3—4 = G major chord, bar 4—5 = B major chord. But between G and B major there exists here no 3rd-relationship

, but rather the triple 5th-relationship

, the functions are: C

G

B

F maj, F min, G maj, C min, G major

This group — harmonically revised and formally widened — is repeated:

reinterpreted as D (in E major mediant type)

Forty bars follow in E major (2nd theme with after-phrase) = nullifies the mediant again! =

. The return (“bridge”)

= E minor G major C major.

As always, strong modulatory activity begins in the development. Remarkable is the strong counter-mediant emphasis in contrast to the mediant shifts of the exposition. Very notable further on is an enharmonic faux pas, as this becomes more and more frequent later in Schubert, Chopin, Liszt, Wagner, Bruckner and Reger, and finally completely abolishes tonality: original positions: development

Neapolitan Recapitulation

corresponding enharmonic re-evaluation

(F )

variant Neapolitan

(G )

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Beethoven, Op. 106 (Hammerklavier Sonata), 1st movement, 1st theme in B major (strictly diatonic): bar 34

The re-evaluation of the tonic-mediant in a dominant (harmony) yields an auxiliary mediant shift downwards, i.e., G major in this instance. The spatial relationship between the principal and mediant sections is analogous to the Waldstein Sonata: ca. 1 : 2. The return takes place through the unexpected entrance of the variant

in G major in B major

The development begins with E major and lingers greatly in this narrow key-cycle. D major enters suddenly as , two diminished 7th-chords make a cadence in the sense and unexpectedly the ‘cantabile theme’ enters in B major , which breaks off from the F major sound and in a transitional passage (development of the main motif) leads to the recapitulation in B major , which modulates to G major in the 14th bar. The enharmonic breach: D F (instead of G ) major to B major . . . is architecturally intelligible enough: execution introduction development: (diminished 7th-chords)

B major

the mediants fluctuate towards both sides D major F major

F major (double mediant)

mediant of the 3rd dominant

G major

G major

B major

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Mediant-fluctuating parenthesis:

Beethoven: “In What Obscure Tomb”, WoO 133 (song) [middle[start and finish of the main section] section] O laß bei nacktem

Schatten fried - lich ruh’n mein Herz

Asche nicht mit eitlem, eitlem Schmerz

und be - net-ze weinend mei-ne

Recapitulation

The Italian, theatrically exaggerated middle section begins of course with F major. The two alternating chords in the 6th bar of the middle section represent the

and D

major chords

The harmonic course presents itself in the following way (transposed metharmonically, as the original positions are presumed to be 2 octaves lower). The bars correspond to the original example above. cadence

cadence etc. tonally in A major

standardised (A major)

C major A major F major

F major

C major

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Tempered ultra-contradominants as bridges to the counter-mediant (alias tempered contradominant as bridge to the variant parallel). Beethoven: “Adelaide”, A - -

- - -

-

de- la - i - de!

Op. 46 (song)

recurring F major: original B major: middle section

via G and F to

B major

(leading-note triads)

(leading-note triads)

Excursion to the consonant primary collective changed chord (i.e., mediant leading notechanged chord) A minor respectively. D major Franz Schubert: Symphony in B minor

Andante con moto (simplified)

A minor

NB) or

D major

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The polar counterpart is the considerably more common variety:

or standardised to C major: C major

G minor

C minor or

NB) the complicated polar minor form:

Counter-mediant as chromatic replacement of the parallel chord: Slowly

Franz Schubert: “Thou art my rest”, D 776, Op. 59, No. 3 Thou art my rest / Thy glance alone / Has given sight / To these my eyes, —

Verse 1 and 2

Verse 3

Perhaps the following function is rather more logical: potential cadence:

for the C minor chord is more an A major replacement, then the A minor chord of this variant and the C major chord of its parallel sound are thus contradominant variant parallels.

Counter-mediant as leading-note change of the variant: One little word round and round./“Yes” is that one little word, / The other is “No”,

Franz Schubert: “The Lovely Maid of the Mill”, D 795, Op. 25 Text by Wilhelm Müller No. 6: “The Curious One”

7 bars in G major then proceeding further in B major (g)

(g)

is the same

Pay attention to the symbols of the functions:

and

‘Yes’

as opposites!

G major parenthesis cadence

‘No’

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Franz Schubert: “Swan Song”, D 957 (1828) Text by Ludwig Rellstab No. 6: “In the Distance”

Some examples in the minor:

Ha - ting his mo-ther’s house, desert-ing his friends

A minor:

(back to A minor) same 3rd!

Characteristic Schubert octaves between the outer parts! original positions: replacements: dominants and dominantparallel variant - parallels (!)

cadence:

(The name corresponds exactly to the essence!)

murmuring stream, blustering wood, starkest crags,

or

Franz Schubert: “Swan Song” Text by Ludwig Rellstab No. 5: “Abode” murmuring stream

potentially: equal function! C minor (E maj: F min)

B major

enharmonic same 3rd metharmonic same 7th But in these cases, the minor forms lie above the major forms! (metharmonic)

Very slowly

You double of mi - ne,

Franz Schubert: “Swan Song” Text by Heinrich Heine No. 13: “The Double” pal - est old fel -

(Recurring theme:)

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(ditto)

twice - low— Why are you aping my lover’s anguish

so many a night, in days of yore?

(recurring theme) same 3rd

or

D major : D minor = same 3rd. A major : B minor = metharmonic same 7th or enharmonic same 3rd. These last three quotations (incidentally, pieces of enormous expressive power) afford a first glimpse of the utmost novelty of Schubert’s mediant harmony: all “progressions” come to a cadence and variant “turns” are suppressed, i.e., all bridges are broken off, tonal distance has moved into immediate proximity, agglomerations collide with each other like eruptions. Anyone who can read, recognises here very distinct symptoms of concise so-called “expressionistic” diction (called ironically, but not inappropriately, “telegram style”). No.

A minor

C major

C minor

A major

No.

C minor

G major

E minor

B major

No.

D major

B minor

F major

D minor

Franz Schubert: “I Send You Greetings”, D 741, Op. 20, No. 1 (1822) Text by Friedrich Rückert slowly You whom love has bestowed on me, / You, who have been taken from my

lov-ing heart! / With this

gush

of

tears

G min | D

uniformly valid for B major suddenly

F

B maj

(auxiliary mediants) D major

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same 3rd (ditto)

D major

(as previously)

D minor

became mine, with

B , F maj; B min; A maj

(ditto)

I

my

A

(as previously)

soul . . .

G

G maj; C, G min; D, F, B maj

am with you,___ you are with me,__

/

I

hold

(metharmonic)

Centre: you

F major

closely

E major

in

my

arms

G

/

I

(as previously) send . . .

F

referring to B major potentially referring to D major

A single chord serves as bridge between the G minor and D major chords:

(D major:) (= G minor)

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Franz Schubert: “Swan Song” Text by Heinrich Heine No. 8: “The Atlas” Unbearable things I bear, and with it all / My heart will break within.

B minor introduction to the cadential chord: C major

G minor

F maj to B min,

consecutive octaves between the outer parts, typical of Schubert. G minor: F major = again

all quotations (as in nearly ) “enharmonic same 3rd:” from Schubert

B

A

or metharmonic same 7th E (ditto)

happiness

infinite

/

Or

else

(as previously) infinite

small centres:

everything referred to G minor:

or B major

counter-mediant variant or variant-parallel variant

(ditto)

sadness, you proud heart / And now you are so abject!

tonal cadence of the original key E minor

G minor (collective changed chord) later very common.

with suppressed E minor triad: = i.e.,

or

respectively

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Franz Schubert: “Swan Song” Text by Ludwig Rellstab No. 2: “Warrior’s Foreboding” Here

my heart feels

it’s

all

a - lone, here

my heart feels it’s

or auxiliary mediant mediant

F minor

all

a – lone/ Sad

tears - - - - - -

of

woe well

up - - - .

or

F minor (chromatic chords, shift backwards of minor triads)

How the

the

raging

rafters

Franz Schubert: “The Young Nun”, D 828, Op. 43, No. 1 (1825) Text by Jakob Craigher storm howls in the tree-tops! / How

groan,

the house

shudders!

/

How

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(semitone shift ) the

thunder

rumbles

and

(G minor) and

the

(C minor)

(E night

is

the

lightning

flashes,

major) dark

as

the

grave!

(B minor)

In this and the next example (both of the utmost ingenuity!), the semitone shifts of Liszt and Wagner are very strongly anticipated. Enharmonic re-interpretations which are necessary for readability, unfortunately blur greatly the logical harmonic progression. This phrase is as follows, transposed into E minor and brought to original positions!

(dark the night)

(cadence) B minor A minor with lowest parts uniformly tonal leading note triads

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minor 3rd shifts up:

B min

G min

travel, the road that I must travel, / on which no - one ever returned.

C min

Because of the consistent auxiliary mediant shift, the D minor chord had to come after the G and B minor chords, but these would lead to A minor instead of G minor. C and D minor are the two tritonic poles of G minor: D minor

G minor

C minor

The tonal return results from the melodic line: Basic function:

The Schubert examples shown favour quite markedly the minor forms. Fundamentally, Schumann, Liszt and Bruckner have hardly gone beyond Schubert’s mediant harmony functionally; Schubert, understandably, runs through the whole scale of chromatic relationships, from the auxiliary mediants right up to the chromatic chords . . . New close relationships are no longer possible, as within the 12 semitone system all opportunities for resolution are overdrawn. In many cases (e.g., all those quoted here), Schubert’s tonality trembles down to the roots but in the end still holds its own! And it is exactly the maintained validity of tonality which accounts for the alluring excitement sensed in the oscillations beating in and out, because only in coming from a firm pole do they gain recognisable distance and differing amplitude. An example of this: the C and E major chord works considerably more colourlessly as a progression in A minor than in C or in E major, for in the first case both chords have taken up close relations (parallel and cadence) to a common centre (A minor), and whilst in the second case a sound has been sensed as pure mediant-jumping = T : and respectively. respectively, or

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An accumulation of mediants of the same type very quickly creates tonal extravagance, which destroys itself: in E major the G major chord acts as mediant chord of tension. But already in strongly emphasised C major, the progression rated as

major would easily be

(rebounding). The over-excitement of

remains as a rule a

problematic paper value. Only the introduction of the tonal 5th-relatives yields a strong contrasting factor for the mediants, for they act ‘relatively’ stronger as ‘absolutes’. (See the comments on Schumann’s F major Novellette and Wagner’s ending to Tristan.) It is very remarkable that, with Schubert, the mediant distant relationships, which act as mighty disruptions of tonality, are always the last reserves and enter as symbols of immense emotional disruption, as dramatic peripeteiae, as crises, after which catharsis follows immediately as a rule (i.e., the tendency to a diatonic cadence, in the sense of tonality); often indeed conflict results directly from the juxtaposition of tonal “centrifugal” and “centripetal” elements. See example No. 580

centripetal D major

centre:

F major G minor conflict

B major F major

B major centre

B minor A major centrifugal D major

All these strong tensions between the poles are lost when the centre loses its equatorial validity. Even if the absolute tonal progressions remain, they still change their relative values, on which they depend. Ho - ly

ho - ly,

God

of

Hosts———

Hector Berlioz: “Requiem”

(manifold divided strings with solo cymbal) (= E

major)

Mediant swings. Expressive and colouristic effect (mysterious, enraptured) A -

men.

A - men.

A - men.

A - men.

A - men. (as previously)

4 harps added, with broad arpeggios

principal chords (diatonic) mediant chords (chromatic) Both these very bold examples for their time already anticipate Liszt’s “incense” . . . .

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Robert Schumann: 1st Novellette, Op. 21 (1838)

(F major)

This excessively popular piece is somewhat confused tonally (as, incidentally, Schumann’s harmony and metrics more often prove to be on analysis). ‘A’ falls by a major 3rd to ‘C’ (F to D major). The same ritornello ‘A’ falls once again a major 3rd to ‘B2’ (that would be D to B major) and once more repeats the fall from ‘B2’ to ‘A’ (that would be B to G major). More graphically, starting with F major:

(enharmonic faux pas)

The same group progression F to D to A major is also shown in the main ritornello ‘A’. It is actually recognised — ex abrupto — as , but the linear progression of the harmony absolutely does not permit this interpretation! The stations {or stages} are the following:

No mediant relationship exists here between F, D and B major (let alone A major), for the chain is actually of a principal triadic nature. But as the (tonal) journey goes a completely incomprehensible distance, this eightfold 5th-lowering will later be repeated once more. This piece is like a nearly infinite journey without previously determined goals. Very simply understandable from station to station, the end gets lost as seen from the beginning, or the beginning gets lost as seen from the end in the dim distance . . . . . . . . Anyone who can read, recognises here so readily a typical case of romantic nature: all limits of reality have shifted out into infinity . . . . . .

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The technical description of such boundless expanse encounters insurmountable restraints: the equally tempered 12 semitone system eliminates of course all metharmonic and enharmonic conflicts (comma differences), but it also transfers pathways by wide distances, as it transforms straight lines into rebounding curves. The presentation of absurd distances is thus left solely to fantasy (NB: this is also a typical romantic element!). One more work (admittedly of a completely different nature! A comparison of inner values must not take place) displays such a flight into the boundless distance, as auxiliary mediants line themselves up in near endlessness: the ecstatic ending of the inimitable Tristan. (Details in the relevant examples later.)

Another double example from the same little work by Schumann: Counter-mediant parentheses: (Schumann, Op. 21, ex. 589 contined)

counter-tritonic progression

The G major triad (ditto)

is present instead of the diatonic G minor triad.

:

= same 3rd. (as previously)

auxiliary mediants

1st time = tritonant = 2 dominants and 1 mediant in the same direction 2nd time = auxiliary mediant = 1 dominant and 1 mediant in opposing directions Dominant Med. Dom. Dom. G

B

F

C

tritonant

B

D

F F

auxiliary mediant

Mediant

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“Hiking Song” (“In good health, still drunk”): B major; middle section: G major; recapitulation: B major “Dedication” (“Thou, my soul”): A major; middle section: (has become a very common E major (F major!); recapitulation: A major shift principle in Schubert)

mist

-

G minor

pale,

Robert Schumann: “With Myrtle and Roses”, from Liederkreis (Song Cycle), Op. 24, No. 9; text by Heinrich Heine yet the old glow quickens them anew

A major D min A major D major

enharmonic parallels

chromatic chords

same 3rd

The tremendously expressive, impressionistic-romantic chordal progression is literally shown (even tonally) in Schubert’s “The Double” (see ex. 579). As the narrow ‘key’ (D major) is strongly fixed, G minor and A major act as mighty deflections

of the fluctuating cadence.

Without

more precise tonal direction, G minor : A major is interpreted as simple parallel-sounds (G minor : B major).

from what are you so pale and wan?“

Carl Löwe: “Herr Oluf” (Ballade) And should I not be pale and wan

mediant primary collective changed chord G major— E minor (with omission of the corresponding middle sounds) or making cadence:

in G major

or making cadence:

in E minor

primary collective changed chord.

Considerably more entangled is the fully accomplished mediant harmony of Chopin. Just as for Schubert and Schumann, Löwe and Liszt, he also uses major and minor 3rdshifted parentheses frequently or brief pendulum-swings. Chopin, however, often makes this already familiar form especially appealing by showing individual chords with a particular suspended veiling (thus dissonant colourings): the mediant sequences, to which

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the semitone (chromatic and leading-note) displacements also belong, move closer and closer together in time, and create a symbol of feverish restlessness and flickering colourfulness. The great striving for tritones and countertritones respectively drives tonality towards complete annulment. Purely harmonically, Chopin’s frayed, nervous, sensually exaggerated harmony displays several points of contact with a part of the harmony of Liszt and Wagner, which together later gave Scriabin direction and led to apparent totally new consequences . . . First a relatively simple example: Agitato

Chopin: Prelude, Op. 28, No. 1 (1838)

The original functions, up to making its cadence, are strictly diatonic; but the suspended veilings — provided they are chromatic — are of mediant nature (mediant 3rds). A B B (diatonic) E F major complexes F C C C G The resolution of the suspension — with regard to tonal analysis — is merely the annihilation of the stimulating mediant. If the stimulus of double sounds turns out unweakened, then we stand in the middle of the “modern forms of aggression”: Upper part in itself F pentatonic tritonic complexes

Lower part

without chromaticism:

“bitonal”

C major tonal

strong atonal impact Mediants pressing on: Largo

Chopin: Prelude, Op. 28, No. 9 etc.

D maj

G maj

E maj

A maj

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(ditto)

(metharmonic)

(metharmonic) (counter leading-note chord parenthesis)

The deviation to G major is just as logical as one to A major! The return A — B — E major admittedly seems unbelievable, compared simply with G — B — E major. The modulation to the G major tonality would be fulfilled in the following way:

This enharmonic hybridism is by no means responsible for the diminished 7th-chord (as is commonly alleged), but is merely conditional on the previously determined goal (i.e., A major as counter-mediant of the parenthetical C major triad or G major as tonic-modified mediant). The A major triad relates to the E major triad in a way complementary to the relationship between the A major and D major triads: Counter5th counter5th countermediant

A maj chord

D maj chord

D

”

”

G

”

”

G

”

”

C

”

”

E

”

”

A

”

”

E major

Mediant Fifth Fifth

A

Counter-tritonic chord (weak)

A maj chord

D maj chord

F

”

”

B

”

”

B ”

”

E

”

”

E

”

A

”

”

”

D major Tritonic chord (strong)

These tritonic forms are characteristic of Chopin; they occur reciprocally as resolutions in either direct (conditioned) or indirect (conditional) sequence and often achieve potentially simple cadence sounds: cadences

whole tone shifts

“bitonal” cadences

in C major at the same time and in G major

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If C major is valid as T, then the lower line holds (which then represents alterations)

as counter-tritonic

If G major is valid as T, then the upper line holds (which then represents alterations)

as tritonic.

Example above in another setting (compare with the different “little bundles”)

without 5ths:

B F

E B

A E

D A

leading note chords

G D

C G

dominant chains

chromatic chords

“Neo-French” music emanating from Debussy (which today is, however, already “yesterday’s”) is based predominantly on such tritonic relationships. The 3rds and 7ths of the tritonic chords stand in a reciprocal enharmonic relationship:

E B

A

F

In tritonic sequences, a notation of enharmonic change is quite often ignored, whereby the functional interpretation is then of course changed:

referred to C E G B : mobile sounds

alteration

original positions:

Instead of:

tritonic F C

tritonic F B

tritonic E B valid function

is often notated F F chromatic chords

or

or C

B B chromatic chords

C

F B dominant chords!

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in which cases completely different evaluations always result. o refers to the movement or alteration form of . From now on it is clearly recognised that the tonality has become a hybrid, necessitated by the destructive tendencies of tritonants. Instead of the robust and powerfully moving fifths making cadences, there has entered a fragile, degenerate form: a semitone trickery. This gradual semitone scanning is absolutely typical of Chopin. It results from an overstrung, almost sickly refined sensitivity. Three recognisable paths lead out from these charming, exaggerated forms of resolution: 1) to Wagner’s Tristan, 2) to Debussy’s work and 3) to Scriabin’s work. semitone descending shift

(with veiled movement and alterations forms)

Basic function Tonal analysis:

D E B min maj maj

C A E min maj maj

Tritonant

C D min maj

Tritonant

A maj

B min

Tritonant

G D maj maj Tritonant

semitone descending shift Dominants: Dominants:

F C Tritonant

B

E F Counter tritonant

A major

B

E

Tritonant

A major

Counter tritonant

mediant shift (or C)

From bar to bar: auxiliary mediant shift, from double bar to double bar: tritonant shift:

(or G)

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Treatise on Polarity Chopin: Prelude, Op. 28, No. 4

Semitone scanning (chromatic movement chords) Original E

E

Original B

etc.

later

original E

A E

NB: F, A , E and B blur the functional type, which is identical to the starting chord of “Tristan”:

Semitone shifts with tritonant cadences:

Chopin: Scherzo No. 3 in C minor, Op. 39

Presto

G D

C

F F

B

(G

C etc.)

B

E E (F

to D

A

A B )

(A ) (D )

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Chopin: Scherzo No. 1 in B minor, Op. 20

Prestissimo

F major

shift

Plan:

D A

D A

G

G

C G

F cadence in B maj

(A maj:) Tonality B minor: All tritonant double chords as

and

respectively (alteration in the bass) or

are

referred to the succeeding consonance. Shifts (semitone downwards) Presto

A

F min

Chopin: Prelude in C minor, Op. 45

G

G

(A F maj) min

potential centres: C i.e., in C minor:

C

F

E min

F

(G E maj) min

B

E

D min

B

A

C min

D major

G minor

(A minor)

) enharmonically re-evaluated = imperfect (or half) cadences

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Chopin: Prelude, Op. 28, No. 22

Molto agitato

twice

twice D major

D major

metharmonic same 7th

variant principal (!)

D D

F F

Counter-mediant parenthesis with shifts:

(D minor)

C C

Chopin: Prelude, Op. 28, No. 17 (F major)

(A major)

A A

(C )

(B

(F )(B )

major)

(E ) (E ) (A )

(D )

(ditto) (D ) (G )

(C )

(F ) major

A minor

E major

or

in A major

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Similarly, like the previous example: notated:

E

A

E

B

E

B

E major

i.e.,

E

A

E

C

F

C

F major

wholetone shift

C

F

C

F major

D

G

D

G major

shift (semitone deeper)

ditto C

F

B

cadence (tritonant)

E

B

E major__

cadence cadence (counter-tritonant) (as in the preceding example)

C

E major

chromatic chords auxiliary mediant

mediant

Unusually complicated and of the most ingenious boldness is the 8th Prelude. It is abundant in mediant shifts and tritonant complexities and could serve as one of the most typical examples of the nervous, unravelled/closely analysed, exaggerated harmony of Chopin. Molto agitato

Chopin: Prelude, Op. 28, No. 8

simile

to:

to: 263 - English

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(G )

(F )

(D )

(tonal) (!) A

(!) B

D minor

(A major) B major

D minor

(!)

as at the start

To avoid notation in A major, Chopin wrote in the more easily read B major, but this resulted in the tonally neighbouring B major becoming a confusing C major with a glaring tonality jump: C —C major before the recapitulation! In all cases, the exact notation reaches into territories of the most distant keys: following — by comparison — in F and F minor without enharmonic reinterpretation:

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A major The main stages are: F minor (F major) F minor = D minor

Automatic creation of mediants through metrical support-points in the closed 12-tone circle:

Liszt (everywhere)

(or F major)

or or

or G,

F major Franz Liszt: Sonetto del Petrarca, No. 104 (Années de pèlerinage, 2. année: Italie, No. 5) (G )

Shift with contraction: Agitato (G )

D B

F

Each

A

C

is introduced through its diminished 7th-chord of the 2nd order

E

F (G ) major

.

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Lento Franz Liszt: “Chapelle de Guillaume Tell (Années de pèlerinage, 1. année: Suisse, No. 1) ultramediants

Tessellated juxtaposition of mediant-shifted phrases (typical style of Liszt): Franz Liszt: Christus: “The Wise Men” (for orchestra)

B minor

D major

A major

E major

enharmonic

C major

G major

Model: D (C ) major B minor (A minor)

E major

B major

B major

G major

C major

A major

auxiliary mediant ascent

E major

E major

in similar form everywhere

counter-mediant rebound to the contradominant

Quite contrary to principal modulation through substitution-repercussions, these mediant shifts very easily appear stingy and theatrically bombastic.

From the same work (ex. 617), a few more extraordinarily expressive examples of mediants:

Christ: Sad

is

Franz Liszt: “Christus”, 3rd part: Sad is my soul soul

my

(G ) all minor key forms: (strong mediants)

C minor

D minor

A minor

B minor

ultramediants

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Christ:

let pass from

G major

(G )

me

A minor

let pass

from me

A major

B minor

The tonal group G major : A minor returns at the end in reverse sequence, but it is notated as A minor — A major with widely altered fifth. A minor : G major = metharmonic same-7th:

; A minor : A major = same-3rd:

(already known

from Schubert). A minor : D major =

: T || A minor : C major =

: or

:T

Both forms later become known as primary collective-changed chord:

Closed, widely spanning tonality circle (E major) in evening sunlight . . . . ,

in

correctly:

Franz Liszt: “Lorelei” (1861) evening sunlight . . .

A minor

E major A major

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her golden jewellery

sparkles, she

A minor

combs her golden

C major

G major

hair

C major F minor

shift tonal to the end

F maj E maj

Swing to the tritonant:

E

Moderato

but

C

A

what

C

is

E Franz Liszt: “I’d like to meet you again” (Song) my fate,

(5th-bass)

your

beloved

(5th-bass)

soul

shall

I

bless,

(5th-bass)

your

soul

is

my

star,

your

(C )

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beloved

soul

shall

I

love.

Taken precisely, the first phrase ends in F major and the second in D major. But as the second begins in the final key of the first, then the complete chain runs from E major via F major back to E major.

Original positions 2 octaves higher 3 auxiliary mediants

3 dominants

2 mediants

2 dominants

It is remarkable, in Liszt, that the mediants and auxiliary mediants nearly always occur in 3rd or 5th bass-forms {i.e., 1st or 2nd inversions}. Strikingly similar (and yet in principle conflicting) is the following example from Brahms: Johannes Brahms: “Ever More Quiet”, Op. 105, No. 2 (1886) Text by Hermann Lingg, 1857 no - one wakes and lets

you in,

weep

I

bitterly,

weep

a - - wake

and

bitterly.

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If

you

once again.

would

Come,

see

O

come

me

soon!

tonal to the end

The first section reveals a metharmonic faux pas, the second an enharmonic faux pas: A minor

B

referred to A minor— C major:

major instead of A major variant of A minor

cadence . . .

The artificially closed 12 semitone system comes ever more clearly into the foreground, where the enharmonically different intrinsic values have become illusory. Similarly is the case in the following example: Agitato J. Brahms: Rhapsody in B minor, Op. 79, No. 1

(F )

(C )

F maj B min

A maj

D min

C maj

F min

(D maj — G min!)

C major

F minor

C major

D minor

F minor, continuing thus

C major is used instead of D major! Trio I = D minor via B major — G major, D and F

*}

A maj

to

G major.*}

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G major is used as tempered contradominant to B minor! Such enharmonic exchange is not nullified in a different setting but is again repeated in the same way! B minor becomes — literally — D minor (!) . . . . The decline of tonality proceeds blatantly, even though Brahms’s ‘sound in itself’ still seems completely unbroken. C major, of course, cannot really be evaluated as a brightness-increase via B major any more, when in the process it proves to be D major, which further pales into F major . . . . . Such cases of enharmonic faux pas without switching back become ever more common from the middle of the last century.

J. Brahms: “Of Eternal Love” nowhere

a

light and

counter-sound

nowhere

space

parallel counter-sound E B G D (G C) sequence min maj min maj (maj min)

potential E major, yet without a perfect tonic in C minor (original):

In the case above there is no reason, for the time being, for an enharmonic reinterpretation (for instance, E major instead of D major) as 1) the making of a cadence is easily understood and 2) a return to the original tonic follows. And yet, in a further shift at some point, enharmonic re-evaluation must be risked — for purposes of readability:

Auxiliary mediant shift upwards. Dominant-mediant interjections ( ) within tonal groups: J. Brahms: Variations & Fugue on a Theme by Handel, Op. 24 (1861) etc.

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E major, not C major even though E minor

not

always auxiliary mediant shift downwards

E major : B major = ultramediant.

(powerful C accentuation!)

Stereotyped shifts in full bloom:

Richard Wagner: Rienzi’s Message of Peace

D

F

A

C major

auxiliary mediant shift upwards enharmonic

E

D

D

enharmonic

C

C

B major

semitone shift downwards

-a- = counter leading-note chord, acting as chromatic introduction -b- = the contradominant parallel, acting as diatonic introduction Both forms relate to each other as same-3rd Wagner’s extraordinary preference for mediant shifts already reveals itself in this early work. Admittedly, his technique here — in contrast to Chopin and Schumann — is still raw, primitive and unpleasantly stereotyped. 272 - English

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Strongly widened tonality through mediant shifts: countermediant

tritonant

triple dominant

R. Wagner: Overture to Rienzi auxiliary mediant mediant (minor)

referred to E minor

referred to F minor

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B major is possibly just as good as C major. The mimicry-form would lead to the C major chord. Yet the original notation is absolutely logical for a tonal return: G , F, E major = twofold leading-note chords: G major:

E major

Auxiliary mediant shift upwards in the minor: Andante to

brighten

he

who

R. Wagner: Rienzi’s Prayer thinks

humbly,

to

auxiliary mediant shift upwards

D major G minor elevate that

which

sinks

into

the

dust . . . . .

F major B minor

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The same principle broadened: Andante maestoso

R. Wagner: Tannhäuser

B minor (C major) B major

(tonal return)

Equally:

Tannhäuser 2nd time

E

G

2nd time

D major

B

Allegro vivace

E

Tannhäuser

C

G major

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E

C major

Through enharmonic change, artificially closed tonality: R. Wagner: Lohengrin

Auxiliary mediant parenthesis with strong retaliation before the tonal return shift: R. Wagner: Lohengrin ditto

etc. long E major group

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formal repetition

tonal repetition

G major to the end

referred to A major

G

E A

G major

The intervening chord between the E major and A major group is the diminished 7th-chord, which functions in the sense of a G major chord: E major

G major

A major

ultramediant or

Quasi infinite circuit of mediant shifts upwards (following without enharmonic reevaluations, which exist in the original): Wagner: Tristan

G (A major) original

D

B major (Ring closure

F

through enharmonic change)

A

C major (B major) original

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Semitone shifts upwards (as enharmonic changes) Sheltering

- ghing

in

our

stars

breast

of

the

sun,

illuminating

delight. Of

your

magic

B

-bracing,

from

your

Wagner: Tristan

major

eyes

lau - -

softly

em

-

A major

sweetly

fading

B major

( C major

A minor : D minor with lower-7ths counter-tritonant

in A major) C major

B minor : F minor with lower-7ths tritonant

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Both forms occur equally often; meanwhile, enharmonic reinterpretations are very often involved, so that a differentiation between the two types — ex abrupto — is not always immediately possible. The 2nd example comes out of an A major range, as if it were thus B minor F minor ‘tritonantic’; but that A major episode is to be rated in the superior A major as B major (

), in which tonal group the B minor chord becomes C minor tonality (C , E ,

G , A ), its succeeding F minor then becomes counter-tritonant (= 1st example: A minor | D minor). in A major = C minor : F minor E major = B minor : F minor E major T

D

In C major, the sequence B minor : F minor is simple enough: c

Doomed

head, _______

doomed

Wagner: Tristan head

The A major chord is assessed by most theorists as an enharmonically changed Neapolitan 6th-chord (D , F , B ) referred to A major . . . but when enharmonic reinterpretation is assumed anyway, then of course the perception as a root tone (

) lies considerably nearer:

C minor : leading collective-changed chord note chord (primary form)

But the sequence A , A major is completely understandable, e.g., in C major | potentially made a cadence

and

| or

respectively.

That is to say, referred to C minor (= valid tonal group): A major A major

F minor

D major F minor

G major

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Mediant-grouped chromatic scale:

R. Wagner: Valkyries (Sleep harmonies) as at the start

(C major) A major

E major

C major

G major

E major

dominant chord

chromatic chord

chromatic chord

auxiliary mediant

dominant chord

chromatic chord

leading-note chord

auxiliary mediant

dominant chord

dominant chord

leading-note chord

chromatic chord

dominant chord

chromatic chord

leading-note chord

auxiliary mediant

How relatively simple this colourfully surprising tonal chain manifests itself in the functions.

Tonality centres whole-tone deep shift

R. Wagner: “Siegfried” (Traveller’s harmonies)

5th-relationship ” tritonants

mediants

or

or

without standardised tonal centre

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The sequence above is functionally intelligible from any arbitrary tonal centre: e.g. mediant centres

B major: (relative best chance) even if T itself does not appear:

G major: E major

But none of these interpretations has indisputed validity, as a closed tonality does not exist here at all. This group is ‘free tonally’ and portrays a symmetrical chordal interpretation of a symmetrical tonal sequence: pronounced Chopin type: The notes are evaluated as intervals thus:

E major chords

leading-note chord

D

tritonant

A

mediant

A C auxiliary mediant

E

dominant

tritonant

B

mediant

auxiliary mediant

i.e., B D

The destruction of tonality advances inexorably!

tonal origin:

The tonal chain above portrays the strict polar counterpart of those “wandering harmonies”. How unconsumed this expressive sequence sounds in the minor key, which is conspicuously close to Reger’s style!

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The same with lower-7ths and embellished:

B

G

B

F

C

A

C

Mediant minor-key forms: D minor

G

F minor

R. Wagner: Parsifal G

E

G min, B maj F minor

D minor : F minor = counter-mediant || G minor : E minor : G minor = auxiliary mediant B major : F minor = mediant leading-note change or primary collective-change (characteristic of Parsifal)

D minor

Referred to A major:

C minor

D minor — C minor — A major — A minor leading-note

A minor

(thus very simple!)

mediant chords

minor-key forms!

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Shift upwards from minor-key auxiliary mediants:

R. Wagner: Parsifal

(E min : E maj) A

E minor

A

B major

D major centre

D major

G minor C

F major centre

B minor

A major general tonic

auxiliary mediants

The short-term movement form E minor : E major =

= same 3rd.

Shift downwards of minor-key mediants: violently

R. Wagner: Parsifal

or

or (thus ultramediant of the original tonic)

Shift downwards of minor-key mediants: Clar. as above:

B minor G minor E minor

E minor C minor A minor

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R. Wagner: Parsifal

(B !)

(G minor)

G maj A maj

C min G min A min

A maj B maj

C min D min G min A min

E min

F maj

E min

C maj F min

E maj A min

R. Wagner: Parsifal

shrieking,

raging,

(stationary): (making cadence):

storming,

raving

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in

R. Wagner: Parsifal need

greatest

or leading-note chord _______________

chromatic chord

I wrongly believed his eye already near

or leading-note chord

Same 3rd:

(cadence)

R. Wagner: Parsifal further in B major

(referred to D minor) (B major)

Collective-change (primary):

R. Wagner: Parsifal G maj

F maj

or

(in C: [?]

) or ultra countermediant

mediant

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The collective changed chords — like all minor-major resolutions — are functionally reciprocal (i.e., polar corresponding):

Solemn pendulum-swings of the primary collective-changed chord (tritonant sequence): R. Wagner: Parsifal

Lanterns of the (Holy) Grail

making cadence:

(NB: functional signs without enharmonic compromise.)

These tonal sequences have created an exceptional school; particularly Hugo Wolf, Bruckner, Elgar, Reger and Strauss — thus the last great representatives of the fading consonant harmony — maintained them with special devotion. But this extraordinarily expressive tonal change already exists with Mozart in a very distinct place: W.A. Mozart: Fantasia (allegedly C minor!), KV 475 (1785) (better: Chromatic Fantasia) A maj

G maj

F min

E min (or D minor) via B Major, B minor to G major

or

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The tonal progression is:

from here on see example above

What astonishing audacity and what intuitive assurance in the mightily widened commaperfect tonality! Mendelssohn also writes the same sequence in the “Variations sérieuses” . . . E major C minor F. Mendelssohn: Op. 54 (1841) G minor

Ultramediants, same third: A major D major

grief gone to sleep,

that the devout master Hugo Wolf: “Sleeping Jesus-child”

D major

D minor

D maj A maj

exactly as before

A. Bruckner: 3rd Symphony, 1st movement

A. Bruckner: 9th Symphony, 1st movement

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A. Bruckner: 9th Symphony, 1st movement

in D minor:

(E )

(G !)

re above: Enormously widened tonality of the D minor centre: F major chord

D minor

C major chord

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Tritonants (major chords):

Very slow

(as above, 3rd movement)

tubas

in E major

Always

through

gentle

light.

tears

do

I

Hugo Wolf: “Seclusion” of the sun

see

referred to C minor: Often

am

I

hardly

conscious

(potential G major) (referred to C minor) and

feel

pure

joy

quivering

delightfully

main key A

D maj E minor

Tritonant same 7th

G major

E minor

primary collective -changed chord

The architectonic main centres are: E major C minor

G major

C minor E minor E major

Tonic representatives

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Far more tonally impressionistic is the following song; scent and colour are determined by the written draft: Hugo Wolf: “He’s it”

Very lively, jubilant

flutter through the breezes;

Sweet

famil-

accompaniment simile

iar

scents

rove full of

expectation the

countryside.

(4 bars C major)

(tritonant to G major) Violets

dream

Listen!

already,

from afar,

want

a

soon to

quiet

come.

harp-tone.

Spring, indeed you’re it!

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Max Reger, Op. 59, No. 9: Benedictus (organ)

auxiliary mediants

tonally further

Unmistakeably Liszt’s types. The first 4 bars form an interesting cadence-loop:

E

B

D

G

A major

(looks less “interesting”) Max Reger, Op. 46: Fantasia and Fugue on B.A.C.H.

referred to B minor C minor: E major referred to B major primary collective-change

Semitone shift downwards of minor chords

Max Reger, Op. 76: From my journal

D major: leading-note chords leading-note chords chromatic chords

D minor:

thus relatively simple

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Shift up of Neapolitan cadences without tonic:

Max Reger, Op. 116: Violoncello Sonata

D minor

to A minor

to D minor

to B minor

Shift downwards of Neapolitan cadences without tonic. (G !)

(as for 667) {i.e., Reger, Op. 116}

(as previously)

D minor

primary collective-changed triads

(as for 667)

(A major:) Neapolitan to

(E Major)

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This quotation is very typical of Reger’s style: The conventional original phrase reads:

1st

bar

diminished 7th-chords as feminine introductory forms (accentuated, thus having a stronger effect).

2nd

Between the first and second bars a parenthesis is interpolated, which brings about a counter-mediant close conclusion: 1st bar

2nd bar

4th bar

3rd bar

and which forms at the same time a chain-introduced Neapolitan cadence on the final dominant made tonic. F major

B maj E major

(drawn together)

The metrical count now appears quite by itself:

The metrical form (stretching a two-bar group to one of three bars, or packaging a fourbar group into a manageable group of three) is that of Brahms, the notation ( as ) that of Chopin. (See the four Scherzi.) Thoroughly Chopinesque are also the semitone glidings {lit. scannings} following the examples above:

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{still, presumably, Reger’s Op. 116} (simplified)

(A minor) (embarrassingly denoted:) D minor

G minor

in A major: (D !) as above

A major C minor

A minor

D minor

A maj

metharmonic auxiliary mediant bridge

Further, some samples of typical Reger songs, in which mediant harmony comes out virtually orgiastically: Expressive

rests my eye

On your portrait

Max Reger, Op. 70, No. 12 in the black frame___, there

now and then and sometimes I

call

your

name

(E )

a) auxiliary mediants || b) mediants || c) tritonants || d) secondary collective-changedtritonant variants || e) leading-note chords || f) metharmonic 7th counter-sounds

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Max Reger, Op. 66, No. 4

Simple (!) You are good to me,

a secret sounds

Beginning!

Groups

G major

C major

in my soul in deepest dream.

F major: G minor chords (= same 7th) B major: C minor chords (= same 7th) E minor: B major chords (= tritonant variant)

E minor

E major

You are good to me,

the beds are now blooming

The main motif here is always understood as minor chord with its tempered contradominant.

etc.

The connections, e.g.,

present themselves as

metharmonic same 7th.

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If the beginning of each motif is understood as being in a major key, then this yields ultra-counter-mediants within the same mediants and as connections:

No matter what the understanding is: the furthest distant relatives are always present, which casts very much doubt on the tonality. Max Reger, Op. 70, No. 3: Knight advised thus to the squire: (Bierbaum)

Very lively

Sit

in the saddle,

ride,

ride

to

freedom,

free

(minor 3rd relative) auxiliary mediant

auxiliary mediant

Sit in the saddle, ride, ride

enharmonic countermediant

to

auxiliary mediant

freedom,

primary collectivechange

free

metharmonic 7th-changed chord

the fairy

(major 3rd relative) mediant primary tritonant mediant primary tritonant mediant mediant collective- variant collective- variant change change (“ride home with her. . . .”)

primary collective-changed chords

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Max Reger, Op. 70, No. 2: The King at the Coronation (Mörike)

With great power and force Wedded

to

constant

you

at

the

altar, o

fanfares on the piano

Fa - - - - ther - land,

thus

am

I

yours.

Again the same theatrical mediant bombast. (Compare with the sympathetic setting of the noble Mörike poem by the master, Hugo Wolf!) The above tonal chain purged of enharmonic disguise: (sudden turn)

mediant tonality destruction

tonally D major E major chord never appears again

(sudden turn)

(sudden turn)

(sudden turn)

(same 3rd as D)

or

fluctuating F major B minor

mediant E major

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Max Reger, Op. 70, No. 16 The Declaration of Thirst (Fick) “Always sweating”

as

__

the

hellfire

is

nothing_______

quenched,

remains to me

in

nothing,__

the sack

Typical Reger: splendidly bold in harmonic outline (free tonality), but immoderate in the means of presentation, and wild in the accumulation of mediant fallacies.

The harmonic framework is presented in legato style in the following form: (transposed B

.

A . C

.

H)

1) primary collective-changed chords. 2) secondary collective-changed chords.

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Richard Strauss: “Rosenkavalier”

can be interpreted as: tonal

free tonal

D minor A major F minor C major or E minor B major G minor D major

centralised referred to F major G

primary collective-changed chords E minor B major F minor

primary collective-changed chords

C major

decentralised referred to G major (original signature}

Such enharmonic ambiguities are exceedingly common in modern music. The primary collective-changed sound of the B major chord can be referred equally well to the G minor chord as to the F minor chord; and the F minor chord can be referred equally well to the B major chord or A major chord as collective-changed sound.

making cadence or

substituting

making cadence or

substituting

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Quite similar cases are in the following example: centralised F major C major Richard Strauss: “Elektra”

decentralised G major C major or leading-note mediant-(shifted) (B major range)

(D major range)

F major : D minor or G major : D minor — both cases are possible in the same way. D minor : G minor or E minor : G minor — ”

”

”

”

”

”

”

” !

Richard Strauss: “Alpine Symphony”

until the end of this episode

tubas and trombones B minor B major primary collectivechanged sound

The D minor and G minor sounds stand instead of the tonal D major and G major chords, as their same-3rd they act: G

B

D

D

F A

G

B

D

D

F

A

More detail will be given later on the matter of these very strongly colouring representations. (See also the example from Elgar’s “Apostle”, p. 302.)

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Richard Strauss: “Till Eulenspiegel”

typical tonal sequence

Motif

primary collective-changed sound as: C minor : F major

referred to E major secondary, primary collective-changed sounds B major E minor A major

twin! fifth-bass!

Collective-change:

primary

primary

primary

primary (D minor)

C minor or

as in a)

secondary primary

(F major)

D minor : A major (A major)

NB) As primary collective-changed sound: re-evaluated enharmonically concordant = and

by comparison: dissonant original form:

respectively

G

B

D

E

G

(B)

The primary collective-changed sound enters into the cadence instead of the dominant, as its same-3rd (possibly re-evaluated enharmonically) it acts: a) and d)

as apparent C minor : F major = C major : F major

b) and c2) as

″

E minor : A major = E major : A major

c1)

as

″

G minor : C major = G major : C major

c3)

as

″

F minor : A major = E major : A major

E

G

B

c4)

as

″

E major : C minor = F minor : C minor

F

A

C

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The secondary collective-changed chord occurs in the cadence instead of the contradominant, where it acts as its same-3rd (perhaps re-evaluated enharmonically): b) = B major : E minor = B major : E major e) = D minor : A major = D minor : A minor Edward Elgar: “The Apostles” (Oratorio)

4 choirs with harps

or

same 3rd

instead of A major

B major chords

E major : A minor, as well as F major : B minor = secondary collective-changed chords or varied tritonants respectively. The tonal sequence F major : B minor

or

stands in reverse order

B minor : F major in Liszt’s B minor sonata:

Liszt: Sonata in B minor

or

i.e., exchanged genders = B major : F minor — Here

Liszt: “The Blessed Elizabeth” the voice of com-pass-ion is dead:

Almost impossible to disentangle seems the following crazy tonal chain from Salome (with tonal weight-burdened motif of the startling hallucinations of the tormented Herod):

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A sustained ‘C’ (tone of the Jochanaan motif) seemingly complicates tonal perception even more, but just this persistent voice is the calming influence, which represents the last remnant of tonal centralisation, and to which all sounds have to relate, i.e., from which all sounds derive relative differences in appeal. Richard Strauss: “Salome”

(F )

(A )

(A ) (potential B minor)

(tonal)

same 7th

quasi

cadence

primary collective changed-chord secondary

7th countersound

cadence

or

variant

Everything referred to C major:

(potential B minor)

or

The expressive effect of this chain becomes confused and crazy, chiefly through the tonal conflicts with the inflexible ‘C’, as of course the programmatic presentation demands; but must be thoroughly comprehended in tonal analysis and function. It is not at all the “chance construction”, of which some modern theorists very conveniently speak. . . . . . S. Karg-Elert (1919): “Music, for 5 instruments and voice”, W 62 (E)

(violin) you are not dying the easy death of a

flower

(flute)

(alto)

primary

secondary

primary

secondary

primary

(viola) (bass clarinet) (v’cello [on the bridge])

collective-changed chords = whole-tone movement upwards 303 - English

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(schematic example)

C

secondary

primary

primary

secondary

B

F major D minor

secondary

A

E

secondary

D

primary

major chord C series minor chord series

collectivechanged chords

The potential chromatic scale is formed from alternating 3rd and 7th tone changes: (Tones:)

The minor sounds are to be understood in the major chain as varied counter-chromatic chords(!) of the introductory-dominants! with upper 7th

diatonic original form:

A

B

C

major

variants:

A

B

C

minor

counter-chromatic chords

A

B

C

major

variant chromatic chords

A

B

C

minor

with lower 7th

Notated strictly as leading notes, these secondary collective-changed chords present themselves as introductions in the following way:

and only in this dissonant form are the introductions true collective-changed chords, thus movement sounds of unambiguous kinetic tendency. They are, as already mentioned often, amalgamations of incomplete leading-note and counter leading-note chords.

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E B G E

F C A F

= double chromatic

If the two minor-sounding collective-changed chords in the major with their lower 7ths form the lower 9ths, then they become major-sounding upper 9th chords of the leading-note and counter leading-note chords: E C A

E C A F

E C A F E D

E C A F D

C A F

C A F D

C A F D E B

(meth.)

C A F D B (meth.)

See for this the following example: (stylised version)

Valse élégante

Primary collective-changed chords with lower 7th and lower 9th counter-leading-note sounds as upper 9th-chords: A D

C

B E

E

(enharmonic)

E A

A

(enharmonic)

G C

B

C

minor chords major chords F major chords

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Treatise on Polarity S. Karg-Elert: “Op. 152” 2nd String Quartet with Voice: “Clarification” {the Gerlach catalogue has no entry as Op. 152; nor does this work appear elsewhere in the catalogue} likewise

minor tritonants

viola

A minor D minor F major B major

A minor F major

(ditto) v’cello

D minor A minor B major F major

potential leading-note twins in tritonant sequences:

violin II

ditto

viola }pizz. v’cello} mediant shifted F

C

B

E

D

F

G

C minor chords

minor tritonants

Sequence of minor chords B on a whole-tone basis: Range:

C F D B

D

E G E C

F

G minor chord.

= 2 whole-tone shifted mediant pairs.

Tritonants and tritonant variants (secondary collective-changed chords): (S. K.-E.: “Op. 152”)

D minor : G minor F minor : B Minor F minor C major

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(G )

E minor B major A minor E major

ditto

ultradominant chain (minor)

F maj : B maj : F maj.

F minor : B minor

C : G min. D : G min. D : A min. E : A minor

F : C minor ditto ...……………….… C : G min.

G : D min. E : A min. E : B minor (D ) (A ) chord

chromatic and tritonant bitonality respectively: e.g., from bar 8 on: C

F

G

C

D

G

D

A

minor

A

E minor

Re the above example: Bar 1 = F

D

G minor chords

B

Bar 2 & 3 = E

B

F

bitonal

A

E

auxiliary mediant group. Skipped components = tritonants.

C B

F

major chords

with upper lower 7th tritonant

minor chords

variants (= secondary collective-changed chords)

(tritonants) Bar 7

identified

Bar 8 up to the end F

C C

G G

D D

A A

minor chords E minor chords

tritonants.

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B minor

F major

quasi: B F B min

Bar 1—2 and 9—10 = A Bar 3—4 and 7—8 = D

B

F

G

D

B

E major

F B F major (movement sounds)

G minor chords E

C

A

F major chords

you are besmirched in mind and deed hasty

v’cello

auxiliary mediant auxiliary mediant

(minor tritonants) more excited

tritonant

ditto

ditto

what is it?

ditto

(Christ) lean back

and delicately drink quickly

ditto

Minor-key tonal chain, founded on whole-tones: E

.

F

.

G

A

.

B minor chords

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14th Chapter The Seventh Counter-sounds (Counter-concordances) Their acceptance simplifies tremendously the harmonic analysis of complicated cases. Just as for each triad or tetrad, where the parallel or leading note-changed chord can occur as diatonic or chromatic substitute — moreover, their mediants or counter-mediants — also their variants — and furthermore their parallel variants or variant parallels — so also can every sound be represented by its 7th-countersound.

and respectively = concordant. Opposite position to the principal = counterconcordant. Capital {upper case} letter = = major chord; = minor triad (exactly as with all other function-letters). These counter-concordants often occur as far simpler formations for the ‘tempered ultracontrants’, ‘counter-mediant variants’ and ‘variant parallelvariants’ (!). Cadences:

potential cadences: (instead

of:

or

(instead ) of:

or

)

of:

or

(instead ) of:

or

)

(instead

of:

or

)

of:

or

)

Cadences:

potential cadences: (instead

(instead

One should understand correctly: the sounds in question can in many instances be accepted as counter-concordant substitutes, as when e.g. in the above cases those very sounds again follow the tonic, yet their values as third-relatives still remain if their affinity to some other sound in this evaluation proves to be more closely related:

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1st time

F. Mendelssohn: Variations sérieuses perhaps

simpler E major: E E

G G

B B

D D = G minor

The same sequences (NB) occur in “Parsifal” (Wounds of the Redeemer) and in Debussy’s “Faun”, and they are easiest understood by assuming the inclusion of 7th-changed chords making cadences: R. Wagner: Parsifal

7th-change at

G

C

F

B

B

E

A major chords

very slow Claude Debussy: “(Prelude) to the afternoon of a faun”

Horns (muted)

E

A

D

G

C

C

D

(B )

(C)

A

E

(C) minor

Incidentally, this last-mentioned, precious work is filled with 7th-counter-sounds, which lend it a quite specific charm: very moderate

(ditto)

(ditto)

solo flute (horn) arpeggio

2nd bar:

A

C

E

G

A

C

E

G

3rd bar

(A major instead of B major)

(ditto)

instead of: E

G

B (D)

occurs: E

G

B

D

instead of: = B

D

F

A

occurs: B

D

F A

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Schematic example: chains of fifths alternating with 7th-counter-sounds: instead of: C

occur:

F

C

B

A

G

A major chords F

B A

instead of: C

E

G

C

D

A

D B

C

D

maj chords, A

E

B

F

C A

B

F maj chords, A

D

E

F maj chords, D minor chords A

B

G

C

E major chords minor chords

D B

E

C

G C

F B

A G

F

minor chords

E minor chords

G major chords E minor chords

One recognises in the 7th-counter-sounds the strongest means of tonal widening. Very remarkable is the phenomenon that nature, in grading the tonal relations: 1st fifth, 2nd third, 3rd seventh relationship at the same time moves the limits of tonality outwards, but always makes the apparent tonal chain connection through a comma-differing type of change; in C major and A minor: tonal range through 5th-relationship, excluding the ultraforms: 5th-relationship: 3rd-relationship:

7th-relationship:

The tonal chain, in the middle of which the C major or A minor chord functions as tonic, henceforth runs from the D minor to the G major chord:

{This over-comprehensive sequence begins with the maximum number of flats, decreasing by steps to zero (A minor, C major), then ascends through the sharp-containing signatures to the maximum at the opposite end; at the extreme ends are the theoretical keys demanding double accidentals.}

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15th Chapter Abolished Tonality (“Free Tonality”) a) General Comments The tonality is “abolished”, or in an episode or movement made “free tonal”, if harmonic events no longer allow the recognition of a common harmonic centre. As every cadence

or

acts to specify a key (even if only

briefly), then it is understandable that free tonality in the first regard eliminates any precise cadence, particularly on metrically marked periodic endings.

ultradominant variant

ultradominant variant

primary collectivechanged chords

counter-mediant variant

primary collectivechanged chords

counter-mediant variant

counter-tritonant variant

secondary collectivechanged chords

dominant variant

tritonant variant

secondary collectivechanged chords

Theoretically, functionally, this tonal chain admittedly is easily referable to very many general centres, e.g., C or E major occurring as T, but they are never experienced in such a form! The sounds appear in a linear detached comparison (chain relation) and prefer a distant relationship, which excludes central grouping. It is not the outer parts that cause free tonality, as their tone series easily allows “tonal” interpretation also:

F maj.

D

B min. G maj. B maj. E min. F maj. E min. B minor

E

G

C

F

B

E

A major

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tritonant and its variant (“hybrid”)

tritonant parallel

7th-counter-sound

dominant leading-note-changed chord

And of course, on the other hand, a strict diatonic (tone) sequence can be interpreted harmonically in free-tonal fashion:

b) Parallel Chord Shift As — according to the previous chapter and the schematic table on page 52 to 53 — all sounds are related to each other, then all sounds can also appear in a detached chain, one after the other. The most primitive type of such sound chain appears in the “parallel chord-shift”. Just as the harmony remains unaltered with “position change” and only the chord form changes, the chord form remains unaltered — inverted — at “parallel chord shift” whilst the harmony changes: in polar evaluation of harmony

in polar evaluation of harmony

harmonic D minor (A minor lower tetrad)

harmonic F

A

B

D minor chords

Tone compass projected in a line: (D major) F major (D minor)

4-note range

E F major minor

9-note range

Chord tones: F major F minor E major D major D minor

In general, Debussy is described as the “father of bundled chord parallels”, and it is hardly disputed that, for a while, he was the accepted leader in this direction [Cyril Scott (everywhere), W. Niemann]. But it appears as if this ‘stylish-primitive’ parallel style was around at the turn of the century, as the author of this work had already, around this time, written such bundlings — to the dismay of his teacher — even before the name Debussy was known to him.

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Some different forms are established. They occur in chain relationship: 1. sounds of changing gender within the key (tonal stepwise shift) 2. strictly imitated sound types without regard to tonal closeness 3. changing ” ” ” ” ” ” ” The chains can also emancipate themselves from the melody and merely act as upwards and downwards shifting collateral mass. Claude Debussy: “Sacred Dance” (chromatic harp)

major and minor chords (octave positions)

pentatonic range only fourth-triads (neutral)

bass

F B

pentatonic range

C F

only major chords (third-positions) tonal melody

hexatonic fifth-range: G D A E B F major chords major and minor chords

Claude Debussy: 3 Nocturnes (No. 1: Fêtes, orch. and 16 female voices)

clarinets in E-B

upper part diatonic tetrachord middle ” ” ” lower ” ” ”

(A B C D E) (F G A C ) (D E F G A)

E major : G minor chords, as well as D major : F minor chords = 7th-counter-sounds. The notes B and A (middle part) are linear to A and G . S. Karg-Elert (1899) Lento semplice

2nd movement from the Piano Concerto, Op. 6

4 solo violins

Re 1) English horn

accentuate, quasi parlando major and minor chords, in steps, tonal Con moto

S. Karg-Elert, Op. 71 (I) Violoncello Sonata (2nd movement)

tonal stepwise shifts of tetrads (7th-chord parallel and leading-note twins)

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Re 2)

S. Karg-Elert, Op. 139 (A) [1919] “Youth” for piano, flute, clarinet, horn

Vivace con umore

root position 9th-chords, 5th harmony, strictly imitated shift

Re 3)

S. Karg-Elert, Op. 102 (1914) 12 Impressions (for Art-harmonium) (Monastic Melody)

Lento mistico

different types, harmonically free-tonal, melodically tonal (D minor—A minor) S. Karg-Elert, Op. 108, No. 2 (1923) “Starlight” (for organ)

Adagio

E

F

C major

A

E minor

very conventional transitions in major 6/3 chords, which hardly unsettle the tonality appreciably, particularly if the supporting points are still easily obtainable tonally. Quieto

Flute with tremulant

chromatic shift downwards in 6/4 chords

S. Karg-Elert, Op. 108, No. 1 (1923) “Sunset”

sequence in the enharmonic counter-mediant

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(ditto) {continuation, Op. 108/1}

combination

wholetone shift

semitone shift

sequence in the upper fourth

Languido

W. Niemann, Op. 86, No. 1: Evening on the Nile (“Land of the Pharaohs”)

Melody strictly tonal: E Phrygian. Harmony retouched in tonic and mediant fashion Minor chords (root position) in parallel shift.

.

Wolfgang Ey (born Con. Ferd. Meyer): “Fulness”

Maestoso

bundled 9th-chords (double sounds)

Schematised Forms: A) Assuming the following melodic phrase (unstylised):

Only major chords appear (three to five note chords). Each tone can be 1, 3, 5 or 7. From the second half, change the type of chord:

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(Model: No. 716 _ _ )

major triad root-basses

major triad 5th-basses

major 7th-chord 5th-basses

major 9th-chord root basses

5th-basses triad 3rd-basses

7th-chords

root-basses

7th-chord 5th-basses

root-basses 9th-chords

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From now on, a mere 3 chords of the same name appear together in groups:

If only 2 chords of the same form appear “shifted”, then the mechanising element recedes even further, particularly when the pairs occur in 3-part rhythm.

Such tonal series can be almost endlessly assembled into ever new groupings; admittedly an artistic achievement is not exactly presented by such kaleidoscopic transformations. For instructive reasons — for the purpose of training in free-tonality sensing — a thorough consideration of this tonal form is meanwhile highly recommended. B) The minor-key forms are still missing. For them, the following tone-chain may be used:

Ah ——— dea - r

Au - gu - stin

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Allegro con brio

7th-basses

Risoluto

root-tone basses

Tranquil

soft as velvet

Root-tone basses. Lower 9th-chords = identical with upper 9th-chords in 5th-tones. Analogous to the major-key forms, smaller groups can also change their type here: or

etc.

In the same way, parallel- and leading-note twins can be shifted in all 4 positions and situations. In the diatonic system, the change between both is very common: usual with Grieg Debussy MacDowell Delius

Niemann Grovelez Scott Frank Bridge

to play with any or signature (in the same notation) one wishes: e.g., minor) or

(E flat major : C

(A major : F minor) etc.

with any signature as desired

Lento

Frank Bridge: “Heart’s Ease”

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Eric Coates: “Sleepy-head” (String Orch.)

Major and minor triads shifted in isolated groups:

c) Dissonance Parallels (Atonal Complexes) If dissonance is asserted continually, atonality results. The transition between freetonality and atonality produces an hybrid form, which sets dissonances of clearly recognisable consonance-foundation in the decentralised tonal sphere. (For example, parallel and leading-note twins, triplets, dominant 7th-chords with the 6th, etc.) In this connection, the relationship of the foundation still remains virtual, whilst the additions creating dissonance function more or less only as colourings. Diminished 7th-chords and augmented triads in chain progression close any tonal recognisability, i.e., any unambiguous affiliation to a certain range of keys; thus do they establish an atonal complex. The great romantic masters Schumann, Chopin, Liszt and Wagner a long time ago dared such chains of dissonance without key-relations, before the catchword ‘atonal’ had been coined . . . .

(or A )

The tonal series and the following table of dissonance-types is given: diatonic dissonances

or

(on F or C)

(on E)

or

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auxiliary-med. chrom. trit. counter-mediant Mediant

(relationship just as to the left) ineffective

(D ) (E ) (A ) where ineffective, better to notate simplified insipid

(C )

(G )

(G E )

insipid

simply notated

Tonal forms of minor thirds stacked over one another act as minor third shifts Tonal forms of major thirds stacked over one another act as major third shifts

not as functionally different entities.

Parallel direction is a primitive way to define, as clearly as possible, a particular tonal type in the harmonic event through stereotypical repetition. By transposing the type, the main emphasis is guided to the relationship proportion, as the understanding of the once established and mechanically shifted sound form presents no further problems in the course of the harmonic event, but the specifically charming effect of the sound in itself is strengthened through repetition.

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d) Consonant and concordant tonal interpretation in a free-tonal range As each note has a nine-fold tonal interpretation, if only the simplest forms

are assumed ( , thus ranking as if the same sound), then two tones in succession allow 81 different forms of resolution:

as

E

D

as

G

F

as

D

C

as

B

A

as

F

E

as

B

A

as

F

E

as

D

C

as

A

G

major chords

minor chords

Enharmonic change in many cases is self-evident (C major — C minor instead of D major — C minor, etc.). In the table above, all relationships that take on a semitone step are included. Each can insert its 7th and 9th. The sounds indicated in the cross-linked column produce the chord parallels mentioned previously, yet they can strongly reduce or abolish “mechanical bundling” by jumping the middle and lower parts, but especially so through bass contrary motion:

D -enharmonic 7 (E )

F:E major

ditto (B = C )

D :C major

F : E minor

D : C minor

E 9 : D7 major

G : F 9, etc. major

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The following freetonal or atonal eight-note chain will be adopted as upper part:

(melodic model)

Through the introduction of stronger forms of cadence, the phrase shifts completely in the tonal range:

But already the strict implementation of a similar form of resolution (shift) noticeably widens the tonality:

(tempered contradominant) D minor

D major

(tempered contradominant) B minor

B major

Example fluctuates around a suppressed C minor centre; ( ). Example in the same way around an A minor centre; (the same functions). Should both examples be mixed, then the tonal chain fluctuates around two centres. The sequence is a strong tonal disruption; (NB — take heed of enharmonic change, which has become necessary):

better: A major G major

Detachment from a unified centre of tonality progresses through denial of the primitive employment of the cadence.

counter-trit. counter-trit. variant

7th-countersound

counter-trit. variant

trit. variant

counter-trit. variant

Distant relatives enter caesurally {i.e., through gaps} and thus do not act as definitely as when they form a metrically-matched motif. 323 - English

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Stronger widening of tonality results from the mediant motif. Mediants:

(major)

(minor)

mixed:

A maj D minor

E min B major

or enharmonic

Auxiliary mediants:

C

E

A

C D

(enharmonic)

F

F

A

F

D

D

B

C

E

E

G major

D

E major

(just as mixed as you like)

Minor forms with lower-7ths:

C

A

F

A

B

G

Leading-note chords:

F

E

D

C

D

E

F

G

D

C A

Tritonants:

E

C minor (B ) ”

G

A

B

in 9th-chords:

(major)

B

E (D )

G

C

D

A

F

C

E

A

B

F major

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Minor-key forms as double suspensions coming into concordance:

etc.

effectively: B E (all with lower-7ths)

F

C

B

G A

F

C

B

E minor

Primary collective-changed chords (counter-mediant-variant or mediant-parallel respectively):

F

A

min

D (C )

F

G

E (D )

maj min maj maj

B G

min maj min

F

C

maj

min

D (C )

A

maj

B

E

min min

D (E )

maj

min

G maj

notated as dissonants (changed-tones) as they were: G or A

C

E

A

F

B

F

A

D

G

minor chords major chords minor chords

B

Secondary collective-changed chords ([counter-] tritonant variants with upper and lower 7ths):

B min

E

C

D

maj min maj maj

B

E

maj

D

F

G

min

G

maj min

(G ) F A

maj

A

maj

E

min

min maj

C

G

C

min

maj

min

C

D

min

F

min

min

B

maj

E

A

B

maj min min

A

F

maj

min

A

min

F maj

D (E ) min

A maj

C

maj

E

maj

325 - English

Treatise on Polarity

Metharmonic 7th-countersounds (tritonant and counter-tritonant parallel-chords respectively):

C

maj

E

A

min maj

C

min

D

B

min maj

F

D

min maj

(17-note compass)

C

maj

B

maj

E

min

C

min

A

maj

C

B

min

min

F

maj

A

maj

A

min

E

min

C

maj

D

min

B

maj

F

min

D

maj

The tonality in the examples that implement distant relations is thoroughly decentralised. All phrases quoted here show mechanised shift- (sequence-) forms, i.e., they implement a paired relationship as “motif” in consistent imitation. The functional type thus remains the same within an example. The harmonic event becomes freer whenever each tonal pair displays changing types of relationship. For example (the above groups are inferred):

possibly exchange

Legions of such forms of resolution are possible . . . . Complete decentralisation of the tonal range (results), in which however the individual elements themselves still represent “closed” forms (germinating from a root). If polysonant harmony comes in its place (= double or triple sound, alias dissonance), this results in pronounced ‘atonality’. 326 - English

Treatise on Polarity

To the easily understood tonal chain

comes simultaneously the following sequence, which is in itself just as easily understood: (F )

By omitting individual chord-tones (which however could be kept in other, more complicated cases), the following charming polysonance results:

Polysonance chain — Atonality: Group A Group B A and B, often levelled enharmonically. The tonal-pair sequences acknowledged previously (“horizontal resolution”) major (or F minor) | etc. appear amalgamated (“vertical complex”), and constitute typical new sound forms (polysonances), which are to be rated as specific elements of extreme-modern harmony. consequently, in the above phrase, only three different types of dissonance exist.

The internal progression of function is in fact easy to recognise theoretically, but in reality the effective unambiguity of the cadence- and leading-note influence, as they appear in the tonal sense, is very strongly hamstrung or even completely abolished in the atonal event, for an unambiguous tendency in tonality and note (cadence and leading-note) can take place only where a limited range of tonality and notes draws attention to the individual sound and the individual note in a specific passage and value (“function”). {Thus ends this extraordinary work: no separate conclusion, no summary, no passing comment. One may wonder whether K-E provided such a termination, which the overtaxed publisher suppressed, but this is idle speculation, of no help to those of us who wish to grasp his concepts. It is now just over 75 years since the work was released. History has treated it with contempt. To read and understand it is no simple matter. The time has surely come, however, for English readers with little or no knowledge of German to make their own decision. H.F., August 2007} 327 - English