Noting music, marking culture: the intellectual context of rhythmic notation, 1250–1400 9783775131957, 3775131957

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MUSICOLOGICAL STUDIES & DOCUMENTS URSULA G0NTHER General Editor

NOTING MUSIC, MARKING CULTURE: THE INTELLECTUAL CONTEXT OF RHYTHMIC NOTATION,

1250-1400

AMERICAN INSTITUTE OF MUSICOLOGY ARMEN CARAPETYAN t founding Director

MUSICOLOGICAL STUDIES & DOCUMENTS 46

DORIT TANAY

NOTING MUSIC, MARKING CULTURE: THE INTELLECTUAL CONTEXT OF RHYTHMIC NOTATION, 1250-1400

AMERICAN INSTITUTE OF MUSICOLOGY HANSSLER-VERLAG 1999 68.746

© Copyright 1999 by American Institute of Musicology Hanssler-Verlag, 0-71087 Holzgerlingen, Order No. 68 746

CONTENTS

Abbrevicttions

vi

Acknowledgements

vii

Introduction

PART I: ARS ANTIQ_UA

Conflicts and strategies of conciliation 2

The natural foundation of rhythmic notation

PART

II :

ARS NOVA

67

3

Rhythm mathematized and demystified

4

Musical time and time in general: the challenge from mathematics

102

5

Jacobus of Liege and William of Ockham: the challenge from logic

146

PART

III:

ARS SUBTILIOR

6

The horizon of God's power and the scope of musical innovation

185

7

Late-medieval sophism: a new context for the Ars subtilior

207

PART

8

IV:

THE E ARLY PROTOTYPE OF NASCENT MODER

ITY

God, man, nature, and music

24 9

Bibliography

2 77

Index

2 95 V

ABBREVIATIONS

AjMw

Archiv fur Musikwissenschafi

AIM

American Institute of Musicology

CHLMP

Norman Kretzmann, Anthony Kenny, and Jan Pinborg (eds.), The Cam­ bridge history of later medieval philosophy (Cambridge: Cambridge University Press, 1982)

cs

Edmond de Coussemaker (ed.), Scriptorum de musica nova series a Gerbertina altera, 4 vols. {Paris: Durand, 1864-76; repr. Hildesheim: Georg Olms, 1963)

CSM

Corpus scriptorum de musica

DSB

Dictionary of scientific biography, ed. Charles C. Gillispie, 16 + 2 vols. {New York: Charles Scribner's Sons for the American Council of Learned Societies, 1970-90)

GS

Martin Gerbert (ed.), Scriptores ecclesiastici de musica sacra, 3 vols. {St. Blasien: Typis San-Blasianis, 1784; repr. Hildesheim: Georg Olms, 1963)

JAMS

journal of the American Musicological Society

MGG I MGG 2 New Grove

Die Musik in Geschichte und Gegenwart, 1st edn, ed. Friedrich Blume, 14 + 2

vols. {Kassel: Barenreiter, 1949-79)

Die Musik in Geschichte und Gegenwart, 2nd edn, ed. Ludwig Finscher, 9 + 12 vols. {Kassel: Barenreiter; Stuttgart: Metzler, 1994- ) The new Grove dictionary of music and musicians, ed. Stanley Sadie, 20 vols.

{London: Macmillan, 1980)

vi

ACKNOWLEDGEMENTS

O

o F THE PLEA s u RE s of completing a book is recording the debt of gratitude owed to so many individuals, who contributed in so many different ways to the work. This book represents a learning process that started at my alma mater, the Department of Musicology at Tel Aviv University, and was initiated by Shai Burstyn and Judith Cohen. Shai Burstyn introduced me to the field of late-medieval music and to the broader contextual approach to musical techniques and procedures. His preoccupation with the ideational space of music-making in the fourteenth and fifteenth centuries sparked my imagination and fueled much of my research. Judith Cohen supplied me with countless insights and responded to earlier drafts of my book with wonderfully thoughtful and learned advice. Her uncanny historical instinct shaped my historical perspective and greatly enriched the book. I thank the Faculty of Arts at Tel Aviv University, and especially Nurith Kenaan-Kedar, for unwavering support and loyalty. The topic of this book evolved out of my doctoral dissertation. The erudition and generosity of my teachers in the Music Department and the Medieval Studies Program at the University of California, Berkeley, have become my ideal as a teacher and scholar. I am especially thankful to have been supervised by Richard Crocker and Amos Funkenstein. In this book I continue and develop Richard Crocker's theme of the impact of the Aristoxenian mathematics of continuity and infinity on music theory and adopt his notion of the linear development of musical style. I became a historian of ideas because of Amos NE

vii

Funkenstein. Central ideas of my book were developed through long conversa­ tions with him and reflect his unique interpretation of intellectual trends in the Middle Ages. My deepest regret is that he did not live to see the work published. In different and complementary ways, Jehoash Hirshberg and Lea Dovev played important roles in shaping this work. Jehoash Hirshberg read early ver­ sions, providing perceptive suggestions and criticisms. This book would never have come into being had he not insisted on it, and it would have been the poorer without his tremendous enthusiasm and illuminating research in fourteenth­ century music and its harmonic language. Lea Dovev generously lent me her extraordinary talent, sharpened the philosophical claims presented here, and contributed immensely to the rhetorical presentation of this work. Throughout the years I have benefited from my close association with the Cohn Institute for the History of Science and Ideas at Tel Aviv University. Making the Institute my home, Jehuda Elkana, Sabetai Unguru, Rivka Feldhay, and Menachem Fish provided the best possible intellectual setting for my . interdisciplinary research. I appreciate now more than ever their boldness and erudition, and I am deeply indebted to their unique approach to scholarship. Producing this book required the assistance of many other people. I would like to thank the series editor of Musicological studies and documents, Ursula Gtinther, for her judicious comments and penetrating observations. I am particu­ larly grateful to Jeffrey Dean for helpful copy-editing and for supplemental scholarly information; he took especial pains to ensure that the translations of theoretical quotations have been rendered with due precision and to resolve a number of corruptions in the texts as they have come down to us. I am also grateful to Noah Efron for polishing my English, thereby rendering medieval philosophy and mathematics more accessible to modern readers. I thank Stanley Boorman for preparing the index. Earlier versions of some of the material in Chapters I and 7 appeared in Musica disciplina and thejournal of the history of ideas. A modified version of one aspect of Chapter 5 appeared in Science in context. I have benefited from Shulamit Volkov's highly perceptive comments on these earlier versions and am thankful for her kind offer of financial assistance from the Konrad Adenauer Foundation. Most important are Amir, Gilad, and Galya. This book is dedicated to them. Boston, April 1998

viii

INTRODUCTION iS!iiWWWP+&&thr

T

in late-: medieval musical thoughr is probably a reflection of the apparently troublesome and highly techni­ cal nature of its subject-matter: note shapes and mensural signs, and their practical value. A lack of intellectual enthusiasm is indeed understandable if the sole target of research into this subject is extraction of practical knowledge from the veil of abstract theoretical language. The discovery of notational rules has long been the driving force behind studies of medieval theoretical sources, which accounts for the divorce of the practical content from the broader, enveloping theoretical discourse. The apparently extraneous and abstract dis­ course has been deemed unimportant for a musicological narrative on the evolution of music history in general, and of the history of rhythmic notation in particular. It is true that in their reflections on notational concepts medieval theorists did use a theoretical language that is completely alien to the discourse of contempo­ rary scholars. Medieval definitions of time-units, in terms of processes through which values become endowed with perfection or imperfection or undergo alteration, do not really make sense to us today. A linguistic barrier engenders misunderstanding, and discrepancies between our perception of the content of late-medieval musical thought and the sensibilities, intentions, and meanings that medieval scholars conveyed through their idiomatic figures of thought. Important recent studies have contributed to our efforts at reconstructing the cultural setting of late-medieval theories. The possibility of penetrating the HE RELAT IVELY MODEST INTEREST

INTRODUCTION

rhythmic notions of the Ars nova and Ars subtilior is unthinkable without the pioneering researches of Ursula Gtinther, which laid the foundation, delineated the mathematical deep-structure, and uncovered the cultural background of rhythmic notions between the mid fourteenth century and the early fifteenth. 1 Max Haas's substantial and authoritative work on the medieval system of teaching music theories illuminates the pedagogical context of music teaching and notes the tendency in elementary schools to base rhythmic theories on basic grammar. 2 Haas highlighted the anomalies, already apparent in the late thir­ teenth century, that resulted when rhythmic modes were enlarged and supple­ mented by patterns involving semibreves and minims, which could not be accommodated to the grammatical base that had suited the basic modal system so well. These anomalies forced theorists like Johannes de Muris-whose Notitia artis musicae was intended for younger students at the elementary level-to search for alternative disciplinary frames of reference. Haas showed that basic materials taught in the higher-educational system of the arts faculty were applied to the explication of the increasingly complicated rhythmic practices of the later Middle Ages. In this way, metaphysical considerations associated with the commentaries on the known works of Aristotle infiltrated music treatises and supplied a conceptual system of constraint, within which musical innova­ tions and breakthroughs had to be justified. According to Haas, the new, imported materials were permitted on the proviso that they would suit the level of young students by avoiding controversial and complicated issues, or inde­ pendent critical development of metaphysical positions. Anna Maria Busse Berger has explored a different path. In her Mensuration and proportion signs: origins and evolution, 3 Busse Berger showed illuminating connec1 See for example Ursula Gunther, "Der musikalische Stilwandel der franzosischen Lied­ kunst in der zweiten Halfte des 14. Jahrhunderts, dargestellt an Virelais, Balladen und Rondeaux von Machaut und datierbaren Kantilenensatzen seiner Zeitgenossen und direkten Nachfolger" (Dr. phi!. dissertation.University of Hamburg, 1957); "Die Anwendung der Dimi­ nution in der Handschrift Chantilly 1047", AJMw 17 (1960): 1-23; "Der Gebrauch des tempus perfectum diminutum in der Handschrift Chantilly 1047'', AJMw 17 (1960): 277-97; "Die Mensuralnotation der Ars nova in Theorie und Praxis", AfMw 19-20 (1962-63): 9-28; "Das Ende der Ars nova", Die Musikforschung, 16 (1963): 105-21; "Unusual phenomena in the trans­ mission of late fourteenth-century polyphonic music", Musicd discip!ind, 38 (1984): 87-n8. 2 Max Haas, "Studien zur mittelalterlichen Musiklehre: I. Eine 0bersicht iiber die Musik­ lehre im Kontext der Philosophic der 13. und friihen 14. Jahrhunderts", in Hans Oesch and Wulf Arlt (eds.), Aktuelle Frdgen der musikbezogenen Mitteldlterforschung: Texte zu einem Btu/er Kolloquium des jdhres 1975, Forum musicologicum: Basler Beitrage zur Musikgeschichte, 3 (Winterthur: Amadeus, 1982), 323-456, at 349-90. See also idem, "Die Musiklehre im 13. Jahrhundert von Johannes de Garlandia bis Franco", in Frieder Zaminer (ed.), Die mitteldlterliche Lehre von der Mehrstimmigkeit, Geschichte der Musiktheorie, 5 (Darmstadt: Wissenschaftliche Buchgesellschaft, 1984), 89-159. 3 Anna Maria Busse Berger, Mensurdtion dnd proportion signs: origins and evolution (Oxford: Clarendon Press, 1993).

2

INTRODUCTION

tions between the musical system of measuring time and other practical measur­ ing systems prevalent in the economic-commercial culture of the period. Busse Berger analysed and compared the practical content of a large number of theoretical sources and scrutinized the temporal relations between superim­ posed or juxtaposed time signatures. Her book discloses the dependence of rhythmic thought on the broader practical mathematical knowledge of the Middle Ages and exemplifies the valuable premise that a full understanding of notation requires one to consider the conceptual world of the theorists and musicians that used it. In this book I shift the focus from both pedagogical strategies and day-to-day commercial-economic computations to the speculative, mathematical, philo­ sophical, and theological thought of the late Middle Ages. My study has to do with the impact of specific mathematical and philosophical doctrines on late­ medieval reflections about the articulation of musical time and its representation. Briefly, it seeks to analyse and clarify the seemingly obscurantist metaphysical screen which apparently eclipses the purely practical content. This is, then, an attempt to reconstruct the sensibilities, values, and modes of understanding that constitute the metalanguage of musical thought, focusing on the theoretical language, words, and locutions rather than on practical intents. It interprets, rather than translates or paraphrases, well-known rhythmic treatises that have been hitherto either admired rather than mastered (abov_e all, Johannes de Muris' Notitia artismusicae) or deemed reactionary and narrow-minded, and thus imma­ terial to the history of the Ars nova, like Jacobus of Liege's manifesto against the

My hermeneutical setting is that of contemporaneous mathematical and logical discourses (non-practical, non-commercial), as well as the new philo­ sophical and theological trends of the fourteenth century. In a way, this is a continuation of Haas's pioneering foray into the philosophical context of musical thought, exploring the presence of material taught in the arts faculty within scholastic musical treatises. In contrast to Haas's approach, however, I argue that the musical treatises incorporated mathematical and philosophical ideas beyond the boundaries of traditional intellectual commonplaces. Some­ where in the gap between my perspective and Haas's there lies, I am sure, a fascinating research-space for further investigation. Since I am focusing on the mathematical and philosophical writings of the fourteenth century, I do not pursue further Haas's investigation of documents associated with the teaching of music in the elementary curriculum or in the medieval faculty of arts, such as the special questiones concerning music or commentaries on the central musical textbook, the authoritative lnstitutiomusica of Boethius. Curiously enough, recent studies on such questiones show that an important change occurred between the thirteenth and fourteenth centuries in the treatment of traditional musical issues. The shift involves the application of modernL

3

INTRODUCTION

new conceptual languages and new analytical tools to the elucidation of tradi­ tional musical problems. The point of interest lies in the realization that these new perspectives and insights, these very same conceptual-analytical tools, played a similar methodological role in the independent treatises on mensural music that I propose to interpret here.• This correlation is fascinating; it indirectly supports my thesis that fourteenth-century musical thought in all its articulations was steeped in the new and sometimes revolutionary ideas of the time. Yet comparisons of the independent treatises on the one hand with the textbook commentaries and the musical questiones on the other will be possible only when research on both kinds of sources has advanced much further. The subject and aim of my inquiry demand both a• cutting across separate historical disciplines and a bringing together of notions and ideas that belong to music, mathematics, physics, logic, grammar, semantics, philosophy, and theo­ logy. While I make no claim for a profound and exhaustive scrutiny of any of these fields, I do plead for the merit and fruitfulness of a broad view of possible interconnections between different cognitive spaces of the late Middle Ages. It is here, I admit, that my approach is most vulnerable: it sacrifices meticulous analysis and comprehensive exposition of a single issue or discipline in order to advance beyond a rephrasing of theoretical discourse and reach the level where one can suggest explanations of the cultural sensibilities underlying theorists' intentions and modes of shaping musical concepts, within the conceptual universe that constitutes late-medieval scientific reflections. Deliberately relin­ quishing a sharper focus, I shall instead attempt here to integrate the history of musical thought within the fascinating broader realm of the history of ideas. Setting late-medieval musical thought in the context of the history of science and ideas may help to rectify the asymmetry between a prolonged legacy of expounding late-medieval musical practice (most notably the practice of the motet and chanson) along the whole gamut of its social, religious, and political setting, and the scant attention given to the theoretical ground of this practice. The disparity should not surprise us: there exist no direct indications guiding musicologists to consult historians of late medieval science and ideas in order to cast doubt on the conception prevailing today that rhythmic theories are merely technical literature. Even the known fact that some major music theorists were famous mathematicians has no significant bearing. Johannes de Muris, for example, was a highly esteemed astronomer and mathematician. Yet his math­ ematical works do not clarify the mathematical parts of his musical treatises, and therefore one can not expound his musical thought by means of his other scientific achievements. Despite such unfavorable circumstances, certain recent researches on medieval science have stressed the unitary character of medieval 4 Cecilia Panti, "La scienza musicale nella prospettiva 'Occamista' di un anonimo magister artium del tardo medioevo", Studi musicali, 19 (1990): 3-32.

4

I NTRODUCT I ON

learning, and pointed to the existence of a multiple, parallel embodiment of similar vocabulary, similar research projects, and similar analytical and concep­ tual tools in otherwise different and autonomous disciplines. To understand late-medieval philosophy or theology, historians of medieval intellectual history argue, one must understand late-medieval mathematics and physics. To grasp late-medieval mathematics or philosophy, the argument con­ tinues, one has to master medieval logic. Music, I intend to show, is no exception. To understand the Ars antiqua, Ars nova, Ars subtilior, or even the controversy between antiqui and moderni as unfolded in Jacobus of Liege's manifesto, we must understand late-medieval intellectual culture at large. This study, therefore, is not meant to be a manual of notational theories, in which one would legitimately expect to find a systematic presentation of the evolution of rhythmic notions in the high Middle Ages. 5 Nor is it a history that would show the linear interdependence and progressive modifications of rhyth­ mic notions. My intentions are both less and more ambitious. I will cite only a few, albeit central, sources. But I would like to say more about their content, about their underlying presuppositions, about the set of problems that chart their goals, and about the conceptual frame that circumscribes how far they can proceed. Assessment of the harvest of musical thought may clarify the role of the Ars nova in promoting the reformative and revolutionary trends that justify viewing the fourteenth century as the birth of the phenomenon termed "moder­ nity", in the sense of the incipient demystification of nature. The old relationship between fourteenth-century philosophy and modern thought has recently been claimed to reveal some unexpected linkages not only to post-medieval philosophy but also to twentieth-century thought, through two distinct yet closely related issues: (1) the sweeping concern with the meaning of signs, language, and representation; and (2) the correlated question of the meaning of nature. 6 Present-day scholarship demonstrates a high interest in language, which has given rise to new, typically modern branches of inquiry, such as semiology and narratology, while other disciplines, such as literary criticism, hermeneutical philosophy, structural linguistics, anthropology, Lacanian psychology, and his­ torical hermeneutics converge on the analysis of language. Late-medieval philo­ sophy, too, betrays an unexpected concern with inquiry into universal grammar, with semantic problems, and also with logical inferences. Medieval philosophers asked questions that still absorb us today : how do words signify, what do they ' For a systematic presentation of the history of rhythmic notation in the fourteenth century, see Alberto F. Gallo, "Die Notationslehre im 14. und 15. Jahrhundert", in Zaminer (ed.), Die mitulalterliche Lehre von du Mehrstimmigkeit, 257-356. 6 John Marenbon, Later medieval philosophy {II50-1350) : an introduction (London : Routledge & Kegan Paul, 1 987), 85-6. See also Ludger Kaczmarek, "The age of the sign : new light on the role of the fourteenth century in the history of semiotics", Dialogue, 31 (1992 ) : 509--16.

5

I NTROD UCT I ON

truly signify, and what kind of relation exists between the signifier and the signified? The fourteenth century was the age of the sign. Late-medieval scholars stressed the centrality of logic and linguistic analyses as the most efficient key to solving difficult problems. 7 Medieval philosophers evidently adhered to the very premiss that twentieth-century philosophers have so forcefully discarded : the premiss of classical referential semantics that words are signs of something else, being pictures or images or imitations referring to prior objects in the world. Although contemporary theories of language have liberated words from their referents and developed the new functional semantics in which meaning de­ pends on the functional relations among the words themselves, both cultures, late-medieval and our own, shifted the focus towards semantic behavior and the truth value of written scientific protocol. This similarity is cogently expressed in the Introduction to the new edition of the Cambridge history of later medieval philosophy: By combining the highest standards of medieval scholarship with a respect for the insights and interests of contemporary philosophers . . . we hope to have presented medieval philosophy in a way that will help to end the era during which it has been studied in a philosophical ghetto, with many of the major students of medieval philosophy unfa­ miliar or unsympathetic with twentieth-century philosophical develop­ ments, and with most contemporary work in philosophy carried out in total ignorance of the achievements of the medieval on the same topics. It is one of our aims to help make the activity of contemporary philosophy intellectually continuous with medieval philosophy to the extent to which it already is so with ancient philosophy. 8

In the late Middle Ages, much like today, concern with language was allied to the problem of the relation between language and reality, between words and concrete natural objects, or between words and pre-existing ideas or concepts. Up until the fourteenth century, scientific discourses referred to pre-existing universals that mediated between the real world of singulars and our perception of it. Late-medieval philosophers associated with the movement called Nomin­ alism challenged this Christian-Platonic doctrine and argued that words re­ ferred to concrete, transient, and contingent singular things that neither imitate nor participate in the higher and absolute reality that the universals inhabit. Thereb)"they significantly eroded the traditional Platonic view, which insisted on the ontological and conceptual priority of universals. Challenging the

Marenbon, Later medieval philosophy, 85-7. Norman Kretzmann, Anthony Kenny, and Jan Pinborg, Introduction to CHLMP, 3. See also Claude Panaccio, Les mots, /es concepts et /es choses: la simantique de Guillaume d'Occam et le nominalisme d'aujourd'hui (Montreal : Bellarmin, 1992 ). 7

8

6

INTRODUCTION

mystique of a higher abstract reality, which our world merely reflects and towards which it points, late-medieval philosophers inverted the previous un­ derstanding of the world and the language which describes it. The "beyond" or "above" that was under attack concerned not only the mathematical and concep­ tual "order" beneath the face of nature, but also the net of theological construc­ tions that separated man from an immediate relation with God. In a word, the fourteenth century began what postmodernism has claimed to accomplish : the dedivinization and demythologization of the world. Allegedly beyond the horizon of medieval culture, this theme nonetheless owes its earliest overt pronunciation to late-medieval philosophers. The new attitude towards mathematics that evolved is of special importance for this study. The Pythagorean system of ethical and esthetic values as allied to numbers and forms lost some of its edge and its power to dictate the nature of mathematics and the kind of mathematical relations or foi:.ms that the world "ought" to reflect. The serious attack on the idea that our world is a mere shadow of higher levels of being, beyond the world of life we inhabit, entailed the realization of the absolute reality of our world. Hence it encouraged a new scientific attitude marked by concentration on contingent and sensible particu­ lars rather than on abstract and immutable universals, and it was guided by a new empiricist attitude. Thus, within a typical medieval climate of thought and in response to cultural exigencies and sensibilities that we shall explore throughout this book, late­ medieval scholars initiated serious discussions about the validity of the archaic myth-creating or mystical view that things in the world point to other things that they resemble or reflect by imitation or participation, and advanced the idea that nature is autonomous ; for if our world is not a shadow, than it must be independent and autonomous. In the Nominalistic world, shorn of "living" concepts or universals, singular beings of here and now are no longer a reification of their corresponding higher universals. Consequently, terms or concepts denoting universals were reduced to mere name-tags, that is, pure signs that no longer refer to concrete beings but rather to linguistic constructions. The late medieval preoccupation with the exact meaning of signs, and its close relation to the simplification and demythologization of nature, offer a new perspective from which to discuss the music of that age. This new scientific climate also provides an appropriate cultural setting for confirming the evolution of a new concept of music in the fourteenth century- an Ars nova that truly deserves to be called revolutionary. But how so ? It may appear that these two fourteenth-century themes - the all-embracing concern with linguistic signs and the denial of the mathematical and conceptual "beyond" - should have no bearing on the history of music. Closer scrutiny, however, will reveal this impression to be deceptive. We may start by noting the nearly obsessive concern with musical signs 7

I N T RO D U C T I O N

throughout the fourteenth century. Signs concerned music theorists as much as they concerned philosophers or logicians. Before Nominalism, in the Ars antiqua, musical signs were governed by the realm of the beyond that their alleged perfection inhabited, and for the sake of which, or in relation to which, rhythmic figures suffered alteration and imperfection : the analogical realization of transcendent perfection through rhythmic patterns was a current theme of rhythmic treatises. In the fourteenth century, and with special vigor towards its latter part, composers and theorists developed new desiderata : the utmost rhythmic variety, as attested by the invention of an exhaustive arsenal of new rhythmic objects, and the utmost simplicity and immediacy in the figural representation of those new rhythmic objects, as disclosed by the new nota­ tional style of the Ars subtilior. In this study I show how these radical changes in musical desiderata may be explained within the context of the ontological and epistemological revolution of Ockhamist Nominalism. 9 Ockham with his eponymous "razor" cut away all the metaphysical construc­ tions screening our cognition, such as the world of concretely existing ideas or universals (the Platonic ideas) which for generations had served as the ontologi­ cal backbone of the world and the indispensable instrument for its cognition. Reducing the world to concrete, particular, sensible beings, Ockham revolution­ ized the practice of science by shifting the focus from definitions of universals to discursive reports, to written protocols about particulars and their unique qualitative and quantitative properties. The issue of representing reality in all its variable manifestations by means of a suitable language shifted to the forefront of philosophical interest. To gain full command of the flight from the theoretically endless multiplicity of variant cases that nature can exhibit, or that one can imagine, to their verbal description, Nominalists worked out a theory of meaning that defined for every term, in any possible linguistic context, the nature and extent of its referent. Nominalist philosophers invested a tremendous amount of energy in making language as transparent and unambiguous as possible. It is within this environment and philosophical agenda that I situate the development of rhythmic notation in the later Middle Ages. The unprecedented 9

Nominalism was the dominating philosophical school of the 14th c., and actually com­ prised more than one school : there was the conservative Augustinian branch, termed "right­ wing" Nominalism ; the moderate central school of Nominalism headed by Ockham and termed "Ockhamism" or "moderate Nominalism"; and finally the radical movement, the so­ called "left-wing" Nominalism. Here we shall be concerned mainly with "moderate Nominal­ ism". This school of thought is associated with Ockham, but it continued to exist and to influence the shape of late-medieval thought even after Ockham's death. See William J. Courtenay, "Nominalism and late medieval religion", in Charles Trinkaus and Heiko A. Oberman (eds.), The pursu;t of hohness ;n late mcdt"cval and Rma;sssancc rd,g;on (Leiden: E. J. Brill, 1974), 26-59, at 32-6.

8

INTRODUCTION

interest in developing notation and the quest to represent musical processes in all their possible temporal manifestations resonate in Johannes de Muris' vision­ ary demand that "Everything that is uttered singing with a normal, whole, and regular voice, the knowledgeable musician must write by appropriate notes (Omne quod a voce recta, integra et regulari cantando profertur, debet sapiens musicus per notulas debitas figurare)." 10 Muris' emphatic call -extracted from his famous Notitia artis musicae of 132 1 - has been hitherto unnoticed. Yet it expresses, in one condensed sentence, the new empirical spirit of the fourteenth century and its immense concern with variability and suitable representation. Johannes de Muris, however, failed to carry out his own program. He sought an ideal that he himself not only did not, but could not put into effect. The disparity between his actual achievement and his demand for a system of notation suitable for all possible rhythmic progressions can be partly explained by his traditional and hence ambiguous and inefficient musical semiotics : Muris developed his rhythmic ideas within the given preconceptions and assumptions of the Franconian system. His new ars notandi used the same operative principles that had guided his predecessors : his new rhythmic progressions still had to be "understood" via the metacategory of rhythmic perfection. Although he broad­ ened the rhythmic system of the antiqui and integrated the hitherto forbidden duple values, Muris none the less did not revise the underlying principles of Franconian notation. His discourse on note values remained sympolic and analogical, and hence unclear and ambiguous, where rhythmic figures were still connected with the ideal reality of perfection, imperfection, and the like. Against this background and in the light of the Nominalists' demand for a transparent scientific language, which Johannes de Muris did not provide, the controversy between Jacobus of Liege and Johannes de Muris gains an utterly new meaning. For it is precisely this vulnerable point of Muris' discourse that Jacobus attacked, and for good reason. Unlike Muris, Jacobus was aware of the Nominalist contribution to the improvement of scientific protocols. Using Ockhamist preconceptions and applying current Nominalist methodology, he exposed the deficiencies in Muris' discourse. Jacobus was the earliest theorist to reflect on and criticize the obfuscating mystique of the idea of rhythmic perfection. He went so far as to advise theorists to purge their language of all such vague terms as rhythmic perfection and imperfection. Against the common view that Jacobus of Liege was a narrow-minded conservative, I argue that he should be hailed as a most perceptive and thoughtful analyst of musical thinking about time and its representation. Although he is generally viewed as the last spokesman for the Ars antiqua, I hope to redeem him for the history of 10 Johannes de Muris, Notitia artis musicae, in Johannis de Muris Notitia artis musicae et Compm­ dium musicae practicae; Petri de Sancto Dionysio Tractatus de musica, ed. Ulrich Michels, CSM 17

(AIM, 1972), 47-107, at 94.

9

I N TRODUCT ION

fourteenth-century music by relating his ideas to the most advanced scientific methodology of his age. With this new image of "Jacobus the progressive" the controversy between a ntiqui and modern i will shift ground: to be construed not as occurring between innovators and reactionaries but rather as an expression of two different branches of fourteenth-century Nominalism. Whereas Muris' theory is linked to the new non-Pythagorean, demystified mathematics of the fourteenth century and its theological background, Jacobus' thought belongs to the logic, epistemology, and methodology of Ockham's Nominalism. Both Muris and Jacobus will thus be portrayed as "demythologizers" -Jacobus in his critique of the mythical association of perfection with note shapes, and Muris in his contribution to the dedivinization and naturalization of music. Curiously, though Muris himself did not question the premise of the franco­ nian notation, the new notational practice of the Ars sub til iorliberated signs from dependence on the mediation of metaphysical rhythmic perfection. The demys­ tification of rhythmic signs coincided with the culmination of the frenzy to measure every possible rhythmic idea, so as to exhaust the scope of free play within the parameter of musical time. A similar convergence characterized the new type of fourteenth-century sophistical logic, which concerned propositions describing extremely complicated mensural situations involving intricate mo­ tions measured against the continuum of time. Since linguistic representation and utmost mensural refinement appeared together in both types of sub til ita tes, I will compare the musical Ars sub til ior with the logical sub til ita tes d e motu and offer a new context for understanding both the specific nature of the rhythmic complexity involved in the Ars sub til ior and the evolution of a new method of representation for which nothing existed "beyond" the actually notated figures. The presence of the "beyond" in actual interpretation of musical notions was not confined to the realm of mensural notation. In fact, the most deeply rooted preconceptions about music, its beauty, its correctness, have to do with the tyrannical presence of the reality of the "beyond". Classical and early medieval notions of music were confined within the smothering embraces of two distinct metaphysics. There was, in addition to the "beyond" of divine perfection, the "beneath" of mathematical perfection. Both functioned as realities of a second order that dictated the musical notion of correctness and its preferred esthetic goal. In music, as in mathematics and physics, the long legacy of ethical, emotional, and esthetic values conferred upon mathematical and physical enti­ ties began to collapse. The mathematics underlying the rhythmic innovations of Johannes de Muris reflects a gradual but by no means complete loosening of the grip of the mythical Pythagorean mathematics, which dictated what ought to be exemplified in a world governed by the static mathematical harmony of the spheres. Muris' new mathematics was a measuring tool, capable of quantifying complex natural processes that involved change and variability through a math­ ematical language that was partly stripped of ethical or esthetic value. IO

I N TRO D U C T ION

Muris' rhythmic theory thus presupposed a new understanding of mathemat­ ics itself. With this new mathematics it was not the divine, absolute, and timeless that was drawn into the music, but rather real time, full of surprises, that with its dynamic articulations conveys the whole web of hopes and anxieties that our individual time consists of: in its rich and varied rhythmic texture and its new, flexible, and shifting metric patterns, the music of the fourteenth century seems to bear the countenance of time. One of my principal arguments in this book is that a close reading of major fourteenth-century musical theories, especially of the texts by Johannes de Muris and Jacobus of Liege I have been referring to, shows that, in the realm of music as in the realm of philosophy and theology, serious doubts about the validity of metacategories and metaphysical specula­ tions became articulated in theories as well as in fact. The analysis of major musical theories supports broader historical accounts that chart the development of late-medieval naturalism and empiricism . The inception of the "disenchantment of the world" does not mean seculari­ zation, for one of the salient features of fourteenth-century theology was its preoccupation with mensural questions that were to be solved by mensural theories that shared with the new musical and mathematical discourses a large set of terms, theorems, and operative principles applied to a common subject matter: the infinite and the continuous. In theology, 'the infinity in question was that of God himself, his wisdom, free will, and power, or the infinite distance between him and other beings, or the infinities (small or great) that he could actualize -to mention just a few examples. 1 1 In mathematics, infinity came to the fore with regard to irrational numbers and irrational proportions, while in physics infinity and continuity arose everywhere, in endless attempts to measure processes against the continuum of time or space. In music, the infinity under scrutiny was that embedded within the continuum of time. My thesis should thus not be mistaken for the prevalent argument about the Ars nova constituting a shift towards secularization, as highlighted by recogni­ tion of the duple meter and the growing dominance of secular musical forms. On the contrary, it revises, refines, and in fact reverses this argument. It was the new Nominalistic theology, and above all its central doctrinal distinction be­ tween God's absolute and ordained power ( potentia absoluta- what God could have done without involving contradiction, and potentia ordinata-what he actu­ ally did do in our world) that advanced the demythologization, dedivinization, and naturalization of the universe. 12 The recognition of duple meter and the 11 John E. Murdoch, "From social into intellectual factors : an aspect of the unitary character of late medieval learning", in John E. Murdoch and Edith Sylla (eds.), The cultural context of medieval learning, Boston studies in the philosophy of science, 26 (Dordrecht : Reidel, 1975), 27 1-348, at 289-303. 12 See Heiko A. Oberman, "The shape of late medieval thought : the birthpangs of the

II

INTRODUCTION

new rhythmic language of the fourteenth century will be set not within the context of a seemingly deteriorating religious practice, but within the context of more extensive attempts to narrow the gap between ontological or categorical oppositions, and above all to create a rapprochement between the sacred and the profane. 1 3 Under this crucial reconciliation many other related trends may be subsumed, such as the gradual closing of the gap between perfection and imperfection, harmony and disharmony, motion and rest, variability and uni­ formity, rationality and irrationality, the finite and the infinite. The intimate relationship between fourteenth-century theology and the mathematics and physics of the time is the recurrent background against which the musical achievement of the Ars nova will be redefined and reinterpreted. The case of music illuminates from a new angle the all-pervasive presence of a new understanding of the continuity- rather than the absolute opposition - be­ tween the realm of the good and the ordered and the realm of the evil and the disordered (with all their possible derivatives) in fourteenth-century culture. The new conceptual flexibility affected the understanding of the notion of beauty, harmony, and perfection. This relaxation of rigid and restrictive definitions offers a new perspective and a new explanatory framework to account for the mediation between rhythmic perfection and imperfection in the fourteenth century and the consequent mediation between consonances and dissonances in the fifteenth century. This study interprets the musical developments of the fourteenth century with new figures of thought : "Music in the age of mensura mania", or "Music in the age of the sign", or "Music in the age of growing conceptual relativism". Indeed, these fi gu res of thought may better explain the salient feature of fourteenth-century music as music that was running around measuring rhythms in any possible way, encompassing all possible rhythmical variables, from the large-scale metrical organization down to the single note-value. The age of "measure mania" (as the fourteenth century is often called) 14 created music that was equally absorbed in mensural inventions. "Measure mania" describes not only the fact of far-reaching developments in rhythmic mensural procedures, but also testifies to the campaign against metaphysics and its abstract constructions. To measure means to describe something concrete, rather than something universal, and to express in a most specific and realistic way how such a concrete subject is, say, whiter, faster, or longer than other, similar particulars. This measure mania is coupled with a preoccupation with symbolic representation of modern era", in Trinkaus and Oberman (eds.), Tht pursuit of holiness, 3-25, at 11-15. See also idem, "Reformation and revolution", in Murdoch and Sylla (eds.), Tht cultural context ofmtditval learning, 397-435, at 408-11. 11 1

•

Oberman, "The shape of late medieval thought", 6-15. See Murdoch, "From social into intellectual factors", 340-41. 12

I N T RO D U C T I O N

mensural data, and is not, therefore, the passing fancy of some eccentric and marginal naturalists or empiricists : it converged with the new and central Nominalistic ontology and epistemology. This is a new and positive portrait of the Ars novtt. New, because recent musicological works either diminish (if not reject) the revolutionary character of the Ars novtt, 15 or, under the spell of Huizinga's legendary book The wttning of the Middle Ages 1 6 adopt the image of the declining Middle Ages in their interpreta­ tion of the history of (mainly Burgundian but also French) music from around 1350 to 1 5 00. In light of the positive implications of my own interpretations, I hope to revise the view of fourteenth-century music as reflecting a waning culture, offering in its stead the view of an age of the gradual naturalization of music and the resulting inception of the pre-modern phase in music history. In the first section of the book, I shall address the problem of the clash between the Aristotelian qualittttive metalanguage of rhythmic theories in the Ars ttntiqutt and the quttntittttive metrics that constitute the language of these theories. I shall then present an analysis of the ambiguous and cumbersome system of mensural notation in the light of contemporaneous theories of semantics and grammars that in themselves project the Aristotelian ontology and physics onto linguistic constructions. Here, as throughout the book, quotations from primary sources are given in Latin and followed by the English translations. Chapter 3 focuses on the revolutionary anti-Aristotelian and anti-Pythagorean mathematics of the fourteenth century and its relations to the innovations of the Ars novtt. Chapter 4 expands upon the previous discussion of the relation be­ tween music and mathematics, focusing on the musical and mathematical analy­ sis of continuity and infinity, the inner composition of a continuum, whole­ to-part relationships within a continuum, and the problem of defining and locat­ ing a minimum within a continuum. Chapter 5 deals with Jacobus of Liege and his progressive ideas, shifting the focus from mathematics to the logic and scien­ tific methodology of the fourteenth-century Nominalist movement. A new read­ ing of Jacobus as an Ockhamist thinker provides the basis for an analysis of the controversy between him and Johannes de Muris and other moderni. Chapter 6 concerns the theological impetus behind the new scientific attitude of the four­ teenth century as reflected in the mathematical and musical theories of the time. Chapter 7 deals with the Ars subtilior and its relation to logical-mathematical subtleties in which the mathematical and logical-linguistic analysis of the prop­ erties of time as a continuum reaches its utmost refinement. Finally, Chapter 8 deliberates on the role of the fourteenth century in the transition from the Mid­ dle Ages to the early-modern culture of the fifteenth and sixteenth centuries. For a survey of the relevant current literature, see below, pp. 68-9. Johan Huizinga, The waning of the Middle Ages, trans. F. Hopman (London : E. Arnold, 1955) . 15

16

13

P A RT I A R S A N T I Q UA

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Aristotelian Scholasticism and music

entitled "The influence of Aristotle on French university music texts", Jeremy Yudkin reminds us that "We cannot fully understand the writings of thirteenth-centtity theorists on music, unless we appreciate the extent to which they incorporate the vocabulary, logical systems, and mode of thought of Aristotelianism." 1 For Yudkin, as for many other scholars, recogni­ tion of the Aristotelian resonances behind much of the vocabulary used in thirteenth-century scholarly writing allows us to read these documents with "an eye that is keener and an ear that is more sensitive to the nuances of the text". 2 I agree. Still, to date, the maps of meanings produced by efforts to relate musical theories to Aristotelian philosophy have neither led to new conceptual territo­ ries nor generated dramatic new insights. Yudkin's description of the impact of the Aristotelian method of classification upon the division of musical notions and the organization of musical treatises echoes Lawrence Gushee's analysis of the Aristotelian influence on medieval mensural treatises. 3 Yudkin's particular interest is in identifying the precise

I

N AN ART I c LE

;

1 Jeremy Yudkin, "The influence of Aristotle on French university music texts", in Andre Barbera (ed.), Music theory and its sources: antiquity and the Middle Ages (South Bend, IN: University 2 of Notre Dame Press, 1990), 1 73-89, at 178. Ibid., 189. 3 See Lawrence Gushee, "Q!!estions of genre in medieval treatises on music", in Wulf Arlt, Ernst Lichtenhahn, and Hans Oesch (eds.), Gattungen der Musik in Einuldarstellungm: Gedmk­ schrifi Leo Schrade (Bern: Francke, 1973), 365-433. Gushee notes the influence of the Aristotelian idea of hierarchy on the overall organization of the treatises and on their order of presentation,

17

CONF LIC T S AND S T R AT E G IE S OF CONC ILIATION

sources in the original Aristotelian texts of some of the borrowed terms and notions that were appropriated for musical usage by scholastics. Such taxonom­ ical identifications, Yudkin believes, help illuminate the Aristotelian meaning of every single term and enhance our understanding of the musical texts. Tracing the Aristotelian genealogy of thirteenth-century musical concepts leads Yudkin to several conclusions. He notes, for instance, that Johannes de Garlandia's De mensurabili musica of c. 1260 applies an Aristotelian scheme of classification, dividing the genus of organum into its three species : discant, copula, and organum. He also observes that Garlandia's definition of discant, aliquorum diversorum cantuum sonantia secundum modum et secun­ dum aequipollentis sui aequipollentiam the sounding together of various diverse melodies according to mode and according to the equipollence of its equipollent part, 4 includes the term modus, which according to Yudkin is intelligible only in the light of its previous history in Aristotelian logic. There it described the combi­ nations of different categorical propositions in a syllogism, and involved the notion of possibility. In addition, Yudkin notes, it was also at the root of an important concept in the new (non-Aristotelian) speculative grammar, the modi significandi that mediated between being and meaning. Garlandia's use of modus, then, echoes the word's use in logic and grammar. 5 Yudkin's conclusions seem unexceptionable at first, but become more tenuous when subjected to close scrutiny. First of all, the two origins he proposes for Garlandia's modus call for two utterly different interpretations of musical theo­ ries. Furthermore, even if either (or both) genealogies were demonstrated con­ clusively, it is not at all obvious what we might learn from the attribution of namely, from the simple to the composite, from the general to the particular, or from essence to substance. Yet for Gushee the effect of this influence is not completely positive : "the Aristote­ lian component has a negative aspect. The modern reader rapidly discovers that common sense is a poor guide for the interpretation of the theory of music ; and thereafter, that reading Aristotle does not solve all problems, inasmuch as Aristotelianism is not necessarily Aristotle" (p. 425). " �otation and translation from Yudkin, "The influence of Aristotle", 183. Yudkin's "sound­ ing together" translates "consonantia", the reading of MS B and the indirect witnesses to Garlandia's text; see Johannes de Garlandia, De mmsurabili musica: kritische Edition mit Kommcntar und Interpretation der Notationslehre, 2 vols., ed. Erich Reimer, Beihefte zum AfMw, 10-11 (Wiesbaden: Franz Steiner, 1972), 1 : 35. The awkward word "equipollent", which will appear frequently in the later part of this book, is a technical term of logic, "said esp. of propositions which express the same thing, notwithstanding formal diversity" ( The Oxford English dictionary (Oxford: Oxford University Press, 2 1989), s.v., A.3.c), in other words "equivalent". Here it means "of equal rhythmic value, even if of different note-values", e.g. an imperfect long and an altered breve, both lasting two tempora; see below. ' Yudkin, "The influence of Aristotle", 183-4. 18

A R I S T O T E L I A N S C H OL A S T I C I S M A N D MU S I C

origin. Admitting that Johannes de Garlandia borrowed the term modus from Aristotelian modal logic or from contemporary grammarians does not in itself enhance our understanding of the given musical theory of rhythmic notation. This is not to say that identifying the sources of musical terms is not valuable. But such identifications, made by invoking concordances, are not enough and cannot be seen as ends in themselves. Mere identification of antecedents tells nothing about possible cross-fertilization between music and its cultural con­ text. It does not go far enough. It is the historical significance- including the immediate effect and the future consequences - of basing a rhythmic theory on Aristotelian matrices that calls for an interpretative analysis. In undertaking such an analysis I make two assumptions. First, I assume that any notational theory is a semiotic system. Understood as a semiotic system, notation emails philosophical and mathematical subtexts. Even if theories of notation may seem to be no more than practical guide-lines for composers and performers, they involve such primary notions as the very idea of regulated order and its substantiation through the interrelationships of the whole and its parts. Theories of notation, then, inform us about what constitutes a "work" they exemplify by their level of specificity the degree of available openness in a given style, as defined by possible margins of variances. Second, I assume that Aristotelian philosophy, like all the other schools of thought that will figure in this book; functions not as a repository of technical terms, but rather as a world-view within which terms must be decoded. As world-views, philosophical schools provide not just terminology and procedures for reasoning, but also systems of ethical, cultural, and esthetic values, as well as notions of correctness defined through conceptual constraints or regulative principles. Thus, in the following analysis of theoretical sources, I will not isolate Aristotelian notions, as Yudkin has already done so successfully, nor will I relocate them in their original settings. My purpose here is different. I wish to explain contradictions and ambiguities in theories and, above all, the cumber­ some and enigmatic signs and syntax of mensural notation by invoking the Aristotelian universe in which they find their logical base. My aim, then, is on the one hand to decode the peculiar semiotic behavior of the mensural system of notation so as to illuminate its more opaque qualities, and on the other hand to disclose anomalies and discrepancies arising from heretofore overlooked ten­ sions between Aristotelian philosophy and mensural theories. Before proceeding further with my argument, however, let me review briefly some crucial points in the history of the reception of Aristotle in the Latin West. The reception of Aristotle in the thirteenth century marked a turning-point in the intellectual history of Western Europe. Aristotelian notions guided philoso­ phers and scientists away from the prevailing apodictic-descriptive science towards argumentative and demonstrative modes of discourse. Thirteenth­ century philosophy, called Scholasticism, spoke an Aristotelian idiom even 19

C O N F L I CTS A N D STRATE G I E S O F C O N C I LIATION

when i t differed from Aristotle. Aristotelianism became the official language of medieval learning in general ; it provided the dominant discourse for philoso­ phers who sided with Aristotle, but also (and more interestingly) for philoso­ phers who disagreed entirely with the Aristotelian world-view, or who criticized and even labored to amend some of his doctrines. The ability and willingness to speak the language of Aristotle was the identifying characteristic of intellectuals as a social group, whatever their variegated and conflicting philosophical points of view. Since Aristotle stimulated more than one system (and even opposing systems) of natural philosophy or theology, we must be alert to the immense differences between all possible usages of Aristotle's principles and notions, and especially to the crucial difference in using them to preach his world-view, or to amend or dismiss his doctrines. Stated differently, the modern scholar must distinguish between true acceptance of Aristotelian tenets and the mere use of an Aristotelian idiom for the sake of talking in the most fashionable mode of discourse. In this latter, stylish use, Aristotelian notions were sometimes mis­ interpreted and sometimes extraneous and immaterial to the issue under scru­ tiny. As a result of the great variety of ways in which Aristotelian philosophy was put to work at the time, evaluating Aristotelian influence on musical theories is neither simple nor one-dimensional. We must always be mindful of the range of usages of the Aristotelian idiom, spanning from mere rhetorical embellishment or conventional usage to sincere intentional commitment. The complexity of assessing the influence of Aristotle on thirteenth-century thought about music becomes still more pointed when one recalls that the reception of Aristotle- the translation and transmission of his works - did not occur in a void. Some of Aristotle's works on logic (the Categoriae and De interpretatione) had long been known in the West through the translations of Boethius (fifth century AD) and other Latin and Arabic philosophers. Other Aristotelian arguments had been transmitted through Platonic-Augustinian channels. Aristotelianism, in fact, appeared on the scene after a Christian philosophy steeped in Platonism had already been well accepted. Platonic notions were integrated with Christianity in a seamless way that made it impossible to distinguish the former from the latter. In the same vein, the revival of Aristotle was not accompanied by total and blind subservience to his philosophy. From the outset, Aristotle was criticized or amended - at times with the use of his own methods. Some of his maxims clashed with central creeds of Christian dogma (for example, his belief in the eternity of the world contradicted the Christian concept of creation ex nihilo ). Others of his tenets suffered from internal inconsistencies or deficiencies that his critics wished to correct. Corrective theories provided space for original and innovative developments which advanced science beyond the scope of Aristo­ tle's thinking and sometimes against his explicit pronouncements and impera­ tives. 20

A R I S T O T E L I A N S C H O LA S T I C I S M A N D M U S I C

I n the domain of Christian philosophy, Aristotelianism was coordinated with traditional Christian-Platonic dogma and assimilated within one or another comprehensive philosophical synthesis such as what has sometimes been called the "Thomistic Compromise". Such syntheses reflected a cautious rather than a sweeping absorption of Aristotle. Thomas Aquinas, in his philosophical reconciliation of the Aristotelian and the Christian-Platonic traditions, uprooted Neoplatonic positions from their original grounds and reasons and presented them as consequences of Aristotelian notions and principles. 6 The abundance of quotations from Aristotle so characteristic of Thomas's theological and philo­ sophical writings should not blur the radical differences between his system and Aristotle's. Philosophical or theological systems like Aquinas' were multilingual ; they were often subtle combinations o f a variety o f earlier traditions. Similarly, musical theories in the age of Scholasticism were also composites, in which different and even conflicting schools of thought coexisted and were inter­ twined in different ways by different theorists. The rhythmic theories of the school of Notre Dame, for example, merged the Augustinian tradition of quantitative rhythmic patterns - which underlies the six rhythmic modes -with the Aristotelian qualitative language employed in Aristotle's physics. The crys­ tallization of the Franconian system signaled a yet more advanced phase in the influence of Aristotle on musical thought. In this system, the operation of rhythmic entities was regulated according to Aristotelian physics, which in turn provides the key to the understanding of their semiotic behavior. Because there is a clear demarcation between pre-Franconian and Franconian teaching, I ques­ tion Yudkin's emphasis on the close dependence of Franco's Ars cantus mensurahi­ lis on earlier treatises, and his consequent underestimation of the Franconian conceptual shift. In assessing the reception of Aristotle in the domain of musical thought, we must, therefore, be aware of the many layers of medieval texts. We will encoun­ ter superpositions or juxtapositions of several systems, tacked together in a vari­ ety of ways, sometimes approaching integration and synthesis and at other times creating artificial or ad hoc composites. We must also be sensitive to the many different types of motives for invoking any particular Aristotelian notion or doctrine. As I have already observed, the fact that a theorist referred to Aristotle does not automatically imply that he was an Aristotelian or that his theory should be interpreted exclusively within the ideational universe of Aristotle. This observation is crucial : it bears on the status of authorities in general, and will be essential when scrutinizing both the old theories of the Ars antiqua and the new theory of the Ars nava. 6 See Charles A. Hart, Thomistic metaphysics (Englewood Cliffs, NJ : Prentice-Hall, 1959) ; see also Robert J. Henle, Saint Thomas and Platonism : a stu4Y ofthe Plato and Platonic texts in the writings of Saint Thomas (The Hague: Nijhoff, 1956).

21

C ON F L I CTS A ND STRAT E G I E S OF C ON C I L I AT I ON

The fact that the musical treatises of the thirteenth century reflect the Scholastic propensity for the harmonization or at least fusing of different, sometimes conflicting, teachings creates a thorny problem for scholars wishing to identify the precise sources of musical notions. To the extent that thirteenth­ century music theories concern the rationalization and organization of time in music, they may allow the convergence of independent concepts of time as canonized by different segments of medieval society : the Christian notion of time, biblical time, the Scholastic-Aristotelian notion of time, the mythical notion, the symbolic notion, and many others. As we shall see, the heterogene­ ous concept of time affected musical thought first and foremost in the very disposition towards tolerance of such heterogeneous conceptions. Complexities and inconsistencies in musical theories may be at least partly explained by disentangling the coexistent traditions that constitute theories of music in the high Middle Ages. Thus, instead of a neat map of direct paths of "influences", I will delineate a rather vague chart of intersecting intellectual milieus not always so well marked as to guarantee a safe historical passage. Pre-Franconian theories, neither entirely committed to the Aristotelian mode of thought nor entirely subservient to the Augustinian rhythmic model, com­ bined both schools of thought into a loose system. They synthesized the quantitative discourse of the rhythmic modes, as adapted from Alexander de Villa Dei's Doctrinale, with a conceptual scheme of qualitative contraries follow­ ing the procedure of Scholastic-Aristotelian discourses. The Franconian system incorporated a sweeping projection of Aristotelian physics and metaphysics onto all rhythmical phenomena, whereby mensural music became an exemplary universe of Aristotelian order. This is not a merely "philosophical" observation ; what I a m suggesting here i s the possibility that the way the Franconian figures or signs signify becomes meaningful and defensible only within the context of the Aristotelian universe. Without sensitivity to specific Aristotelian demands, Franconian notation appears riddled with shortcomings and ambiguities. With­ out attending to the exigencies of the Aristotelian system, historical interpreta­ tion of Franconian semiology is reduced to claims about the immaturity of rhythmic notation. Within its Aristotelian conceptual climate, Franconian nota­ tion turns out to be satisfactory, reasonable, and responsive to the intellectual expectations of theorists searching for rhythmic notation in an age dominated by an Aristotelian world-view. From the perspective of the history of musical style, the Franconian theory was the first to set up a conceptual frame for articulating time in music that could replace the traditional verse-like or metrical principle of organizing musical time. The Franconian solution, through its tacit distinction between meter and rhythm, determined for generations to come the principle of rhythm as split into two levels of organization, which maintain a subtle balance between disorder and order, contingency and necessity, time and timelessness. As I will 22

A R I S TOT E L I A N S C H O L A S T I C I S M A N D M U S I C

show, the theoretical model that served Franconian theory i n its search for controlling, delimiting, and rationalizing rhythmic contingency was homolo­ gous to the Aristotelian and Thomistic model of mundane contingency and its relation to transcendental necessity. With the notion of rhythmic perfictio, the Franconian theorists shifted the locus of the principle of order to a level that both transcended and at the same time took part in the actual deployment of rhythmic values in real compositions. This dual position was attributed to the divine Idea or the ultimate Good in its relation to the earthly world, in Aristotelian metaphysics as well as in Thomistic theology. Rhythmic modes, the regulative principle of time until the middle of the thirteenth century, were instrumental to music that aimed at the absolute and denied the earthly, that is, time and motion. The principle of rhythmic modes, in other words, was expedient for evoking the changelessness or circularity that approaches the idea of music without motion. By dividing musical time by means of the notion of perfictio, while allowing real . time to be articulated through the relatively variegated rhythmic surface, theorists captured the es­ sence of the Christian-Platonic notion of real time as intimately bound with timelessness. A deeply rooted fear of time, as the symbol of contingency, capriciousness, and death, seems to be alleviated here, but not at the expense of the experience of time. The Franconian style invokes both time and timelessness. In musical practice regular periods of perfection are relaxed through a more variegated rhythmic surface. But in theoretical reflection this tension between time (motion and change) and timelessness was detached from its original Augustinian-Platonic basis, to be passed through the prism of Aristotelian insights. A ristotle, time, motion, and world order Motion as a source of bewilderment, disorder, or distress is a major Greek theme, one that survived in the Latin West until the scientific revolution of the seventeenth century. This bewilderment is common to what is otherwise a quite diverse group of Greek philosophers, who abhorred motion and saw it as an evil and a horror. Aristotle believed that the principles of nature consist of pairs of contraries and lists a number of such pairs extracted from several philosophers of the Greek tradition : limited/ unlimited, one/ many, rest/ motion, odd/even, good/bad, and so on. A vertical division of the contraries results in two groups, that of the positive and good things - the column of the principles of order­ and that of the negative and indefinite things -the column of the principles of disorder. In contrast to his predecessors, Aristotle argued that the most abstract_ and primary pair of contraries is related not to any attribute of a being, but to its very existence ; that is the contrariety betweenform (being) and privation. Privation, 23

C O N F L I C T S A N D S T R AT E G I E S O F C O N C I L I A T I O N

unlike negation, presupposes the possibility o f becoming a being. Becoming can not take place ex nihilo. Therefore, Aristotle imposes the notion of privation which does not refer to not being absolutely and per se, but to being only in potentiality. Being precedes privation, and is also absolutely preferable. Being, for Aris­ totle, is the realm of the things that are knowable - it is the realm of order. Non­ being is the realm of the unknown and the indefinite ; it denotes disorder. Accordingly, Aristotle took this view : "Again, in the list of contraries one·of the two columns is privative, and all contraries are reducible to being and non­ being, and to unity and plurality, as for instance rest belongs to unity and movement to plurality."7 Yet Aristotle recognized the true existence of motion, plurality, and other manifestations of disorder here and now in the physical world. These he subjected to a rigorous system of controls that delimits, regulates, and predetermines the structure of all natural processes. The becoming of an essence or of a predicate is always specific : it is engendered from its privation, that is, from its partial contrary. There is no contingency in Aristotle's classification of absolute and mutually exclusive essences. Aristotle's cosmos is fully ordered, and is governed by strict regulative principles. Aristotle explicated and clarified his world-view in the Metaphysics, using two parables : the parable of the army and the parable of a household. Through the first parable, Aristotle examined the way in which the "highest good" (God) is integrated into the world. Aristotle asks whether the highest good is manifested as "something separate and by itself, or as the order of the parts". In other words, is the principle by which the world is ordered transcendental or immanent ? And he answers : "Probably in both ways, as an army does ; for its good is found both in its order and in its leader, and more in the latter, for he does not depend on the order but it on him."8 The second parable compares the world to a house and illustrates the unity and correlations among the world's manifold orders of beings. A house consists of essentially varied elements. Yet each element contributes to the well-being of the whole. Free men are like celestial bodies, the "behavior" of which is absolutely regular and ordered. Slaves and animals are like entities in the sublunary world, which live at random and are contingent ; that is, they contain an element of disorder. But the sublunary entities and the celestial bodies together comprise the universe ; the two spheres co-operate for their mutual interest. The relation between the highest good and the world in general, and the relation between the celestial and sublunary world in particular, is governed by 7 Aristotle, Metaphysics, D 2, 1004b26-30. For the English translation used throughout this book, see The basic works ofAristotle, ed. Richard P. McKeon (New York : Random House, i941). 8 Aristotle, Metaphysics, M i o, 1075•1 2-14.

A R ISTOT L E , TIM E , M OTION , A ND WOR LD ORD E R

the principle of imitation. Mimesis- the motivation to resemble God, the prime mover - is shared by all beings, the most noble and the most inferior. The prime mover, or the unmoved mover, is the cause and principle of all motion in nature. It is eternal, ultimately perfect, immutable, and one. The celestial bodies imitate its eternity, oneness, and motionlessness in two ways. First, they are composed of the fifth element, ether, which is pure. The fifth element never mixes with sublunary, impure elements. Second, they travel in eternal circular motion, at a velocity and in an orbit which is continuous and absolutely uniform. The motion of the celestial bodies was postulated to be circular, for the circle is the most perfect of all geometrical forms. A uniform circular motion is the closest approximation to a state of rest, or the motionless state of the prime mover. In the sublunary world entities are made of matter, which is naturally generated and naturally ended. Their yearning to. imitate the world's principle of order is reflected through the eternal cycles of all natural phenomena. These cycles embody the preservation of the species. All beings have a natural drive to give birth to a being carrying and preserving their own image. This striving to perpetuate one's essence reflects an attempt to imitate the unity and eternity of celestial bodies and of the highest good - the prime mover. The principle of imitation is associated with the principle of regularity. Natural phenomena inevitably recur through the same processes : a man gives birth to a man, a cloud to a cloud, and so on. Hence, cyclical motion is the principle of becoming in nature. As such, it is the manifestation of the world's order, regularity, and unity. In Aristotle's philosophy, circular motion is also the basic unit of measuring time. The sun's uniform, regular, and known motion measures all natural processes of changes, be they changes in place or in quality. Aristotle feared infinity, and argued that the infinite is neither a substance nor an attribute or dimension of a substance. In the world, infinity can exist only in potentiality, and only by a continuous process of diminution. Infinity is ruled out in any context, be it physics or metaphysics or even logic or the realm of discursive reasoning. Thus, the primary principles of nature guarantee a system­ atic and logical theory of physics. The limited series of predicates that a given subject may acquire makes it possible to define each subject per se: the prime mover delimits the chain of causes, and thus makes a causal explanation of motions and changes possible. All such natural processes, according to Aristotle, can be reduced to processes in which one entity A becomes another B (wood becomes a table, water becomes ice, etc.). To understand fully the processes of becoming, one must consider Aristotle's four "causes": the essential cause, the material cause, the efficient cause, and the final cause ( causa formalis, materialis, efficiens, finalis ). The essential cause defines what sets B apart from A and all other beings. The

C O N F L I C TS A N D STRAT E G I E S O F C O N C I L I AT I O N

form i s the essential quality o f B; thus the form defines B as a clear and distinct being. Forms can be essential or accidental. They are eternal, uniform, un­ changeable, and indivisible. The material cause is the matter from which a being is constructed. Matter is a substratum common to both the entity that existed and the entity that has come to exist. For example, wood and a table have a common material cause. Matter can be realized in an infinite number of different forms. This gives rise to the principle of potentiality. Nothing comes from nothing (ex nihilo nihil fit). Therefore, if A becomes B, then B must have existed passively in A. B is thus transformed from the state of being in potentiality to being in action. Any becoming, thus, is being in action, coming from being in potentiality. Aristotle defines the state of being in potentiality as merely "privation" ( privatio ). Matter is filled with possibilities. While forms or essences are universal and necessary, matter embodies the indefinite aspect of nature. Matter redeems the form from its universality. The fusion of the form with matter engenders a given specific entity. The relation between form and matter is that of hypothetical necessity : the existence of matter does not necessarily entail its uniting with a form. But if form exists, then matter is necessary. The efficient cause is nature. Nature, Aristotle explained, creates like an artist, and the artist creates like nature. Art proceeds through two stages: thinking and doing. The artistic product exists first in the mind of the artist as a form (either the essence of the product or its objective shape). The artist then contemplates what is needed in order to make that form. This stage is followed by that of making. Thus, art recapitulates nature. Just as in nature man creates man, in art a house (the idea of the house in the mind of the artist) creates a house (the artificially made house). Art and nature are analogous in several ways, according to Aristotle. Nature, like an artist, preconceives its goal rationally and prepares all that is needed for its accomplishment. Just as a painter prepares preliminary sketches, so nature first creates the contours of a fetus, and later its parts acquire their characteristic properties. Both nature and art yearn for a certain good. Every art and every inquiry, and similarly every action and pursuit, is thought to aim at some good. This leads to the dominant and most important of all the four causes, the final cause. Aristotle's world is necessary and predetermined. Every physical process, every becoming, is determined a pn·ori and conditioned by its telos. This condi­ tioning is the principle of materialization. A telos is that for the sake of which a thing comes to be, or its natural end-state. A telos is the essence of the end­ product, or the result of the becoming. The final cause accounts for the absolute interdependence between matter and form. Furthermore, it often coincides with the formal cause, for the form and that for the sake of which a thing comes to be are one.

AR ISTOTL E , TIM E , M OT ION, A ND WOR LD ORDE R

Here we are presented with the most striking aspect ofAristotelian physics. If to account for physical phenomena means to specify their four causes, then a physical condition or process is explained by the very definition of its final stage - by specifying its essence (fanna) and matter. For scholastic Aristotelians, nature is reduced here to its etymological root - the verb natus est, denoting the essential quality endowed in birth. The exact nature or essence of each natural entity is defined by its proper genus and specific difference (differentia specifica), which is unique to the being and distinguishes it from all other species classified under the same genus. Since it is a distinguishing characteristic, each specific difference can appear in one species only. For example, "rationality", which distinguishes human beings from all the other species of animals, cannot be the specific difference or the essential quality of any species other than human beings. This specific difference is the essence of a substance -shared by all the individuals of a given species. It is, therefore, a universal concept. The predicates that individuate a substance are grouped, according to Aristotle, in nine cate­ gories : quantity, quality, relation, place, time, state, possession, activity, and passivity. Aristotle's ontology postulates that every being is either a substance or a predicate of a substance. A substance and its predicates are ontologically dependent : predicates, qualities, and universals cannot exist apart from the subject or substance which manifests them. Order and rationality are found not only in genera and species. All classes of beings are interrelated and form a unified and rationally organized cosmos. Aristotle's teleology differs from other mechanistic Greek theories, which argue that processes depend only on the efficient causes, the effects of which may be either unpredictably necessary or simply contingent. The atomists give such an explanation of physical events. According to the atomists, the position of every atom is the result of clashes with other atoms, the position of which, in turn, was determined by prior clashes. These clashes can be traced infinitely. Aristotle, however, rejected purely mechanistic theories of this sort, insisting that natural entities are created always in the same way and never from an arbitrary constellation of efficient causes. His stubborn resistance to mechanistic accounts of physical processes highlights the altogether qualitative nature of Aristotelian physics. It is this principle of actualization or fulfi11ment of a pre­ determined goal - and as a rule the actualization of a specific essence, quality, or nature - that guides Aristotle's physics. When an entity fulfi11s or reaches its final qualitative state for the sake of which it has come to be, it reaches the state of perfection. In the Aristotelian system, mathematics was ruled out of physical accounts on the ground of the vice of metabasis, the injunction against mixing categories or concepts that belong to different genera, and even different species. Accordingly,

C O N F L I C T S A N D S T R A T E G I E S O F C O N C I L I AT I O N

not only was physics incompatible with mathematics, but also one mathematical discipline was conceived as categorically distinct from the other. Geometry, for example, was incommensurable with arithmetic because the former was con­ cerned with continuous magnitudes, while the later was confined to discrete magnitudes. Aristotle therefore counseled against quantifications or measurements of temporal processes, because time was not a mathematical object. With the exceptions of static structures and uniform simple and periodic motion (which can be construed in a sense as motion as well as rest), mathematics was deemed of no use for studying nature. The force of the prohibition was so strong that it ruled out comparison of motion along a curve and motion along a line, since lines and curves belong to different species. Owing to this taboo, Aristotle forbade numerical expression of intensive changes in given qualities, or changes in duration or velocity. Notions such as rate of change, acceleration, retardation, diminution, augmentation, and the like were far beyond the horizon of Aristote­ lian physics, for Aristotle maintained that "there cannot be motion of motion or becoming of becoming, or in general change of change".9 Aristotelian physics denied the key notions of early modern mathematical physics, namely the notions of quantification in general and the quantification of intensive or qualitative changes in particular. It also denied the very core of an applicable rhythmic theory. This being the case, we might ask what kind of a physics it was, or stronger still, was it a physics at all ? To the extent that its main goal was to reach definitions by cataloguing entities into their classes, Aristotelian physics was a dictionary or inventory of beings and their various predicates. Its proce­ dure of explanation was simply to describe the four causes that together comprise the definition of the essential qualities of beings. The Aristotelian universe, as a necessary context of hierarchically ordered essences, was governed by three regulative principles : that of imitation (mimesis), that of repetition or recurrence, and that of materialization.

Musical theories of rhythmic measurements in-the age ofScholasticism As my brief description has demonstrated, Aristotelian physics was unable to provide the theoretical foundation for a theory about the quantification of durational values in music. How, one may wonder, did music theorists treat rhythmic measurements in light of the anti-mathematical attitude of Aristotelian physics so prevalent in their day? The Aristotelian metalanguage that served music theorists clashed from the outset with the basic discourse concerning the relation between longa and brevis 9

Aristotle, Physics, E2, 225b 15.

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within the six rhythmic modes. There has been heated debate about the origins of the rhythmic modal system cultivated by the Parisian school at Notre Dame. Much of this debate has focused on the question of whether or not this system was modeled on the Augustinian theory of quantitative poetical meters. 1 0 Leo Treider made the important observation that the crysta11ization of the modal patterns in practice and the conceptualization of the modal system in theory were two different processes. Accordingly, he argued that the undeniable fact that "the explanations of the modal system bear the conceptual stamp of the tradition of metrics"• • does not prove "the assumption of an a priori system" and does not prove that "modal practice was theory-induced". 1 2 Max Haas reaffirmed the association between the rhythmic modes and the poetical meters, stressing the correlation between the abbreviated list of only six pedes in the Doctrinale of Alexander de Villa Dei and the corresponding six rhythmic modes. Haas noted that Alexander's treatise on meters was the standard grammatical text of the elementary curriculum at Paris, thus strengthening the likelihood that mu­ sical theories of rhythm were dependent on contemporaneous grammatical theories. 1 3 The fact that the earliest measured music took the form o f metrical organiza­ tion should not surprise us in the light of the close association of metrics with oral delivery of prose and the oral transmission of music. Since this connection has been established by cultural historians as well as by musicologists, it needs no further elaboration here. However, further examination of rhythmic treatises of the Ars antiqua shows that their authors could not rely exclusively on the prevalent literature of quantitative poetical meters. The quantitative ars metrica had to be conveyed and taught using the dominant Aristotelian qualitative discourse. Two incompatible philosophical languages became interlocked : the quantitative language of the old metrics and the qualitative conceptual language of Aristotle. Together these led to a discordant association between the language of rhythm (breve, long, etc.) and the reflective metalanguage of theory.

10 William G. Waite argued that the theory of the Ars antiqua was modelled after the quantitative meters; see The rhythm of twtlfth-crntury polyphony: its theory and practice (New Haven : Yale University Press, 1 954). Leo Treider disputed Waite's theory and argued that thirteenth-century music was accentual ; see "Regarding meter and rhythm in the Ars antiqua", The musical quarterly, 65 (1979) : 524-58. Ernest Sanders rejected Treider's position ; see "Com­ ments and issues",JAMS 33 (1980) : 602-7. Hans Tischler supported Treider's general argument but criticized some of its details; see "A propos meter and rhythm in the Ars antiqua",journal of music theory, 26 (1982) : 313-29. For Treider's response to both Sanders and Tischler, see Leo Treider, "Regarding 'A propos meter and rhythm in the Ars antiqua'",journal of music theory, 27 1 1 Treider, "Regarding meter and rhythm", 545. (1983) : 2 1 5-22. 1 2 Ibid., 542. 13 Haas, "Studien zur mittelalterlichen Musiklehre I", 381-3.

C ON F L ICTS AND STRATE G I E S OF C ON C I L IATION

The rhythmic theories of the pre-Franconians Early modal theories combined quantitative metrics with Aristotelian thought in two different, though complementary, ways. First, the pre-Franconians defined the relation between a sound and its duration in terms of a "subject" and its "predicate". Second, they imposed the quantitative components of the modal patterns onto a conceptual scheme of qualitative contrarieties. The subject-predicate relationship · in music is described in the Introductio musice secundum magistrum de Garlandia:

Subjectum in musica est aliquarum vocum seu pausationum conjunctio modo debito ac proprie observato. Predicatum est ipsius musice ars legitima proportionate omnibus suis modis diligenter observatis, . . . 1 4 The subject in music is the conjunction of several notes or rests in the right and properly observed manner. The predicate is the rule-bound art of proportionate music itself, with all its modes duly observed. Aristotelian logic is indeed a logic of predicates. It is therefore restricted to propositions that confirm the attribution of a given quality to a given subject. Further tensions between the musical language and its metalanguage are appar­ ent in fhe pre-Franconian accounts of the modal system itself. While the musical language of the modal theories was derived from Augustine's De musica, the metalanguage is that of Aristotle. Generally speaking, Augustinian poetical meters are based on two quantities : the basic unit (the brevis) and its duplication (the /onga). The relation between them is mathematical : 1 /onga = 2 breves. In mathematics there are no opposites or contrarieties - x can be smaller than, larger than, a multiple of, or a certain part ofy, but x is never the opposite ofy or of any other value. Mathematically speaking, longa and brevis are two points on a continuum of values. However, the thirteenth-century theoreticians viewed longa and brevis as the two opposing qualities governing measured music. Anonymus v11 differentiated categorically between short and long note values by saying that a note is either short or long :

14 Jntroductio musiu secundum magistrum de Gar/andia, CS 1: 157-75, at 158. Leofranc Holford­ Strevens has pointed out to me that the passage seems to be corrupt after the word "legitima", and the remainder of the translation is therefore approximate; "modes" must have the broader sense of "ways of being" rather than the specific musical meaning. In the edition the final sentence of the excerpt ends "cui partem philosophie supponatur ars metrice", which is grammatically unsound; Dr Holford-Strevens thinks it ought to read "Cui parti philosophiae supponitur? Arismetricae" (To which part of philosophy is it subject? Arithmetic), casting it in a familiar didactic/ catechistic fonn, and that arithmetic rather than the art of meter is intended.

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Sequitur de proprietatibus figurarum. Omnis figura sive omnis nota, quod idem est, aut est brevis aut longa. 1 5 Following is a discussion on the properties of the figures. Any figure or note (which is the same thing) is either a brevts or a longa. In a typically scholastic manner, Anonymus VII divided the two distinct classes of long notes and short notes into their subclasses : De . longis triplex est differentia. �edam est longa duorum, quedam trium, quedam sex temporum. 1 6 De brevibus triplex est differentia : quedam est recta brevis continens unum tempus et scribitur, tali modo : • �edam semibrevis continens unum tempus et scribitur, sic : • • • • Qyedam plica brevis descendendo, ut hie : " " " �edam plica brevis ascendendo, ut hie : � � M 11 The classification ( differentia) of long notes ( longae) is threefold. One is a longa of two, one of three, one of six time-units. The classification of short notes (breves) is threefold. One is a normal brevzs containing one time-unit and is written in this way : • • • . One is a semibrevzs containing one time-unit and is written so • • • • . One is a plica brevts descending, like this " " " . One is a plica brevzs ascending, like this � � � . For this theorist, long and short values stand in opposition to one another. Long and short are not viewed as relative terms which lie on the same continuum ; the two represent distinct and mutually exclusive essences or species. The notion of contrariety regulates measured music not only on the element­ ary level of its components but also on the higher level of the rhythmic modes. These are divided into two opposing groups : normal mode (modus rectus) versus mode beyond measure (modus ultra mensuram). This grouping, contingent upon the longa's quantity of time-units, is further evidence of the projection of qualitative classification on the originally quantitative theory. An additional quantity makes a new type, or a new species of notes or of modes. Notandum quod quidam modus dicitur rectus ; alius dicitur in ultra mensuram, qui scilicet excedit rectum modum sive rectam mensuram. Dicitur autem ille modus rectus qui currit per rectas longas et per rectas breves. Et est recta longa que continet in se duo tempora solum. Reeta brevis est ilia que continet in se unum. Primus secundus et sextus sunt in recto modo. Tertius modus, quartus et quintus sunt in ultra mensuram. 1 8 15 11

Anonymus vu, Dt musica libel/us, CS 1 : 378-83, at 379. Ibid., 380.

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18

16 Ibid. Ibid., 378.

C ON F L I CTS A ND STR ATE G I E S O F C ON C I L I AT I ON

It is to be noted that one mode is said to be normal (rectus) and another is said to be beyond measure (ultra mensuram) , namely that which exceeds the normal mode (modus rectus) or the normal measure. A mode is said to be normal when it proceeds by normal long and normal short notes. A normal long note is one that contains in itself only two time­ units. A normal short note is one that contains in itself one. The first, second, and sixth are normal modes (in recto modo ). The third, fourth, and fifth are beyond measure (in ultra mensuram ). In Johannes de Garlandia's De musica mensurabili positio, this distinction is expressed by the pair of contraries rectus and obliquus. Reeta longa est duas rectas breves continens tantum. Obliqua longa est que abundat super rectam longam. 1 9 A straight (recta) long note is one that contains only two straight short notes ; an oblique long is one that exceeds the straight long. The notion of contraries determines the order of the various modes within the system : because it is logically prior, the rectus must come before the obliquus. The latter is clearly a derivative concept. Sed aliqui volunt quod quintus noster modus sit primus omnium ; et bona ratio, quia per istum modum precedit omnes nostros modos. Sed quo ad tempora cognoscenda, prius est modus rectus quam obliquus, et sic non valet quod dicitur, quod quintus est primus. 20 But others insist that our fifth mode be the first of all, and with good reason, since this mode precedes all our modes. Yet so far as the cognition of time is concerned, the straight mode is prior to the oblique, and so what is said, that the fifth is the first, is not valid. All of this demonstrates that the pre-Franconians interpreted mathematical relations (2 : 1, for example) as relations between opposing predicates. Notwith­ standing the mathematical inadequacy of such reasoning, it fits well with Aristotle's logic, according to which the category of relation is the first among the four ways one can pose things as A versus B. As the following quotation illustrates, the category of numerical relation is not mathematical, but is rather one of opposition : Pairs of opposites which fall under (the category oij relation are ex­ plained by a reference of the one to the other. (The reference being indicated by the preposition "of" or by some other preposition.] Thus 19 20

Johannes de Garlandia, De musica mmsurahili positio, CS 1 : 97-1 17, at 97. Ibid., 98.

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double is a relative term, for that which is double is explained as a double of something. 2 1 In other words, the mathematical relations of traditional metrics were brought closer to the Aristotelian qualitative category of relation. But in this early phase of Scholastic musical thinking, no comprehensive integration of Aristotelian and traditional Augustinian thought was achieved. Pre-Franconian theories gave reality to the flow of time, thus perpetuating the Augustinian notion of time in which it is measured by notes and rhythmic modes. In these theories, modus was defined as follows : Modus in musica est debita mensuratio temporis, scilicet per longas et breves ; vel aliter : modus est quidquid currit per debitam mensuram longarum notarum et brevium.22 A mode in music is the proper measurement of time, that is by longas and breves ; or alternatively : A mode is whatever advances by the due measurement of long and short notes. Johannes de Garlandia writes in a similar vein : Maneries eius appellatur, quidquid mensuratione temporis, videlicet per longas vel per breves, concurrit. 2 3 Maneries is the name for whatever proceeds by the measurement of time, namely by long and short notes. For the pre-Franconians, time itself was the object of measurement. Echoing Augustine's specification in his De Musica, they quantified and measured rests. Unde regula : omnis pausatio simplex dicitur equalis penultime modi precedentis.24 Hence the rule : every simple rest is said to be equal to the penultimate of the preceding mode. And again : Pausa est tanta quanta est penultima.25 A rest is just as long as the penultimate. Moreover, the pre-Franconians accepted the Augustinian atomistic analysis of time, to which Aristotle was so clearly opposed. They argued that time is Aristotle, Categories, chs. 10, 11b, pp. 24-6. Anonymus v11, De musica libel/us, 378. 23 Johannes de Garlandia, De mensurabili musica, 1: 36. 24 Johannes de Garlandia, De musica mensurabi/i positio, 103. n Anonymus v11, De musica libel/us, 379.

21

22

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CON F L I C T S A N D STR AT E G I E S OF CO N C I L I AT I O N

composed of finite units of duration, and they measured time by a unit known as This unit, Johannes de Garlandia wrote, is "indivisible"; it is an "atom" of time :

unum tempus.

Propter hoc posset fieri quaestio, quid appellatur unum solum tempus. Ad quod dicendum, quod unum solum tempus, prout hie sumitur, est illud, in quo recta brevis habet fieri in tali tempore, quod fit indivi­ sibile. 26 Therefore one might ask what is meant by one single tempus. It should be answered that one single tempus (as taken here) is that in which a nor­ mal breve can be produced in such time as is indivisible. The relatively late treatise of the Anonymus of St. Emmeram, dated 1279, is no exception to the general trend of superficial use of Aristotle in the context of music theory. As Yudkin has shown, the author of De musica mensurata adhered faithfully to the teachings of Johannes de Garlandia and adapted Scholastic "manners" and terminology for matters of procedure, ordering, or layout. 27 Aristotelian notions were so distant from the crux of the treatise's actual content that the author could allow himself to confuse and exchange Aristotelian notions, as in the following example where the author apparently confuses and misapplies the most basic Aristotelian categories of genus and species: And it should be noted that it is not undeservedly that this chapter should be arranged before the other, since there are three very general species from which every genus of song results and occurs, that is to say notational figure, unit of time, and measure. 28 It would be more appropriate to see all the various types of songs as different species of "song" as a generic concept. And what is the supposed genus that derives from the three mentioned species : notational figure, unit of time, and measure ? The Anonymus of St. Emmeram wrote in the most fashionable academic style of his age, but he failed to grasp the precise meanings of the logical system that he applied to his musical objects. Many other examples of the stylish use of Aristotelian notions or of the erroneous application of Scholastic-Aristotelian methodology might be cited from this particular source. More important, however, is the general conclusion that Aristotle's role in shaping the world of measured music in the works of Johannes de Garlandia and his disciples was far from essential : it extended from the artificial imposition of qualitative schemes of classification to mere jargoniz26

Johannes de Garlandia, De mmsurabili musica, 1: 37-8 See De musica mmsurata: the Anonymous ofSt. Emmuam, ed. Jeremy Yudkin, Music: scholar­ ship and performance (Bloomington: Indiana University Press, 1990), 7-11. 21 Ibid., 87. 27

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ing. In practice, musical time was still organized according to traditional metri­ cal procedures. In theory there was a discordance between the metalanguage of theoretical reflections and the language of theory. The rhythmic theories of the Franconians The crucial difference between the enlistment of Aristotelian notions for mere rhetorical purposes and a real incorporation of these notions is what distin­ guishes between pre-Franconian and Franconian theories of measured music. The Franconian "Aristotelianism" is total : music becomes fully subordinated to the rules that constitute the Aristotelian universe, in order to exemplify in the microcosm of music the macrocosmic world of Aristotle. But how was measured music rendered Aristotelian ? To see this we need to examine the way Franco­ nian theorists transformed the accepted meanings of rhythmic terms into new ones, thus creating a conceptual space in which musical entities or concepts were given an Aristotelian stamp. While pre-Franconians defined the constitutive rhythmic components of measured music in terms of the two basic durational units, the longa and the breve, Franco adapted the Aristotelian distinction between being and privation and made it the foundation of measured music. He replaced the longa and the breve with the more abstract and general distinction between actual sound and its privation - the state of rest. Sounds represent "being" and pauses "privation". With this reframing, Franco brought the phenomenon of music under the constitutive rule of Aristotle's theory of being and becoming. Following Aris­ totle, Franco saw actual sounds as logically prior to their privation. He therefore treated sounds first, because being or actuality is prior to potency. Sed cum ipse discantus tam voce recta quam eius contrario, hoc est voce amissa, reguletur, et ista sint diversa, horum erunt diversa signa . . . Sed cum prius sit vox recta quam amissa, quoniam habitus praecedit privatio­ nem, prius dicendum est de figuris, quae vocem rectam significant, quam de pausis quae amissam.29 Since discant itself is governed both by actual sound (vox recta) and by its opposite, that is by the omission of sound (vox amissa ), and these are different, their signs will be different. . . . Since actual sound precedes its omission (because possession precedes privation), we must first speak of notes, which represent actual sound, before speaking of rests, which represent its omission. 29 Franco de Colonia, Ars cantus mmsurabilis, ed. Gilbert Reaney and Andre Gilles, CSM 1 8 {A IM, 1974), 28-9.

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Aristotle, we will recall, believed it was possible to establish a n exhaustive and exclusive determination or definition of each individual being according to its unique essence (differentia specifica or species) and closest genus. Using Aristotelian science as a model, Franco classified all the types of figures used in musica mensurabilis into three groups, which he named species and subdivided further according to existing variants : Simplicium tres sunt species, scilicet longa, brevis et semibrevis. �arum prim a in tres dividitur ; in longam perfectam, imperfectam et duplicem longam . . . . Brevis autem . . . in rectam et alteram brevem dividatur . . . Semibrevis autem alia maior, alia minor dicitur. 10 There are three species of simple figures - longa, breve, and semi­ breve - the first of which has three varieties : the perfect, imperfect, and duplex longa . . . . The breve . . . is divided into a normal (recta) and an altered breve . . . One [kind of) semibreve is called major, the other minor. Franco conceptualized durations as "essential predicates" of a "subject", which was sound itself. Once reasoned into the status of predicates and abstracted into the three options of duration, the longa, the breve, and the semibreve became categorically differentiated as distinct species. Each species had its own sub­ classes, but the long and the short did not belong to a larger whole in the same species, which contained both, and which would have made possible a quantita­ tive comparison between them. Further use of the Aristotelian notion of the contrariety between being and privation appears in the Franconian discourse on the distinction between the two types of longas. The binary, imperfect longa is one deprived of perfection ; imperfection occurs only when one part of the longa is extracted and figured independently (as a breve), whereby the longa becomes binary rather than ternary and therefore loses its perfection. An imperfect longa has no separate reality : it never appears alone, but is always followed by a breve to round off the cycle of perfection. Thus, the binary version of the longa is a state of "accidental" change rather than an "essential" condition (using these terms in the strict Aristotelian sense). Furthermore, being an accidental rather than an essential state, it does not call for a distinct figural shape : it has no separate knowable identity. The formal difference between the Franconian concept of the binary longa and the pre-Franconian understanding of it as perfect and correct is well known. What is less known is how Aristotelian notions dictated the way rhythmic values were conceived. These notions affected musical theory not only on the superficial level of terminology, but also on the much deeper level of semiotic conception.

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1

Franco, Ars cantus mrnsurabilis, 2 9-31.

R H Y T H M I C T H E O R I E S O F T H E F R A N CO N I A N S

Turning from the longa to the breve : a n altered breve, as we know, is quantitatively equal to an imperfect longa, but this mathematical aspect is not relevant, because imperfect longa and altered breve are qualitatively distinct. The breve can be replaced by its three. equal parts or by two unequal parts, but it cannot be made imperfect by its third part. In other words, the rhythmic hierarchy is based on three levels of duration : the longa, the breve, and the semibreve. But these levels operate independently; they do not relate math­ ematically as parts to a larger whole, for each level is governed by its own set of rules. What could be the reason for such an apparently deficient rhythmic system ? One could attribute shortcomings of this sort to lack of experience, or to expected anomalies typical of formative stages of any theoretical enterprise. One might also say that the theory merely reflects the compositional practice, and therefore the problem lies in the musical imagination or compositional strategies of thirteenth-century composers and has no bearing on the intellec­ tual attitudes of theorists. But even if practical considerations were first and foremost responsible for the "state of the art", theorists explained the practice using the terminology and system of reasoning they saw fit, and this was the Aristotelian system. As a matter of fact, Franconian notation turns out to be satisfactory, reasonable, and responsive to the intellectual expectations of music theorists in an age dominated by an Aristotelian world-view. When viewed in the Aristotelian context, apparent inconsistencies make sense : the understand­ ing of rhythmic values as distinct qualities rather than quantities, the representa­ tion of two different quantities by a single note-shape, the understanding that such two different quantities are two possible realizations of the same essence, and the Franconian presumption that these qualities pertain to a heterogeneous rather than a homogenous time-space, all these elements clearly reflect the Aristotelian world-view. The relation between the Franconian concept of time and the Aristotelian notion of time is even more intimate. Two opposing philosophical doctrines of time were known in the Middle Ages, the Augustinian and the Aristotelian. Augustine described time as the expression of duration in general. Time is the representation of the inner awareness of our experience ; therefore, time shows the structure of experience. Thus time measures rest and duration as well as motion. Augustine analysed time mathematically. He believed that time was composed of atomic units. These units constituted the present, which differen­ tiated between past and future time. Aristotle, on the other hand, argued that time is the measure of motion according to the prior and posterior. Aristotle held that when there is no motion there is no time. Accordingly, time derives all its attributes from the concept of motion, for time is an attribute of motion. For Aristotle, time is completely continuous ; the present does not differentiate between past and future time, it is simultaneously the end of the past and the beginning of the future. This 37

CON F L I C T S A N D S TR AT E G I E S O F C O N C I L I A T I ON

explains the nature of a continuum according to Aristotle. For each two points there .are adjacent limits which become one and the same and are therefore contained in each other. 1 1 Indirectly, and as a derivative use only, time measures rest that can be defined as a state of privation of motion. Franco derived his notion of time from Aristotle. We may start by noting the resemblance between Franco's definition of time and the Aristotelian definition described above. Franco wrote : Tempus est mensura tam vocis prolatae quam eius contrarii, scilicet vocis amissae, quae pausa communiter appellatur. 3 2 Time is the measure of actual sound as well as of the opposite, its omission, commonly called rest. In other words, time is the measure of movement and of its contrary. This contrary, commonly thought of as rest, is in fact unactualized movement, or movement in a state of potentiality. The fact that time measures rest as well as motion was neither obvious nor immediately acceptable to an Aristotelian like Franco. To prove that time measures rests, Franco presented an argument modeled after the Aristotelian "arguments from incommensurability". His argu­ ment was as follows : Dico autem pausam tempore mensurari, quia aliter duo cantus diversi quorum unus cum pausis, alius sine sumeretur, non possent proportiona­ liter adinvicem coaequari. 3 3 I say, however, that a rest is measured by time, for otherwise if one were to take two different melodies, one with rests, the other without, they could not be compared to each other proportionally. Franco was referring to a mathematical-physical incommensurability similar to that delineated by Aristotle in his arguments against "movement in the void" or against "a weightless body". Aristotle presented this argument to prove that movement would be impossible in a void. The velocity of an object in the plenum is inversely proportional to the density of the medium in which the object travels. This can be symbolically represented as follows : Vi : Vi : : M2 : M1 . In a void M1 becomes zero, because it contains no matter. But the proportion becomes meaningless when M1 becomes zero, because there is no common ratio between zero and a finite magnitude. It should be said that Aristotle did not posit an arithmetical commensurability (which different media may not have) but rather an ideal or general commensurability of any given finite magnitudes. Franco developed a formally similar argument when he discussed rests and JJ For the 14th-c. theories of the composition of continUJ:t in general, and of musical time in particular, see below, pp. 102-24. Here I focus only on the Aristotelian aspect of Fr�nco's 32 33 theory of time. Franco, An- cantus mmsurahilis, 25. Ibid.

R H YT H M I C T H E O R I E S O F T H E F R A N C O N I A N S

musical time. If a rest i s not measurable by time, Franco argued, then a m;·lody with a rest is like a ratio including zero. A melody without rest, then, is analogous to a ratio between numbers. Hence, if a rest is not measurable by time, then a melody with a rest must be incommensurable with a melody without a rest, just as a ratio including zero is incommensurable with any ratio between numbers. Since such melodies are commensurable in practice, Franco concluded, a rest must be measurable by time. Aristotle's rejection of atomism of any kind 34 and his insistence on the absolute continuity of time were reflected in Franco's definition of the smallest time unit. Unlike the indivisible recta brevis of his predecessors, Franco's tempus minimum, the minor semibreve, was not an atom of time but rather a fraction of the smallest "full" sound : Reeta brevis est quae unum solum tempus continet . . . Unum tempus appellatur illud quod est minimum in plenitudine vocis. Sed nota semibrevium plures quam tres pro recta brevi non posse accipi, quarum quaelibet minor semibrevis dicitur, eo quod minima pars est ipsius rectae brevis. 3 5 A normal breve is one that contains one single tempus. . . . That is called one tempus which is the smallest in the fulness of sound. . . . But note that there cannot be more than three semibreves for each normal breve, of which each is called a minor semibreve, since it is the smallest part of the normal breve. Furthermore, attention should be paid to the difference between the pre­ Franconian and the Franconian definition of modus. Whereas the former defined modus as the measure eftime ("modus in musica est debita mensuratio temporis") Franco defined modus as "the cognition of sound measured by long and short time-units (Modus est cognitio soni longis brevibusque temporibus mensu­ rati )". 36 These two definitions represent the distinction between measuring time itself on the one hand and measuring by time on the other. This distinction epitomizes the difference between the two opposing philosophical views re­ garding time. Augustine regarded time as the measure of duration. Aristotle, on the other hand, regarded time as the means by which motion is measured. Franco's concept of time was the Aristotelian one. Not all musicologists approve such an interdisciplinary analysis of musical theories. Christopher Page, in his book Discarding images, 37 indeed admits the dependence of musical discourse on Aristotelian taxonomy and terminology, but For a discussion of Aristotle's rejection of atomism, see below, p. 103. Franco, Ars cantus mmsurabilis, 34, 38. For a survey and analysis of the 14th-c. theories of 36 Franco, Ars cantus mrnsurabilis, 26. the rhythmic minima, see below, pp. 1 24-30. 17 Christopher Page, Discarding images (Oxford : Clarendon Press, 1993). 34

1

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39

C O N F L I C T S A N D S T R AT E G I E S O F C O N C I L I A T I O N

h e does not favor contextual explanations o f mensural theories that lean o n the impact of Aristotle. For Page, the use o( Aristotelian notions does not affect the contmt of mensural theory but is a self-evident and self-understood fact : It would be very surprising indeed if it were not. . . . because measured notation, comprising a set of rationally ordered particulars with a nu­ merical frame of reference, lent itself admirably to the kind of discursive rigour and taxonomic clarity that Aristotle bequeathed to virtually every learned enterprise of the thirteenth and fourteenth century. With theo­ rists such as Johannes de Muris, this process of intellectual enrichment (if one chooses to regard it as such) could extend very far. 38

Arguing against the view presented here (as also in my doctoral dissertation), 39 Page ignores an important problem that renders the question of the presence of Aristotle in mensural theories far from self-evident : the absolute conceptual incompatibility between the qualitative world-view of Aristotelian physics and the quantitative-numerical frame of reference of mensural theories in so far as they are theories concerning measurement. It is already becoming apparent that the subsuming of rhythmic notation and its theoretical base to the qualitative Aristotelian universe is not merely a simple act without significant repercussions. Furthermore, it is the Aristotelian context that sheds light on the singularities and the anomalies of mensural theories. Discursive strategies cannot be exposed without reconstructing the formative milieu within which musical ideas were fashioned. Page, however, sees the use of Aristotle's notion as mere color, complexion, or something cosmetic : A great deal of the Aristotelian complexion to be found in the music theorists can be regarded as either heuristic or superficial (indeed cos­ metic). Either way it usually appears to be a postfactum addition to what has been independently conceived.40 I believe that Page finds this "complexion" not in the theorists but in their rhetoric, but even giving him thus the benefit of the doubt does not eliminate the logical problem of detaching the predicates (the Aristotelian "complexion") from the subject and objec� that they refer to - as if theorists first wrote a series of subjects unmasked by any complexion, and only later qualified or masked them with some Aristotelian color. Yet even if philosophical, extramusical notions were mere colors, they still call Page, Discarding images, 1 2 0-2 1 . Dorit Tanay, "Music i n the age o f Ockham : the interrelations between music, mathemat­ ics, and philosophy in the fourteenth century" (PhD dissertation, University of California, Berkeley, 1 989). For Page's discussion of my thesis, see Discarding images, 1 1 9-24. 40 Ibid., 1 2 1 . 38

39

R H YTH M IC TH EORIE S OF TH E F R A N C ONIANS

for clear differentiation. In his discussion on the Aristotelian-Scholastic impact on thirteenth-century musical thought, Page misses some important nuances as well as historical distinctions. Above all, one has to distinguish between the Aristotelian and Thomistic Ars antiqua and the anti-Aristotelian Ars nova. As I will show in Chapter 3, this distinction is crucial for an understanding of the dialectical aspects and the multi-faceted character of the point of transition in question. The Ars nova, I argue, should be set in the context of the anti­ Aristotelian mathematical physics of the fourteenth century and also in the context of the new Ockhamistic (not Thomistic) theology. Page seems to be insensitive to such distinctions and conflates the thirteenth and the fourteenth centuries as if they were one monotonic or monoideistic period. 4 1 Returning to Franco's theory, the attribution of perfection to the long of three tempora can be understood in light of the symbolic propensity of Scholastic theology in general and of Thomas Aquinas' theological doctrine of analogia entis in particular. The interpretation of the universe as a symbol of God had already emerged in the early Patristic sources as the inevitable solution to the central theological problem of how God exists in nature. The symbolic mode allowed the attribution of omnipresence to God, yet eliminated pantheistic interpretations : divine immanence, on the one hand, and absolute divine tran­ scendence - the danger of God being nowhere - on the other. Franco's theory of rhythmic perfections reflected long-standing tendencies to argue and explain by appeal to substantial connections. Musical time became connected with divine perfection, but the nature of the connection was problematic, since the divine is timeless while music articulates real time. This tension was resolved by shifting the position or status of the principle of rhythmic order. In the rhythmic theories of the pre-Franconians, the principle of order was an immanent, intrinsic principle : this was the constant inner order of each of the six rhythmic modes. In the theories of Franco and Lambenus, the principle of order was the perfect longa, which consisted of three time units. The ensuing discussion will show that the relation between this perfect longa and the other rhythmic modes resembles the Thomistic relation between world and God. Interestingly enough, we shall see the artful synthesis of "Aristotelian­ ism" and "Platonism" in Thomas's theology reflected in the Franconian discourse about rhythm in music. Thomas's concept of God is rooted in the Platonic concept of the "One" in which God is conceived as the absolute transcendent unity. As such, one cannot name or define God's essential attributes, since by definition God's unity is lost by dividing God into subject and predicate. Because of God's transcendence, no 41 For example, Page misleads his readers when he writes, "Tanay's argument i that the theorists of the ars antiqua and ars nova often speak of measured notation in a way that betrays their acquaintance with Aristotelian methods of classification and explanation." Ibid., 1 20.

41

CON F L ICTS A ND STRATE G I E S OF CONC I L IATION

predicate can be applied to God unambiguously. In order to avoid the conclu­ sion that everything said of God is meaningless (since each predicate is ambigu­ ous), Thomas formulated his doctrine of analogia oztis.42 The world, according to Thomas, is an imago Dei. That is, the relation between God - the creator of the world -and the world is like that between the creator and his creation or the relation between a cause and its effect; the cause imparts traces to the effect, and the effect therefore resembles its cause. Because this relationship pertains, one can reconstruct from the world the essential predicates of God, because things in the world "resemble" God. Things in the world resemble God in so far as they have perfections. Attributes or names of God are "a picture of a picture". Their meaning is neither unambiguous nor equivocal; they describe God only by way of analogy. Analogical reasoning creates room for ambiguity and opacity: through analogy one does not define essences, but affirms similarity between relations.43

Thomas's doctrine of analogia oztis had two sources; one was the Greek mathematical theory of proportions (the Greek "analogy" means proportion), and the other was the Aristotelian triple relationship between words, concepts, and things. According to Aristotle, words denote (name) things. The relation between the thing named and the concept is that of similarity (similitudo ), whereas the relation between the word and the concept is that of meaning (significatio ).44 Thomas's analogia entis is based on the distinction between signifi­ catio and representatio or similitudo. One thing can represent another thing or a concept due to the resemblance between them. The disintegration in practice of the modal system demanded new organizing principles if rhythmic disorder was to be avoided. The new principle however, had to meet two demands: (1) it had to account for and regulate the new music cum littera, in which the modal principles were abandoned in favor of free juxtaposition of perfections and their equipollences, and (2) it had to correspond to the assumption common to all Scholastic thinkers that art mirrors nature, which in turn is created in the image of God and is therefore rational, well­ ordered, and unified, whereby motion and its resulting disorder can be linked with motionlessness and order. Like Thomas, Lambertus postulated that the world (and music within it) is an image of God: Et ideo non immerito ad summam refertur trinitatem, quia res quelibet 42 On Thomas's doctrine of analogy, see George P. Klubertanz, St. Thomas Aquinas on analoKJ: a textual analysis and systematic synthesis (Chicago : Loyola University Press, 1960). 41 Thomas Aquinas, Summa theologica, I• quaest., § 13, artt. 5-6. 44 In addition, the concept of significatio was central to medieval semantics and grammar. For

its history in logic and its possible role in the formation of mensural notation, see below, pp. 50-58.

42

R H Y TH M I C TH E O R I E S O F T H E F RA N C O N I A N S

naturalis ad similitudinem divine nature ex tribus constare invenitur. In vocibus et sonis, et rebus omnibus trina tantum consistit consonantia, scilicet diatessaron, diapente et diapason. Hane igitur trinitatem omnia naturaliter formata consequuntur."5 And therefore it is rightly brought into relation with the holy Trinity; because every natural entity is found, in the image of the divine nature, to consist of three [components]. In voices and sounds and all other things consonance is only threefold, namely diatessaron, diapente, and diapason. All things formed by nature follow the Trinity. Franco's reasoning is similar to that of Lambertus: Longa perfecta prima dicitur et principalis. Nam in ea omnes aliae includuntur, ad earn etiam omnes aliae reducuntur. Perfecta dicitur eo quod tribus temporibus mensuratur ; est enim ternarius numerus inter numeros perfectissimus pro eo quod a summa trinitate, quae vera est et pura perfectio, nomen sumpsit."6 The perfect longa is called first and principal, for in it all others are included, and to it all others are reducible. It is called perfect because it is measured by three tempora, the number three being the most perfect among numbers because it takes its name from the Holy Trinity, which is true and pure perfection. The very essence of the perfect figure, its function, and its position were described by Lambertus as analogous to the divine "One":"7 . . . scire debemus, quod sex tantummodo figure sunt adinvente . . . �arum e(s)t prima super omnes fons est et origo ipsius scientie atque finis, que perfecta longa merito vocatur ; nam a perfectione trine equali­ tatis nomen habere sumpsit . . . Cum igitur perfecta figura manens in unitate sit fons et origo ipsius scientie et finis, propterea quod omnis cantus ab eadem procedit, et in eadem replicatur, et ipsa in numeris consistit temporibus et mensuris, et trinam in se continet equalitatem, videre sequitur quod ipsa prior ceteris esse videtur, eo quod mundi conditor Deus omnia in numero, pondere et mensura constituit, et hoc principale extitit exemplar in animo conditoris. 48 We must know that only six figures have been invented, . . . of which the [Lambertus), Cujusdam Aristotdis Tractatus de musica, CS 1 : 251-8 1 , at 270. Franco, Ars cantus mmsurabilis, 29-30. 47 On Platonism and the Christian philosophy see Arthur H. Armstrong, "Plotinus", in Arthur H. Armstrong (ed.), The Cambridge history of later Gmk and earlier medieval philosophy (Cambridge: Cambridge University Press, 1967), 193-263. 411 Lambertus, Tractatus de musica, 269-70. 4

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first, above all the source and origin o f the science [of music) a s well as its goal, is that which is justly called the perfect longa ; for it has drawn its name from the perfection of the threefold equality . . . Since, therefore, the perfect figure that remains in unity is the source and origin of the science as well as its goal, because every melody (cantus) proceeds from it, and is repeated in it, and because it consists of numbers, times and measures, and contains within itself a threefold equality, it follows that it is prior to all others, because God the creator of the world constituted everything according to number, weight, and measure, and this was the chief pattern in the mind of the creator. Here Lambertus reasons about the divine properties of the perfect long through analogy ; but he analogizes the perfection of the triple long to the Christian-Platonic notion of divinity. His discourse, then, is a fascinating exam­ ple of the Ars antiquds "compromise": it entwines traditional Christian-Platonic notions (e.g.,fons et origo, exemplar in amino) with the Thomistic analogia entis and Aristotle's notion of causafinalis. Christopher Page, however, suggests that for the sources of such discourse one should consult a scriptural commentary, such as a work of twelfth-century biblical scholarship that "elucidates the momentous symbolism of threefold schemes, enhanced by a powerful sense of God as the final unity in whom all numerical speculation can be resolved".49 For this particular passage of Lamber­ tus, Page guides us to Hugh of St Victor's commentary on Noah's ark, which discusses threefold schemes as reflected in the ark : the length of three hundred cubits, etc. 50 Page's idea is interesting and may be correct. But the idea of the Trinity and its reflection in threefold worldly schemes appears ubiquitously in Christian literature and is common to otherwise absolutely different Christian biblical commentaries, philosophies and theologies of the various sects, schools, orders, philosophical traditions, etc. The problem concerns, of course, differen­ tiation. How does one decide for, or more probably against, a possible attribu­ tion? We could, for example, examine the broader system of thought within which discussion of the Trinity and its symbolic reflection was to be developed. The idea of the Trinity is expounded differently within neo-Platonic or Augus­ tinian or Aristotelian or Thomistic or Ockhamistic world-views. This means that the extraction of a discourse about threefold division in a biblical commentary, such as that on Noah's ark, does not prove anything. There are thousands like it. And it always requires more than one symptom to define a syndrome ; some­ thing more substantial and more specific than a shared "fastidious concern to elucidate a symbolism of threefold schemes" (to use Page's words), is necessary in order to establish a meaningful intellectual link. 49

Page, Discarding images,

44

1 24.

0 '

Ibid.

R H Y T H M IC T H E O R IE S O F T H E F R A N C O N IA N S

Interestingly, the three essential predicates o f God - grace, power, and wisdom (gratta, potentta, sapientia ) - are mentioned in Lambertus' text : Cum nihil possit fieri, nisi prius sit in artifice faciendi potentia, nichil sapienter sine sapientia ; et cum nullo indigeat, nihil ab eo fiat, nisi ex gratia. Cum igitur ab eo fiant omnia, manifessum [sic; recte manifestum] est sapientibus quod hec tria, scilicet sapientia, potentia, gratia, sunt in divina essentia, quia ad summum perfectumque bonum plura non sunt necessaria. 5 1 For nothing can be made unless the potential for making it previously exists in the maker, nothing can be done wisely without wisdom ; and because he [God] lacks nothing, nothing is done by him, except by grace. Since therefore everything was made by him, it is clear to all wise men that these three - wisdom, power, grace - are in the divine essence, because more is not needed for the highest and most perfect good. This reinforces the claim that Franco and Lambertus derived their bask terms and principles from the Thomistic doctrine of the analogy of being. The common starting point of thirteenth-century philosophers discussing the relation between God and the world - the central issue of medieval theology ­ was based on the following passage, already referred to earlier, from Aristotle's Metaphysics:

We must consider also in which of two ways the nature of the universe contains the good and the highest good whether as something separate and by itself or as the order of the parts. Probably in both ways, as an army does ; for its good is found both in its order and in its leader, and more in the latter : for he does not depend on the order but it depends on him. And all things are ordered together somehow, but not all alike - both fish and fowl and plants ; and the world is not such that one thing has nothing to do with another, but they are connected. 52 The musical discourse about the relation between perfection and everything else, or between necessity and contingency, seems to be anchored in the Aristotelian world-view and its medieval development. The Franconian princi­ ple of order - the perfect long - is a distinct entity; it exists apart from and independent of any given arrangement of entities. Like the Aristotelian highest good, or the Thomistic God, the perfect long is transcendent, but in another way it is also immanent in relation to the rhythmic variety found in the music. It is both of the system of the rhythmic modes, being one of the six modes, and beyond and above the system, as implied by its divine attributes. Additional evidence for this last point is yet another passage from Lambertus' treatise. '

1

Lambertus, Tractatus dt musica,

n Aristotle, Metaphysics, M io,

270.

45

1075• 1 1 -20.

C O N F L I C T S A N D S T R AT E G I E S O F CO N C I L I A T I O N

Explaining why the perfect long constitutes the first mode, Lambertus used the linguistic construction regit non regitur, which calls to mind the expression mavens non movetur that Aristotle used to describe his first principle or Prime Mover: n Primus modus dicitur, qui tantum componitur perfectis figuris . . . Et hoc patet igitur quod nunquam comprimitur in ligaturis, sed liber excipitur, et solus non patitur unquam a pressuris ; regit et non regitur, imperans non utitur aliorum curis. 5" The name "first mode" is given to that which is composed only of perfect figures. . . . . . And it is therefore plain that this [figure] is never squeezed in ligatures, but received as a free being, and it alone never suffers diminution ; it rules and is not ruled, it commands and does not benefit from the attentions of other notes. Those rhythmic combinations that resemble the perfect figure are therefore accep�able. Thus contingency is both limited and explained. Order is guaran­ teed. Furthermore, through the notion of rhythmic perfection and its realization in music theorists fused two different concepts of time and motion : that of Christian theology and that of Aristotelian philosophy. As a regulative principle of motion in time, the notion of perfection exemplifies the Aristotelian princi­ ples of imitation and recurrence that governed all physical processes in the Aristotelian universe. But the changing rhythmic surface as combined with the uniform, immutable, and divinized perfection allowed real time to interfere with timelessness, in accordance with the traditional Christian-Platonic concept of human time as simultaneously distinct from, yet closely tied with, divine timelessness. We may conclude that the Aristotelian impact on the rhythmic theories of the thirteenth century is far more interesting, complex, and problematic than the mere mapping of musical terms in their exact location in the original Aristote­ lian texts. To be sure, Aristotelianism is but one of the manifold and heterogene­ ous ideational spaces that shaped the concept of musical time and its articulation in the age of Scholasticism. However, since the Aristotelian mode of thought was dominant in the academic milieu of the thirteenth century, its resonance in n Dismissing the Scholastic-Aristotelian universe as the proper locus and resource for explaining the discourse of music theories in the age of Scholasticism, Page claims the priority of medieval aphorism and biblical commentary. Thus, he suggests the following aphorism "Regnat, non regitur, qui nihil nisi quod vult faciat" as the source of Lambertus' expression "regit et non regitur" (see Discarding images, 1 23). I do not agree. While rhythmic notions such as longa, brevis, minima, maxima, etc. have no biblical cognates, late-medieval biblical commentaries spoke the language of Aristotle and applied its vocabulary and conceptual frames to the same extent as did other discursive disciplines. And in view of the continuous dialogue between learned and popular literature, it is hardly surprising that popular proverbs or aphorisms should be colored with Aristotelian language. '4 Lambertus, Tractatus de musica, 279.

RH YTH M I C T H E O R I ES O F T H E F RA N C O N I A N S

scholarly musical discourse is o f paramount importance. Ultimately, a n explana­ tion of the Aristotelian influence would involve going behind the dominant rhetorical style by unravelling the inner dynamics and the interplay among the various cultural and theological forces that shaped the notion of music not only qua music, but also as a symbolic barometer reflecting general approaches to the disturbing categories of quantity, motion, and time. Since these problematic notions constitute the very core of the rhythmic theories of the thirteenth century, the use of the qualitative-taxonomic discourse of Aristotle (and the acceptance of his injunction against quantification of temporal processes) pre­ sents a great challenge to historians who wish to interpret these musical theories and reconstruct their original meaning. But such anomalies and discrepancies are typical of medieval writings in general and reflect the heterogeneous, composite, and discontinuous character of the medieval world-view. For their authors and audiences they represented a standard, rational and convincing theory of the ars cantus mensurabilis.

47

2 T H E N AT U R A L F O U N D AT I O N O F R H Y T H M I C N O TAT I O N

a;;

I

ES R

iiiJ&Y

4W

Natural experience and the medieval mode of representation

to ask what musical notation as a system of signs shared with a related system - that is, the system of discursive language. In more general terms, I shall try to identify common models of signification in the Middle Ages in order to provide a wider context for studying the history of Western notation. More specifically, I shall try to discover what attitudes towards language were implicit in mensural notation and upon what the operation and efficacy of mensural signs were based. Studies of medieval notation have rarely attempted to explain the semantic or semiotic foundation of the apparently cumbersome system of mensural notation. 1 In order to deal with these questions I shall compare notated music with other medieval theories of language and meaning. T Is M Y P u RPos E H E R E

1 E Alberto Gallo made an isolated attempt to discuss mensural notation in terms of semiotics in his "Figura and rtgula: notation and theory in the tradition of musica mrosurabilis", in Hans Heinrich Eggebrecht and Max Liitolf (eds.), Studiro zur Tradition in dtr Musilc: Kurt von Fischer am 60. Gtburtstag (Munich : E. Katzbichler, 1 973), 43-8. Gallo suggests that the two conflicting tendencies to develop and innovate, on the one hand, and to preserve the tradition on the other, were tempered by "the general principle that rules the evolution of musica mrosurabilis, as it does any other 'systeme de signes': the 'analogie' which is at the same time a principle of renewal and of preservation. Thus, seeing that the system of musica mtnSUrabilis was formed initially of the relationships between the two 'signes' longa and brtVis, during later amplifications of the system, the relationships existing between the new 'signes' were arranged in 'analogie' to the initial model" (p. 47).

NATUR A L E X P E R I E N C E AND M ED I E VA L R E PR E S E NTATION

In his classic study of the rise of modern algebra, Jacob Klein argued that in Greek science concepts were closely linked to the natural, prescientific experience from which they were abstracted. In the new science of the seven­ teenth century, however, the relation between scientific concepts and common experience became more attenuated. The new science established its concepts while launching a full-scale polemic against traditional concepts. The new concepts no longer had the natural range of meaning available in ordinary discourse. No longer was the object of the concept-what it is about ­ immediately intelligible. 2 Klein went further still. In antiquity, even mathematical concepts were de­ pendent upon natural experience. Euclid's presentations always used determi­ nate numbers of units. Thus, Euclidean mathematical notation did not, indeed could not, do two things at the heart of the modern, purely symbolic procedures of modern algebra. It did not identify the object represented with its means of representation, and it did not replace the real determinateness of an object with the possibility of making it determinate, such as would be expressed by a sign which, instead of signifying a determinate object, signified possible determinacy (such as the ubiquitous symbols n or x for a quantity). 3 Present-day theories of language stress even more sharply the difference between traditional and current notions of how a language confers meaning, using the distinction between old referential semantics and new relational semantics. Following Plato, it is argued, words were understood as necessarily referring to pre-existing objective things ; therefore, words "correspond" to things and stand for these things in common discourse. Accordingly, signs are exterior to that which they picture or imitate ; as semantic units, they direct us towards something beyond and apart from the signs themselves : namely, the world they reflect. In contrast to this understand­ ing of meaning as already existing in the world, modern linguistic theory divorces meaning from reference and stresses that words constitute reality rather than merely representirig it, whereby meaning ceases to be a given object to which signs refer and become a dynamic function of use - a function of the manifold relations that words have with other words at all possible levels and in all possible contexts of interpretation. These remarks seem to suggest that the most important concept in decoding a system of representation is the concept of intentionality, namely the mode in which concepts and signs mean. Intentionality, Klein insists, reflects modes of understanding, and transcends the boundaries of specific scientific fields. Is Klein's thesis relevant to the history of music? Did the semiotic behavior of medieval mensural notation reflect the general modes of conceptualization prior to the scientific revolution ? If so, then we must concede that the evolution of 2 Jacob Klein, Gruk mathematical thought and th, origin ofalgebra (Cambridge, MA : MIT Press, 3 1 968), 1 17-23. Ibid.

49

TH E NATUR A L F OUNDAT I ON OF R H YTH M I C NOTATI ON

rhythmic notation was not simply grounded in practical musical procedures that were completely autonomous from other cultural activities. Rhythmic notation, if Klein is correct, was circumscribed by much broader non-musical factors that shaped the model and set the boundaries for the development of musical ideas and their symbolic expression. In the most general terms, it is easy to argue that medieval rhythmic concepts, like ancient concepts, were abstracted from natural experience. The regulative principles of mensural notation, the principle of perfecting and imperfecting note values, as well as the principle of alteration, all presuppose that rhythmic concepts stem from natural objects. Notions such as being and becoming, perfect and imperfect, seem extraneous to modern discourse about music, but central to discourse in fields such as physics or even psychology. Furthermore, mensural notation, no less than Greek mathematical notation, refers to something absolutely determinate : namely, the rhythmic modes. Pro­ viding mere prescriptions, mensural notation lacks the generality which is so essential to the mathematical and musical notations of the last three or four centuries. Mensural notation also lacks the absolute simplicity of the new system of scientific notation. It is economical but not elegant : with its economy come ambiguities. In search for the conceptual substructure of rhythmic notation in the thir­ teenth and fourteenth centuries, I will examine the relation between two medieval theories of semantics and grammar and the concurrent theories of mensural notation in order to determine what they share. The first theory concerns the "properties of terms" (proprietates terminorum) ; the second concerns the "modes of signification" (modi significandi), which was central to the so-called speculative grammar (grammatica speculativa).4 Medieval philosophers, following ancient conceptions of language, assumed that linguistic elements and structures had counterparts in thought and in reality; thought, language, and reality were thought to share the same logical coherence. Thus, language was understood as an analogue of reality and not merely an instrument of communication. The school of Chartres of the twelfth century was one of the first to develop theories of language focused upon the logical-semantic aspect of statements taken as the basic units of meaning. This contrasted with an older doctrine of signficatio that investigated isolated words detached from their broader linguistic context. This concentration on the mean­ ing of a word in an actual context gave rise to a branch of logic centered on the properties of terms. Yet the old theory of signification was integrated into the new one. Under the old theory, the meaning of a word (significatio) was con­ sidered to be its nature or essence (fanna ). Hence, the word carried its meaning ' The following survey is based on Lambertus M. de Rijk, "The origins of the theory of the properties of terms", in CHIMP, 1 6 1-73.

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a s a natural property. Each time a word appeared its meaning could be deter­ mined by its natural property - its signification. Within the new theory of pro­ prietas terminorum, logicians recognized a property they called suppositio (standing for) as the most essential property of a term, to stand for its referent in a written proposition. The suppositio assumed a central place in the proprietas terminorum theory. Logicians investigated the possible suppositiones of a given term and explored the various referents which a term could have in different linguistic contexts. They were aware that even unequivocal terms, terms with a single signification, could stand for different things in different contexts. Thus in the proposition "man is a species", "man" represents a being made of flesh and blood, while in the proposition "man is a noun", "man" is a linguistic construct. Thus, a word's actual meaning depends ultimately on the fusion of its basic signification and its relationship to its context. The different suppositions that a term acquires reflected properties other than suppositio and the natural signification, which adjoining words may have. These other properties were identified as ampliatio (ampliation), restrictio (restriction) and, for some philosophers, copulatio (linkage). In discussing the properties of terms, logicians argued that the thing referred to by a given term may be amplified or restricted by adjoining words. Qyantifiers (such as "all", "some", "no"), adjectives, and even some verbs may restrict or expand the number of referents implied by a term. Qyantifiers such as "every" or "any" were thought to have a force of confounding (vis confundendi) the terms which they modify, for they cause their supposition to be confused. 5 Moreover, the tense of the adjoining verb determines whether a term refers to all the objects having the given essence, or only to presently existing objects, or to future beings. In other words, terms were understood in light of their extension. A term taken by itself ( per se sumptus), detached from any proposition, could signify a universal nature, and hence had the power to stand for all the individuals that participated in this nature, whether existing presently or not. for Peter of Spain (d. 1 2 77), a term's ability to assume its unlimited extension was defined as the term's natural supposition. When this extension was re­ stricted or qualified by an adjoining word, it signified accidentally only a limited quantity of individuals. The power to signify a limited quantity of individuals was interpreted as an accidental property of a term. What is already very notable is the relationship between the doctrine of proprietas terminorum and the Aristotelian world-view. Proprietas terminorum is ' Confused supposition occurs when it is not permissible to descend disjunctively to the individuals for whom the term stands: for example, the term "man" in the proposition "Every man is an animal." The supposition of the term "man" is confused because the proposition is true only if it can be verified in the case of every man and not merely this or that man. For further distinctions involved in this theory, see Paul V. Spade, "The semantics of terms", in CHLMP, 188-96, at 195-6. 51

T H E NATU R A L F O U N D ATION O F R H YT H M IC NOTATION

a linguistic theory that projects the Aristotelian ontology onto the linguistic signs : signs, like real entities, have a distinct essence, or nature, and also accidental properties. Here we are reminded of the parallel projection of Aristo­ telian ontology onto rhythmic figures so as to concretize and specify the fundamental, general belief that language and reality manifest the same logical structure. For thirteenth-century logicians and music theorists the reality was that of Aristotle. The doctrine of supposition, then, was a rigorous semantic theory aimed at clarifying linguistic ambiguities or equivocations through the analysis of the different meanings that a term may have in different linguistic contexts. Central to the doctrine of proprietas terminorum was the notion that the actual referent of a given term is determined by the presence or absence of other terms which have the power to alter the supposition of the term in question. We may conclude that the doctrine of the property of terms can be explained as a theory of reference, classifying all the various referents that a term may have in different contexts. Each proposition entails different kinds and quantities of entities of which a given term is truly predicable. Turning from the history of language and meaning to the history of rhythmic notation, one immediately observes that the quantifi c ation of time and its representation by geometrical figures did not follow the model of simple arithmetic in which each quantity is represented by a distinct numerical symbol. The reason for this is made clear by tracing the forces leading to the crystalliza­ tion of the Franconian system. Stripped of its philosophical setting and examined from the practical point of view, Franconian notation can be construed not so much as a conceptual revolution, but rather as an ad-hoc, practical solution introduced to meet the demands of notation in musica cum littera. All that musicians sought was an alternative technique for writing down music based more or less on the same rhythms that characterized the Notre Dame repertory sine littera, but with an underlying syllabic text that prevented the use of ligatures and was freed from rhythmic monotony. As noted above in Chapter 1, the modal practice of Notre Dame polyphony may or may not have derived from Augustine's ars metrica. 6 Documentation for this derivation is lacking from the formative period of modal rhythm (n501250), and there are many obstacles to substantiate direct derivation. We noted, however, one piece of documentation in Alexander de Villa Dei's Doctrinale of n 99 , which indicates that theorists saw a similarity, if not a connection, between the long and short of modal rhythm and those of classical quantitative verse.7 See above, p. 29. For a discussion of this source, see Treider, "Regarding meter and rhythm", 524; see also Haas, "Studien zur mittelalterlichen Musiklehre I", 369-71, 38 1-3. 6

7

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NATUR A L E X P E R I E N C E AND M EDIE VA L R E P R E S E NTATION

The aspect of the similarity that interests us is that the rhythms of both music and verse were considered to be based upon only two species of durations long and short - even though for purposes of rhythmic analysis it might be necessary to invoke both the long and the short in various !durations. 8 The economy in the number of notational symbols used for notating music with text may reflect this basic characteristic of the metric system. It coliresponds to the economy of concepts used for analysing the manifold patterns of quantitative metrical poetry. The unit of quantitative verse, the foot (pes), is composed of two parts, each having one or two syllables. The long syllables, no matter how long, were conceptualized and named simply as "long" (corresponding to their traditional representation by one single symbol, a horizontal stroke -) ; the same was true for the short syllables, no matter how short (represented uniformly by a small semicircle ""'). These two concepts were sufficient for the analysis of common meters, even though sources recording the practice of reciting quantitative poetry betray the use of varying durations of long and short syllables. In musical notation, then, two symbols, one for longa and one for brevis, were sufficient for notating the six rhythmic modes, in which longa could have the value of either two time-units or three, and brevis could have the value of either one time-unit or two. Since modal music, like quantitative poetry, consisted of a fixed number of patterns, two figures were sufficient for notating distinct and easily identified patterns. Moreover, it may be argued that all three systems - poetic meter, modal notation, and the Franconian system of notation - require a contextual ap­ proach. This is because the three systems share a concept according to which the referents of the signs are not single discrete durations but a larger unit, a self­ contained rhythmic movement - a poetic foot or a rhythmic mode. In quantita­ tive metrical poetry, it is obviously a group of syllables which determines the foot. In the modal system of notation, it is of course the context, the arrange­ ment of the ligatures, which determines how the ligatures are to be read. In the Franconian system, too, a single symbol is ambiguous, for the various "figures" change their meaning to meet the imposed rhythmic mode, or to meet with the Franconian principle of perfictio. Franco states clearly that musical figures refer not to distinct note-values but to the modes : Figura est repraesentatio vocis in aliquo modorum ordinatae, per quod 8 For a detailed analysis of ars rhythmica and ars metrica in medieval culture, see Richard L. Crocker, "Musica rhythmica and musica metrica in antique and medieval theory",joumal of music theory, 2 (1958) : 2-23. See also John Stevens, Words and music in the Middle Ages: song, narrative, dance and drama, 1050-1300 (Cambridge : Cambridge University Press, 1986), 413-29.

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patet quod figurae significare debent modos, e t non e converso, quem­ admodum quidam posuerunt. 9 A figure is the representation of a sound arranged in any of the modes. From this it foilows that the figures ought to signify the modes and not the contrary, as some have maintained.

According to this concept, which dates back to the old neumes, musical notation is a means of codifying, standardizing, and conserving a selection of the most popular or conventional melodic inflections (in the case of the neumes) or rhythmic progressions (in the case of both the modal and the Franconian notations). 10 But the notation was unclear. Theorists therefore felt the burden of responsibility, and attempted to elucidate a notational practice that was far from self-evident. I want to suggest that their theoretical explanations reflected what they understood as a proper theoretical explanation of signs and their meaning. In other words, my hypothesis is that theorists formulated a theory of meaning that echoed the medieval mode of understanding semantic operations.

Musical signs, words, and nature: a network of mutual reference

Music theorists, like contemporary logicians, were guided by the principle of the similitude between language and reality. They projected the Aristotelian ontology onto musical notes, and they conceived notes or figures as having not only essential and accidental properties but also the power of acting or being acted upon. Further, in line with contemporaneous semantic theories, music theorists determined meaning by fusing the essential signification of each rhythmic figure with a contextual approach. If natural objects are distinguished and categorized by their distinct essences, so too are rhythmic figures and words or terms. Just as natural entities are endowed with accidental properties, so too figures and words were thought to have accidental properties. The dual under­ standing of how rhythmic figures confer meaning can be gathered from the following quotation from Franco's Ars cantus mensurabilis. On the one hand, Franco clarified, each figure carries its distinct essence which is at the root of every actual meaning of · that figure. Accordingly, he classified note-shapes according to their specific essences : Simplicium tres sunt species, scilicet longa, brevis et semibrevis. Q!!arum prima in tres dividitur ; in longam perfectam, imperfectam et duplicem

Franco, Ars cantus mrnsurabilis, 29. For a semiotic analysis of neumes, see Leo Treider, "The early history of music writi.ng in the West",JAMS 35 (1982) : 237-79. 9

10

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longam . . . . Brevis autem . . . i n rectam e t alteram brevem dividatur . . . Semibrevis autem alia maior, alia minor dicitur. 1 1 There are three species o f simple figures - longa, breve, and semi­ breve - the first of which has -three varieties : the perfect, imperfect, and duplex longa . . . . The breve . . . is divided into a normal (recta) and an altered breve . . . . . One [kind oij semibreve is called major, the other minor. In other words, figures are classified according to the essential quality that they signify. For example, the property of signifying the longa possessed by the figure of the longa is the essence of that figure, and is therefore the fundamental significance of that figure whenever it appears in a larger rhythmic context. Thus proprietas figurarum, like proprietas terminorum, consists in embodying essential signification. But the precise meaning of the figure is determined by the surrounding context : Simplicium autem valoris cognitio ex ordinatione quam habent adinvi­ cem declaratur. Ordinatio vero earum sic accipitur: aut enim longam sequitur longa, aut brevis ; et nota hoc idem esse iudicium de brevibus et semibrevibus. 1 2 The knowledge of the value of the simple [figures] is obtained from their order with respect to one another. Their order is understood in this way : after a longa follows either a longa or a breve ; and note that this judgement is the same with respect to breves and semibreves. Later theorists followed the Franconian discourse ascribing physical character to the rhythmic figures. One example is the late-fourteenth-century Ars cantus mensurabi/is mensurata per modos iuris. 1 3 The anonymous author explained the altered value of a perfect longa, in the context where two perfect longas are placed between two perfect maximas, by reference to the distinction between the essential value and the accidental value of the longa. oportet quod una illarum habeat vim duarum longarum, et non prima quia iam numerata est. . . . Regula iuris est ergo secunda : nee ilia secunda ex se substantialiter valet duas, sed accidentaliter. one of them ought to have the strength of two longae - and not the first, because it has already been counted. . . . The rule of law is, 11

13

12

Franco, Ars cantus mrnsurabilis, 29-31.

Ibid., 3 1.

Ars cantus mrnsurabi/is mmsurata pu modos iuris I The art of mrnsurable song measurtd by the

modes oflaw, ed.

and trans. C. Matthew Balensuela, Greek and Latin music theory, 10 (Lincoln : University of Nebraska Press, 1994). Most of the treatise appears as Anonymus v, Ars cantus· mrnsurabi/is, CS 111: 379-98. For a detailed analysis of this treatise and its relation to Johannes de Muris' Notitia artis musicae, see below, pp. 199-205.

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therefore, the second longa : not that the second is worth two in itself substantially, but accidentally. 14

The Tractatus de figuris sive de notis focuses exclusively on the properties of rhythmic figures. 1 5 This English source, dating from c.1350, 1 6 has three chapters, each concentrating on the properties of a single figure. The first chapter discusses the minim, the second the semibreve, and the third the breve. The proprietatesfigurarum of both the perfect semibreve and the perfect breve include the properties of not being perfectible (for they are already perfect) and yet of being imperfectible. The imperfect semibreve and breve have the properties of not being imperfectible but of being perfectible. For the anonymous author, "to be perfected" and "to be imperfected" are two physical states of existence which relate to each other as being and privation : "all imperfection is the privation o f some perfection (omnis imperfectio est privatio alicuius perfectionis)". 1 7 Moreover, "to be perfected" and "to be imper­ fected" are also states reached through an action performed by an adjoining figure. In other words, figures act upon each other, and one may affect the meaning of an adjoining figure. Regarding the property of a perfect semibreve to be imperfected, the author says : imperficere est de perfecto imperfectum facere. . . . et hoc contingit muftis modis: uno modo per minimam praecedentem, [etc.] 1 8 to imperfect is to make imperfect that which is perfect. . . . and this occurs in many ways : in one way by a preceding minim, etc. Signs and their referents are topics of much importance for Johannes de Muris. In his Notitia artis musicae (132 1), Muris elaborated on the philosophy of musical notation, attending especially to the properties belonging to some musical figures of becoming perfect or imperfect. 1 9 Johannes de Muris' innova­ tive ars notandi reflects the conceptual overlapping characteristic of a period of transition. On the one hand, Muris absorbed and adapted within his new rhythmic vocabulary the Franconian heritage. On the other hand, the constitu­ tive rationale behind his new rhythmic ontology is devoid of the Aristotelian ontology which was the very foundation of the Franconian theory. In the •• Ars cantus mensurabilis mensurata per modos iuris, 154-7 (cf. CS III : 383). u Ed. Gilbert Reaney in Ms. Oxford, Bodley 842 {Wille/mus) : Breviarium regulare musicae; Ms. British Museum, Royal 12. C VI.: Tractatus de figuris sive de notis; johannes Torkesry: Declaratio trianguli et scuti, CSM 12 {AIM, 1966), 40-51. Earlier ed. as Anonymus v1, Tractatus defiguris sive de notis, CS 1: 369-77. 16 For the history and context of this source, see Reaney, Introduction to Tractatus defiguris sive de notis, 35-8. 18 1 7 Tractatus defiguris sive de notis, 42 (cf. CS I: 371 ). Ibid., 45 (c( CS 1: 372). 19 For an analysis of Muris' rhythmic theory and its relation with contemporaneous mathematics, see below, pp. 79-101.

M U S I C A L S I G N S , W O R D S , A N D N AT U R E

ensuing chapters I will describe i n detail the revolutionary anti-Aristotelian content of Johannes de Muris' Ars nova. Here I wish to draw attention to the possible inconsistency and the resulting dualism of Muris' philosophy of nota­ tion. On the one hand, he seems to have captured and advanced the notion of a pure "sign", admitting that note-shapes are arbitrary signs that stand for quantities (that is, for certain numbers of time-units). But on the other hand, he argues that rhythmic signs stand for real entities that he named res musicales, which undergo qualitative change, becoming perfected or imperfected : i Notula musicalis est f gura quadrilatera soni numerati tempore mensurati significativa ad placitum. 20 A musical note is a four-sided figure that arbitrarily signifies a numbered sound, measured by time. Nevertheless, Nee obstat figurae diversitas, quia figura figuram non imperficit, cum omnis figura sit formaliter perfecta. Sed illud, quod no mine uni us figurae significatur, imperficit illud, quod nomine alterius importatur. Figura autem signum est, res musicalis significatum. 2 1 The diversity of the figure is no objection; for a figure does not imperfect a figure, since every figure is formally perfect. But that which is signified by the name of one figure imperfects that which is referred to by the name of the other. For a figure is a sign, the musical object that which is signified. At the base of his discussion lies the assumption that "perfection" and "imperfection" are processes that take place in reality. To use Muris' own idiom, there is a factual transition from one state of being, "being perfect", to the other state, "being imperfect", through a physical subtraction or addition of parts. But Muris understood that these processes of transition take place at the level of the res musicalis signified by the figure and not at the level of the figure itself. It is the entity signified by the figure which is imperfected or perfected by the other entity signified by the other figure. Despite my explanation, the passage remains difficult. Its meaning may be clarified by the distinction between the two forms of a figure which directly follows : Notula ergo duas includit formas : figuram quadrilateram, quae primaria est, et significationem, quae secundaria est. 22 A note includes two forms : the quadrilateral figure, which is the primary one, and the signification, which is the secondary one. 20

Muris, Notitia artis musicat, 75 ; cf. p. 9 1 : "Est enim notula fi gura quadrilatera soni numerati 22 21 Ibid. Ibid., 91. tempore mensurati ad placitum significativa."

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The primary form is the external shape of the figure, used as a conventional sign of notation. This form is a geometrical figure and is perfect by definition. The secondary form is the essence of that figure, that is, its signification. As I mentioned at the beginning of this chapter, it is the signification that is perfected or imperfected. I believe that Muris' understanding of how signs signify echoed the doctrine of proprietas terminorum, since it continues the Franconian tradition in assuming that signs $tand for and thus refer to actual things, and in accepting the Franconian semantic operation involving the fusion of the essential signi.ficatio with contextual interpretation. But Muris proposed a different analogy between music and grammar. To clarify his position, he maintained that there is a structural resemblance between syntactical relations and rhythmical relations : Unde sicut vox ad vocem grammatice non dependet neque causat constructionem, sed modorum significandi rerum proportio, sic figurae ad figuram nulla est proportio musicalis, sed ex proportione rerum musicalium perfectioneque et imperfectione earundem causatur conso­ nantia musicalis. 21 Hence, just as a word does not depend grammatically on another word, and does not cause the construction, but it is the proportion of [i.e., the relationship between] the modes of signifying things, so there is no musical proportion between a figure and another figure, but musical consonance is caused by the proportion of musical objects and by their perfection and imperfection.

Muris' analogy between the theory of language and the theory of rhythmic notation is vague. He implies that rhythmic notation should be construed not within the semantic theory of the properties of terms, as I suggested earlier, but in light of the grammatical theory of the modi signi.ficandi. I believe that Muris made a philosophical mistake. Let me first present the theory of the modes of signification and then examine its relevance to rhythmic notation. Muris' pro­ posal, I will argue, needs refinement. Medieval grammar can be made relevant if matched with the notation not of texted music but of non-texted music. In other words, no comparison can be offered between the function of the modes of signification and the Franconian single-note system of notation. Still, medieval speculative grammar may have some bearing on the Franconian procedure of ligatures. The theory of grammatical modes focused on the syntactical function and meaning as formed by the inflection of words' suffixes or by grammatical inflection. 24 Since a correct reading of ligatures depends on understanding the 23

2 •

Muris, Notitia artis musicat, 9 1-2 . The following survey is based on Jan Pinborg, "Speculative grammar", in CHIMP, 254-70.

M U S I C A L S I G N S , W O R D S , A N D N AT U R E

meaning o f every possible manipulation o f the suffixes and prefixes of given ligatures, there might be an interesting and unexpected resemblance between grammatical and notational modalities. The modistae (the logicians who dealt with speculative grammar) assumed that every word has a phonological component that receives its signification through two stages of imposition. With the first imposition, the sound is turned into a lexeme, called dictio. The dictio is not a specific meaning but rather the basic common meaning. It is the "root" underlying all the possible grammatical forms of inflection. With the second imposition, the dictio gains its specific meaning through the acquisition of several modi significand1: each specifying one of the grammatical categories, such as part of speech, gender, number, case, tense, etc. Thus, the modes of signification are semantic modifiers, by means of which a dictio is turned into a specific grammatical form or a part of speech. Only after acquiring the modes of signification does a word have all the properties needed for playing a particular grammatical function within a linguistic context. Every word-form (i.e., a word with all its semantic modifiers) has more than one mode, and the same suffix or prefix can express several modes. The Latin termination "-us", for example, indicates "noun", 25 singular, masculine, and the nominative case. It follows also that the same mode of signification, say the nominative case, can be expressed by terminations that differ by gender, "-us/-a/-um". Further­ more, while the same case and gender may be expressed by more than one mode - for example, singular, masculine, and nominative designated by "-us", "-er", and "-is" - the same suffix (i.e., modus significandi) may indicate several different cases : the suffix "-is" indicates the dative as well as the ablative plural, and may also indicate the genitive singular. The combination of linguistic strings of different word-forms is governed by the following principle. Two word-forms can be combined only if the modes of signification of the one are identical to those of the other. This is how nouns and adjectives are combined in Latin : every adjective displays the gender, case, and number of the noun it modifies. In addition, words can be combined if the modes of one word-form are proportional to those of the other; one mode specifies an attribute by which the other is specified. Thus, one word depends on the other ; one of the words carries semantic characteristics which presuppose corresponding characteristics in the other word. The modistae took it for granted that there is a natural order by which the various modes of signification should be arranged. They assumed that this order reflected the order in nature, where the agent precedes the action, the substance its accident, action its objects or modifiers, and so forth. This natural order lies beneath the arbitrary order on the surface. With this notion of order in mind, the modistae distinguished between a dependency a parte ante-when the i,

The Latin grammatical category of nomm embraces both "substantives" and "adjectives".

59

T H E NAT U R A L F O U NDAT ION O F R H Y T H M IC N O T A T ION

dependent word-form is prior as far as the natural order is concerned - and a dependency a parte post-when the dependent word-form is posterior in the natural order. The latter does not necessarily coincide with the order of the written words. Armed with this distinction of dependency, the modistae system­ atically worked out a list of all grammatically legitimate linguistic constructions in the Latin language. In other words, grammatica speculativa was first and foremost concerned with congruities between the various word-forms used in linguistic combinations, as analysed on the deeper semantic level. Traditional grammar requires that each mode of signification have an onto­ logical counterpart, its corresponding modus essendi or proprietas rerum. These are accidental properties of the object, as distinct from its essential qualities. Modes of being can be of a real thing, or of an imagined object conceived analogously to a physical substance. Yet it is important to keep in mind that the modistae focused on the formal meaning, and not on the actual referent or application of a given word. This, together with all the other characteristics of grammatica speculativa, suggests a theory striving towards a system of universal language, focusing on the notion of linguistic congruity. It is important to recall that the various modes of signification are semantic modifiers, and therefore belong to words (in the form of various endings or prefixes) and not to real objects. Nature in general and relationships between things in nature in particular were beyond the scope of the modistae. In discuss­ ing what they called the "proportion" or construction between, say, two word­ forms, they dealt with the rules governing the proper linguistic combination of these words, such as agreement in number, case, and gender between a noun and its adjective. These rules, as already argued, operate at the level of deep structure, regardless of the actual arrangement of words in a given proposition. For Johannes de Muris grammatical modi significandi were relevant to the semantic function of musical figures. As we have already seen him state : just as a word does not depend grammatically on another word, and does not cause the construction, but it is the proportion of [i.e., the rela­ tionship between] the modes of signifying things, so there is no musical proportion between a figure and another figure, but musical consonance is caused by the proportion of musical objects and by their perfection and imperfection. 26 In many ways Muris' ideas of the correspondence between the grammatical modi and the musical figures seem to confuse rather then clarify. In the rhythmic notation of perfection and imperfection, the relation between figures depended solely on the actual arrangement of the figures. But this is not the case of the grammatical modi which are coordinated and related according to syntactical · 26

See above, p. 58. 60

MU S I C A L S I G N S , WO R D S , A N D N AT U R E

rules - regardless of the actual deployment of words o n the surface. Further­ more, Muris compared the mod,: which belong to words, to the musical objects, which belong to the world, rather than to the note-shapes. There seems to be a discrepancy between Muris' deep acquaintance with mathematics and his poor understanding of grammar and logic. As I shall show presently, Jacobus of Liege's strengths and weaknesses were exactly the reverse of Muris'. 27 It is possible, however, to trace the influence of Latin grammar on the Franconian theory of ligatures. Franco predicated proprieties not only to simple figures but also to ligatures. A ligature either ascends or descends. Additionally, it may be with propriety (cum proprietate) or without propriety (sine proprietate), or it may have an opposite propriety (cum opposita proprietate), depending upon its initial part. Furthermore, a ligature can be with perfection (cum perfectione) or without perfection (sine perfectione), depending on the final note of the ligature. These differences, Franco stressed, are essential and specific to these ligatures. Unde ligatura cum proprietate essentialiter differt ab ilia quae est sine, ut rationale animal ab irrationali ; similiter et in aliis differentiis prius dictis . . . . A parte autem medii ligaturarum nulla essentialis differentia inveni­ tur. Ex quo sequitur quod omnia media ipsarum ligaturarum conveniunt in significatis. 28 Hence a ligature with propriety differs essentially from one without, just as a rational animal [differs] from an irrational one ; and likewise in the other differences aforementioned . . . . In the case of the middle [figures] of ligatures, however, no essential difference is found, from which it follows that all the middle [figures] of those ligatures agree in signifi­ cance. It is natural to conclude that Franco's theory of ligature echoes his theory of simple figures. Ligatures as simple figures have properties, and both theories postulate the similitude between reality and language. A deeper analysis, how­ ever, may show that Franco's doctrine of ligatures is modeled after Latin syntax in general, and the doctrine of modi significandi in particular. The Franconian ligatures resemble word-forms. They are constructed from a prefix, a trunk, and a suffix. In Latin word-forms, suffixes and, to a lesser extent, prefixes are inflected. This is equally the case for the suffixes and prefixes of ligatures. To see this, one can imagine a series of brevis-like squares mapped on the stave and consider it as the counterpart of a word's dictio. The provision (or not) of different types of tails or of obliquity to the prefix or suffix of a given ligature is an act of inflection. The modi significandi are semantic modifiers through which a word-form acquires its grammatical function. So too are the prefix and suffix of a ligature - these are precisely semantic modifiers. The 27

See below, pp. 155-8 1.

28

Franco, Ars cantus mmsurabilis, 44-5.

61

TH E NATUR A L F OUNDATION OF R H YTH M IC NOTATION

inflection of the beginning or termination of a ligature turns a rhythmically indefinite ligature into a semantically defined one. Such an inflected ligature has the properties necessary to fulfill a particular role within a larger rhythmic context. Just as each word-form has more than one mode of signification (the Latin suffix "-us", for example, has three such modi: it indicates that the word-form is a "noun" that is singular and masculine) so too a prefix or a suffix of a ligature conveys more than one bit of semantic information. Information is conveyed by the position of the suffix or prefix in relation to the other figures; from this one learns whether the ligature is ascending or descending. More information is encoded in the external shape of the suffix or prefix, that is, for example, whether it does or does not have a tail. Moreover, just as the same modus signifi­ cctndi can be expressed by several suffixes, in Latin the nominative masculine singular can be expressed by "-us" or by "-er" or by "-is", so too there is more than one way of notating with prefix or suffix of ligatures a specific rhythmic progression. The following examples illustrate some of the alternatives : L = longa

BL

B = breve

LL r

=

�

BBL L BL

)

f

�

--=

Ji r-_ .-. r-= r-Jl ,(1 ... r-- .,.. -.Ji

finally, the combination of ligatures is not arbitrary, but is determined by the type of "semantic modifier" needed in a given rhythmic context to convey a cer­ tain mode. Modal and mensural music had their own syntax or principles of construction. Legitimate combinations of rhythmic values were determined through a pre-existing list of rhythmic modes and their equipollences (equi­ valences of rhythmic value). Here again we are reminded of medieval speculative grammar. Linguistic congruity between combined word-forms was determined by rules exhausting all possible grammatically correct linguistic combinations. Whether grctmmctticct speculcttivct had any influence on the Franconian notation of musicct cum litterct has yet to be determined. At this stage, because of the lack of evidence from primary sources, only a few tentative speculations can be offered. Grctmmctticct speculativct, the theory of the properties of terms, and the rhythmic theories all postulate that language and reality exhibit the same logical structure. In this sense, speculative grammar could have been a possible model for rhythmic theory. Moreover, the doctrine of the modes of signification is closer to the rhythmic theory than to the doctrine of the property of terms in its concentration on exhausting all legitimate combinations of word-forms. 29 Just as 29 In the musical thought of the 1 3th c. this exhaustion encompassed the six rhythmic modes. Lambertus' unusual list of nine modes codified another three rhythmic progressions, which became conventional and therefore added to the vocabulary or recipes available for musicians of the late 1 3th c. See Gordon A. Anderson, "Magister Lambertus and nine rhythmic modes", Acta musicologica, 45 (1 973) : 57-73- In the 14th c., the exhaustion of rhythmic possibili-

MU SIC A L S IGNS , WO R D S , AND NAT U R E

the modistae strove to exhaust all legitimate linguistic combinations by listing which word-form can be combined with which, so too rhythmic theorists exhausted all legitimate rhythmic combinations, successive and simultaneous. Proprietas terminorum, on the other hand, centers on the extension of terms -that is, on the proper descent to those individuals upon which a given term can be predicated. Needless to say, rhythmic theory has no counterpart to this aspect of proprietas terminorum. Finally, while the surface structure is a crucial factor in determining meaning for rhythmic theorists as well as for terminists, once the idea of syncopation was introduced, rhythmic theorists disconnected rhythm the actual succession of notes, values, or figures - from cyclical metrical organi­ zation, and regrouped figures according to a deeper structural level. In this sense, rhythmic theorists, like the modistae, discussed congruity and legitimate combinations in the "natural" order, which may or may not be reflected at the surface level. I have not claimed a one-to-one correspondence between medieval rhythmic notation and contemporaneous semantic theories. Indeed, differences in subject­ matter between rhythmic notation and linguistic discourse rule out a complete and elegant correspondence. But a comparative study of these two fields can shed some light on the medieval mode of understanding the nature of symbolic signs, and the way such signs intend their objects. This opens up for us a new avenue for understanding mensural notation in terms of the broader attitude towards semantic operations, as it contributes to the reconstruction of the mentality and sensibilities that underlay the development of rhythmic notation in the high Middle Ages. 10 ties and the notational representation of new and increasingly difficult rhythmic combinations became the central theme of rhythmic theories. See below, pp. 1 9 2-205. 1 ° For further evidence on the linguistic character of mensural notation, see below, pp. 1389 with n. 79.

P A RT I I A R S N OVA

3 RH Y T H M MAT H EMAT I Z ED A N D D EM Y S T I F I E D Si)i

2

R

Sb

UiA

Mathematization, de-mathematization, and the legacy of the A rs nova

W

IT H T HE HIND s I G HT

of several centuries, we can see that the mathematization of music in the fourteenth century also involved, in one respect, its de-mathematization. From the ninth century up through the seventeenth, mathematical relationships were inextricable from a web of cosmological and theological symbols. An "autonomous" music, we might say, would have been emancipated from servitude to the numerical interconnections that signified cultural and religious values. Prior to the emanci­ pation of music, mathematics dictated which relationships were "good" and meaningful and which were not; to medieval theorists and composers math­ ematical considerations were often more important than purely artistic ones. The long-standing bond between music and mathematics was in many ways a tyranny of mathematics over music. It is reasonable to expect, therefore, that at some point music might rebel against this tyranny by attenuating its ties to mathematics. Some signs of de-mathematization are indeed apparent in the early four­ teenth century; the rehabilitation of the ill-famed imperfect or binary rhythmic values is one example. But on the other hand, the new "emancipated" music generally relied much more on mathematics than the earlier music of the thirteenth century. The far more intrjcate rhythmic language of the new music demanded the mathematization of music to such a great extent that mathemati­ cal relations, measurements, and mensurational problems became the dominant compositional issue of the fourteenth century.

R H Y T H M M AT H E M A T I Z E D A N D D E M Y S T I F I E D

This i s a n odd state o f affairs. The fourteenth century witnessed de­ mathematization and mathematization side by side and simultaneously. What accounted for these strangely conflicting processes? How did these apparently conflicting tendencies relate to or even support each other? What were the dynamics that led musicians to uncouple mathematical and symbolical connec­ tions, while at the same time employing much more mathematically sophisti­ cated mensurations? What was the relationship between these dynamics and the trend towards secularization, so often attributed to the fourteenth century in general and to its music in particular? The situation is more complicated still, for the Franconian-Scholastic dis­ course undeniably persisted in the metalanguage of the fourteenth century. Notions such as "to be perfect" or "imperfect" or "altered" continued to describe the semiotic behavior of the various note-shapes, and thus perpetuated the thirteenth-century understanding of rhythmic signs as representing natural objects in the Aristotelian universe. What, then, was the difference between the Scholastic Aristotelianism of the thirteenth century and that of the fourteenth century? To address these questions, I will establish three important distinctions in this chapter. First, I will distinguish between the way in which mathematics controlled the Ars antiqua and the way in which it grounded the Ars nova. Next, I will distinguish between the Pythagorean and Boethian mathematics of the earlier period and the very different mathematics of the later period. The differences in the mathematics of each period were paralleled by differences in the music of each time. Finally, I will distinguish between the theology of the earlier period and the increasingly mathematized theology of the latter period. A close analysis of the Nominalistic theology of the fourteenth century will demonstrate the need for a substantial refinement of the historical view that couples the Ars nova of the fourteenth century with the allegedly concurrent trend towards secularization. Heinrich Besseler, in his classic article about Johannes de Muris, believed the theories of the Ars nova conveyed the increasingly profane spirit of the four­ teenth century, expressed in the relaxed attitude towards binary values and duple meters. 1 Ulrich Michels challenged Besseler, arguing that Muris' aim in his Notitia artis musicae was not to legitimize imperfect rhythmical values, but rather to consolidate "something which had long before been there - namely, the Petronian division of the brevis- and he did not touch the perfect mensuration of the ars antiqud'. 2 Michels did not detect new liabilities in Muris' Notitia artis musicae, but acknowledged the enlargement of the old perfect system and its Heinrich Besseler, "Johannes de Muris", MGG •, v11 : 1 13. Ulrich Michels, Die Musiktraktate desJohannes de Muris, Beihefte zum AfMw, 8 (Wiesbaden : Franz Steiner, 1970), 74. 1

2

68

M ATH E M ATIZATION A ND DE - M AT H E M ATIZATION

arrangement into four grades. 3 Lawrence Gushee accepted Michels's view but attributed no concrete positive content to the allegedly revolutionary treatise of Johannes de Muris. In his authoritative article in the New Grove he wrote : the reader will understandably look for those things which might be termed revolutionary, in accordance with the at least partly mythical musical revolution called ars nova. He must look carefully, however. The tact, one might almost say deviousness, with which Muris brought in the idea of fundamental binary relations between the four levels of note values is most striking. 4 Recently Sarah Fuller has endorsed Max Haas's view that the Ars antiqua and the Ars nova should not be interpreted as opposites, but "mesh in an unbroken continuum in which the Nova is a complementary extension of the ars vetus". 5 Most characterizations of the shift from the Ars antiqua to the Ars nova have ignored the crucial complexities that must be considered if the two styles are to be distinguished accurately. How revolutionary was the Ars nova? Why did it boast itself to be a "new art"? �estions of continuity and change can be addressed in different ways. I will argue that although the Ars nova perpetuated the dominant figures of thought of the Ars antiqua, above all the ideal of rhythmic perfection, it nonetheless shifted these figures and this ideal to a new and unexpected context. God's time continued to measure music, but increas­ ingly the effects of his infinite freedom were felt as well. Tremendous efforts to understand and rationalize the irrational infinity of God resulted in a new positive attitude towards the notion of infinity and its derivatives, namely the notions of imperfection, disorder, disharmony, and the like. Hence, through the new prism of God's own infinity, the notion of perfection and its scope gained new meaning and became laden with new possibilities. Another similarity between the Ars nova and the Ars antiqua was their common predisposition towards the Aristotelian discourse. But this idiom was employed in the Ars nova in an utterly different context from that of the Ars antiqua: emphasis was shifted from definition and search for essences to quantitative analysis and representa­ tion of the relation between sensible phenomena. I propose to clarify the pertinent differences - but also the similarity and continuity -between the Ars antiqua and the Ars nova by suggesting a new and, I believe, improved reading of Muris' Notitia. My reading will make it possible to dismiss decisions about innovations and conceptual shifts based on answering questions such as binary values ...... in or out - and will allow for the simultaneous value of in and out. It will also undo the connection between secularization and 4 Ibid., 75. See Lawrence Gushee, "Jehan des Murs", New Grove, IX : 589.­ , Sarah Fuller, "A phantom treatise of the fourteenth century? the Ars nova", journal of musicology, 4 (1985-6) : 23--5 0, at 47. 3

R H YTH M M ATH E M ATIZ ED AND DE M YSTIF IED

duple meter in view of theological concern with the infinity of God or his infinite distance from the worlds or the infinities that he could know or create. For these various infinities stimulated ·great interest in infinite and irrational numbers, the very same numbers that, together with binary numbers, had been considered abhorrent - even abominations - until the fourteenth century. The reading that I propose will narrow the distance between the sacred and the secular, triple and duple, perfection and imperfection : it will show that in the fourteenth century basic categories such as rationality, order, beauty, proportion­ ality, or correctness in general were expanded to encompass (at least partially) their own disturbance or opposite as an organic constitutive part. Calculation and God: mensural procedures in theology, mathematics, and music Many of the new rhythmic ideas expressed in the theoretical works of the fourteenth century had unexpected counterparts in late-thirteenth- and early­ fourteenth-century theology and natural philosophy. That is, many of the rhythmic notions that preoccupied composers and theorists beginning with Johannes de Muris' Notitia artis musicae also circulated outside the realm of music, in expositions and questions on the Physics of Aristotle, in questiones or independ­ ent treatises, ·and above all in the ubiquitous commentaria Sententiarum, that is, the commentaries (written by every bachelor of arts) on the theological Sentences of Petrus Lombardus. Thus, in contradiction to received interpretations of music history, I claim that to a certain extent the ties between music and theology were stronger in the fourteenth century than in the thirteenth, because much of the theology in that period applied the very same mensural procedures and notions that gave fourteenth-century music its salient characteristics.6 Notions such as pars, totum, proportiones, maxima, minima, gradus, a parte ante, a parte post, mediate, immediate, and divisiones figured within theological treatises of the fourteenth century, most notably in commentaries on the Sentences, in a way that was as substantial and decisive as it was in the musical theories of the fourteenth century. It may be arguable whether the new common mensural tools permeating fourteenth­ century musical works were also woven into the intellectual fabric, or the other way round. Preoccupation with mensural problems was by no means merely a local musical matter, and since mensural discourses occur in a large portion 6 The point I am making reflects current modifications of the traditional view of the 14th c. as the final divorce of philosophy and theology; see John E. Murdoch, "The analytic character of late medieval learning: natural philosophy without nature", in Lawrence D. Roberts (ed.), Approaches to nature in the Middle Ages (Binghamton, NY: Center for Medieval and Early Renaissance Studies, 1982), 171-213. See also Courtenay, "Nominalism and late medieval religion".

C A LC U L ATION AND G OD

of philosophical and theological works, it is more likely the case that under­ standing the background and the driving force behind the fourteenth-century "measure mania" (as it has been called)7 may shed new light on the musical developments of the fourteenth century. Just how fourteenth-century theologians adapted and adopted mensural theories, we shall see in three examples. The first example concerns the quantifi­ cation of the notion of perfection as associated with "the great chain of being", and the measurement of the relationships between the whole and its parts and between parts and other parts in the context of the perfection of species. Music theorists of the fourteenth century were equally concerned with the quantifica­ tion of the rhythmic notion of perfection and, like theologians, measured mathematically the relation between whole and parts within the latitude of rhythmic prolongation. Thus this first example will highlight the climate of thought and the analytical tools that music theorists shared with theologians and natural philosophers. The second example illustrates the importation of the mathematics of the infinite to solve debates over the eternity of the world, where the penetration into the mathematics of continuity and infinity reached its culmination. The third example concerns the mathematical methods that were proferred to ex­ plain the increase of grace (caritas) through meritorious acts. As we shall see later, the quantification of rhythmic relations in the theory of the Ars nova advanced from the application of the most mathematically fruitful theory of measuring change in the degree of grace or any other qualitative changes. Measuring the perfection ofspecies

The quantification of the perfection of species involved analysis of the relation­ ship of all created species to God, and the relationship of one species to another, inferior as well as superior. Theologians approached the question mathemati­ cally, searching for an adequate scale for measuring perfection, devising criteria for measuring distances, and probing into methodological questions regarding the determination of the terminus relative to which perfections should be measured : that is, whether measurement should be performed relative to the maximal degree of perfection (accessus vel approximatio ad summum, or ad speciem supremam et perfictissimam) or rather relative to the minimal degree (recessus a non gradu vel a materia prima ). Other mathematical considerations involved the kind of excess to which species were susceptible, or whole-to-parts relationships within the scale of perfection ; that is, whether or not a given perfection contains inferior ones as a whole contains its parts.8 Interestingly enough, decisions 7 8

Murdoch, "From social into intellectual factors", 341. See John E. Murdoch, "Mathesis in philosophiam scholasticam introducta: the rise and develop71

R H Y T H M M A T H E M AT I Z E D A N D D E M Y S T I F I E D

about such questions were taken by drawing analogies with similar problems discussed in physical works or within natural philosophy. Thus, the possibility of measuring by proximity to the maximal degree, for example, was eliminated on the grounds that velocity cannot be measured relative to an infinite velocity. In some extreme but telling cases, philosophers used geometrical representation and proved their theological point geometrically. Murdoch singled out the mid­ fourteenth-century French theologian Pierre Ceffons, who used the infinite excess of a rectilinear angle over a horn angle (one between two curves), or the infinite excess between the angle of a semi-circle (180 °) and other rectilinear angles, in order to notate and demonstrate the infi n ite distance between radi­ cally diverse species. Ceffons also used these calculations to show that infinite increase in the perfection of individuals within a given inferior species will never reach the limits of its superior species.9 Measuring infinity

The consideration of God's absolute free will, omnipotence, and knowledge is sufficient to evoke the notion of infinity as the very core of theological reflec­ tions. However, for the purpose of highlighting the profundity of mathematical thinking within theological deliberations, I should like to elaborate on the problem of the eternity of the world as preconceived by all Greek thinkers against the Judeo-Christian belief in finite creation. Theologians invoked the paradox of the infinite to invalidate the possibility of an eternal world already in the thirteenth century. In brief, eternity entails a complete, that is, actually traversed infinite past time or infinite history ; hence, an eternal world is logically impossible. In the fourteenth century the paradox of the infinite shifted to the realm of mathematical insights : if the world is infinite in time, then its infinity today is greater than its infinity a week ago. Yet in violation of a basic mathematical axiom, the "whole" equals its "parts", since both such past times were infinite. 10 Fourteenth-century thinkers therefore had to cope with an extremely difficult intellectual challenge : translated into modern terminology, the apparent paradox results from the question _of "equality" between an infinite set and its infinite subset (more precisely their equal cardinality), and a resolution to the problem was only achieved by Georg Cantor in the late nineteenth century. Some thinkers, especially the Augustinians, accepted the paradox as given and used it to prove the absurdity of an eternal world. Others, however, .struggled with the ment of the application of mathematics in fourteenth century philosophy and theology", in A rts libtraux et philosophie au Moym Age: Acta du QJ!_atribne Congres International de Philosophie Midiivale (Montreal : Institut d'Etudes Medievales, 1969), 2 1 5-54, at 238-41. 9

10

Ibid., 243-8.

72

Ibid., 2 1 6-24

C A LC U L ATION A ND G OD

puzzle and attempted to resolve the paradox. Thus, within the context of commentaries on the Sentences, there developed the most advanced analysis of the mathematics of infinity : it was grasped that axioms regarding whole-to­ parts relationships, or the meaning of notions such as "greater" and "lesser", could not be transferred from finite magnitudes to infinite ones. Gregorius of Rimini, for example, in his lecture on the Sentences at Paris in 1 344, formulated a distinction between (in our terms) "whole and part in the sense of set and subset and whole and part in the sense of the unequal cardinality of the sets involved". 1 1 In search for a solution to the paradox of the infinite, theologians advanced the problem which is the crux of the mathematics of the infinite : the problem of the "equality" between a continuum and its parts. Even if incomplete, their analyses demonstrate how sweeping and deeply-probing were the mathematical analyses of the infinite within the context of theology. The question of why theologians were interested in solving the paradox, which effectively served the biblical truth as against the philosophical truth, will be taken up later. Measuring the increase of grace

To conclude this brief exposition I wish to turn to the question of the increase of grace ( caritas ) : the question of how a given amount of grace relates to a greater amount of grace -awarded for meritorious acts. This particular question de­ serves our attention, not only as another example of mathematical analysis within theology, but also in view of its bearing on the analogous question of the relation between longer and shorter rhythmic values. There is a striking correspondence between the Franconian attitude towards the relationship between note values and the Thomistic theory of variations in the degree of charity or in qualitative intensification in general. 1 2 Adapting Simplicius' Neoplatonic reading of Aristotle, Aquinas argued that, although different individuals may be endowed with more or less charity, the difference in the degree of charity was not a matter of the quantity of charity contracted on each individual. Accordingly, he distinguished between quantitas virtualis, the intensity of a given form, and quantitas corpora/is, the number of the bodies or Ibid., 2 24. "A multitude is called greater which contains one more times or contains more units (pluries continet unum vel plures unitates)" as against "every multitude which includes all the units of another multitude and certain other units is called greater than that other multitude, even though it does not include more units than it (includit unitates omnes alterius multitudinis et quasdam alias unitates dicitur maior ilia, esto quod non includat plures unitates quam ilia)". 12 The following discussion on qualitative intensification and the quantification of qualities is based on Marshall Clagett, "Richard Swineshead and late medieval physics", Osiris, 9 (1950) : 131-6 1 . 11

73

R H Y T H M MAT H E MAT I Z E D A N D D E MY S T I F I E D

subjects carrying this form. While accepting the general view that the quantity of bodies is measured by counting the number of bodies sharing the form, he argued in favor of the Aristotelian principle of participatio, according to which different degrees of intensity reflect different degrees of participation of the bodies in the immutable, indivisible form. Thomas distinguished between the pair of concepts "small" and "large" (parva vel magna) on the one hand and the pair "more" and "less" (magis et minus) on the other. Small and large refer to the number of bodies carrying a given quality, say the quality of "whiteness"; therefore they represent an arithmetical sum. But "more" and "less" signify the participation of the body in the immutable and indivisible quality. Thomas was aware that other philosophers held different opinions and took pains to reject the possibility that "charity increases in the way a number increases, that is, by the addition of charity to charity". 1 1 Other theologians postulated that a n increase i n charity involves the com­ plete destruction of the pre-existing quality and the acquisition of a totally new quality. Franco echoed this last view and combined it with the general Thom­ istic point of view. Longa and breve were conceived as different species (the species of the longa vis-a-vis the species of the breve) while the imperfect longa in its relation to the perfect longa implies a degree of participation in the quality of perfection. As we shall see below, Johannes de Muris not only grasped the relevance of this matter of variations in degree to the musical question of · conceptualizing the relationship between long and short note-values, but actu­ ally adapted the additive theory (rejected by Thomas) of increase by an apposi­ tion of part to part of the same quality. Muris' quantitative theory of the latitude of rhythmic values bears a surprising affinity to a slightly later theory of the intention and remission of forms developed by a group of English mathemati­ cians known as the Oxford Calculators. These mathematicians were active at Merton College, Oxford, during the first half of the fourteenth century. The Oxonian theories of motion, and especially their analyses of the process of intensification or acceleration, served commentators on the Sentences as a model for theological measurements of distances within the latitude of the perfection of species discussed above. Mensuration and theology

But what was the impact of such measurements on theology? Did it widen the gap between the human and the divine or rather narrow it down ? Or perhaps both, in a complicated dialectical relation ? Consider for example Gregorius of Rimini's attempt to solve the paradox of unequal infinities. In book III of the Physics and book I of De caelo, Aristotle had argued against the existence of an 13

Clagett, �Richard Swineshead and late medieval physics", 132-6.

74

C A LCU L ATION A ND G OD

actual completed infinity, showing the incompatibility of actual infinity and other Aristotelian physical principles. But when the question received a theo­ logical twist, and regarded God's power to create an actual infinity, then it became clear that God's absolute power was not subject to Aristotelian physics. The impossibility of actual infinity was therefore to be rejected not by reference to physical rules, but through the much stronger and universally binding logical rule of contradiction, as exemplified in the paradox of unequal infinities. Now, Gregorius' solution of this paradox, which seems to implicitly validate the non­ Christian belief in the eternity of the world, may have been motivated by the wish to extend God 's infinite power to the limit, so as to make it possible for God, owing to his absolute power, to create an actual infinite magnitude, if he so wished. But the keen desire to magnify God may in a paradoxical way make him so distant that the human mind can no longer understand anything about God -a condition that may eventually lead to religious apathy, to indifference, and ultimately to secularization. Alternatively, the introduction of mathematical analyses might serve the purpose of sharpening and improving theological arguments. If so, such an improved theology might become almost an extension of natural philosophy, against the interest of keeping the nature of theology sui generis. Might it have been the quest for certitude and evidence that nu·rtured the striving for the exactness that only mathematics could deliver, so as to recognize the utmost assistance of mathematical-physical material in the unfolding of difficult theo­ logical issues? This last conjecture implies that in the purely secular advance­ ment of mathematical and physical knowledge the autonomous development of the sciences was allowed, justified, and even encouraged on the grounds of their supposed applicability and usefulness for theology. None of these questions has a single clear and simple answer, but they all illustrate how partial and misin­ formed are musicological interpretations that view the introduction of duple meter as a mere sign of secularization. Theological considerations still played a constitutive role in the age of Muris' Notitia artis musicae but under circum­ stances that are different from those of the Ars antiqua. As I have already noted, it was the very alertness to God's infi n ity that opened up the realm of the infinite, the imperfect, and the irrational as worthy of scientific investigation. How far such investigations were pursued in music as well as in other fields will be considered later in Chapters 6 and 7.

Aristotelian thought in thefourteenth century The quantification of rhythmic perfection in Johannes de Muris' Notitia artis musicae was carried through in the Aristotelian metalanguage of the Franconians. The presence of Aristotelian quotations and the persistence of the Aristotelian 75

R H YT H M M ATH E M ATI Z ED A ND D E M Y ST I F I ED

qualitative discourse in the Notitia is another aspect of the transition from the Ars antiqua to the Ars nova that needs clarification and differentiation. How was Aristotle used in the fourteenth century? What distinguishes fourteenth-century Scholasticism by comparison with the thirteenth-century movement? What could Aristotle, with his qualitative physics, contribute to the "measure mania" of the fourteenth century? These are key questions for a better understanding of the relation between the Ars nova and the Ars antiqua, and for tracing the sources and context of Muris' innovations. Comparing fourteenth-century natural philosophy with its thirteenth-century counterpart, John Murdoch concluded that "there was much less of Aristotle there and in a different way . . . for fourteenth-century philosophy, especially natural philosophy, bore a largely reformulated Aristotle, reformulated in terms consistent with the new logical and epistemological requirements developed in that century". 1 4 In other words, Aristotle was the starting point of fourteenth­ century embellishments that led to directions that suited the new sensitivities of the time. It is not surprising, then, that fourteenth-century scholars were not much interested in Aristotle's account of motion as an actualization of a poten­ tiality or as a change from privation to being or from passion to action. They tended to focus on those arguments in which Aristotle set up the logical condition for measurements -what can be measured and how - and on those Aristotelian arguments that carry a direct or indirect mathematical or mensural content. But why were thinkers so eager to know about measurement? What was the new epistemology that encouraged quantitative aptitudes above all, and in so doing in fact nourished the fourteenth century "measure mania"? A broader analysis of the new Ockhamist epistemology will be attempted in Chapter 5. Here I confine myself to the much discussed fourteenth-century tendency to move away from speculative thinking, to translate abstractions into concrete things, and to shift the focus from universals to particulars. This switch from abstractions to reality marginalized the typical thirteenth-century Scholas­ tic method of mapping things in their exact place within an ontological hierarchy. In the fourteenth century, notably, descriptions became defined in terms of the thing itself, or as compared with other similar particulars. Signifi­ cantly, then, this new epistemology induces measurements, for if one looks at a particular thing and sees that it is whiter, or bigger, than other particulars of the same species, one is driven to express through some kind of measurement how one thing is whiter or larger than another. 15 The attention of natural •• Murdoch, "From social into intellectual factors", 308-9. " The new emphasis on concrete sensible entities does not automatically imply an empirical attitude towards science. As we shall see, science was to a large extent conducted through the application of logical doctrine, whereby the object of inquiry shifted from the natural event to its representation in language. See below, pp. 149-55. Yet, nevertheless, some mathematical and

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philosophers was naturally shifting towards the (scarce) places in Aristotle's writings that concern mathematics in general and measurements in particular. For the purpose of setting the stage for the analysis of Muris' N otiti a, four Aristotelian points are of special interest. The first concerns the restrictions that Aristotle imposed on the mathematical conceptualization of motion and change ; the second involves his notion of the unity and autonomy of the scientific disciplines. The third is Aristotle's account of the relationship between velocity and force, and the fourth is Aristotle's conception of the equality between motions. Given that mathematical and physical objects differ in their very substrate, Aristotle maintained that the one cannot explain the other, thus limiting the application of mathematics to static phenomena only : to bodies at rest or to the periodic motion of the spheres (which, being uniform and cyclical, can be regarded as rest and motion at the same time). Thus, just as physics was kept autonomous with respect to mathematics, so too arithmetical computations could not be applied to geometrical problems because one cannot mix numbers and magnitudes, or the discrete and the continuous. Even within geometry, as I have already mentioned, one can relate figures of the same kind only: lines to lines, curves to curves etc. Under these restrictions qualities could not be measured by quantity. Mathematical accounts of intensive changes were utterly inconceivable, as were questions of understanding rates of change when the motions involved acceleration or retardation. For Aristotle, as we already noted, "there can not be motion of motion or becoming of becoming or in general change of change". 1 6 In his Physics (book v11.4), Aristotle was specifically con­ cerned with motion, with the equality of different motions, and with how motions can be compared. There he limited comparison between motions only to motions of the same kind ; alteration or qualitative mutation cannot be compared with local motion, and, moreover, motion along a circular path can not be compared with linear motion, because circles and lines belong to different species. Both principles are specific instances of the general injunction against cross­ ing genera and mixing categories. In this way Aristotle insisted on the absolute unity and autonomy of each science and prohibited any attempt to carry over methods or principles from one discipline to another. But it was Aristotle himself who qualified the generality of what he called met ab asis-the fault of mixing categories or genera. Aristotle conceded that there were exceptions to his prohibition of transdisciplinary demonstrations and devised a special cat­ egory of "subaltern" sciences (called in the thirteenth century sci entia e mediae, physical works were written with the dear intention of describing reality. See below, p. 8 5 ; see also Ch. 8. 16

Aristotle, Physics, £2, 225 b 15.

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intermediate between natural philosophy and pure mathematics). 17 In Aristotle's list of such inferior sciences, which incorporates demonstrations of their corre­ sponding superior sciences, one finds music subordinated to arithmetic ; optics, mechanics, and astronomy to geometry ; the study of the rainbow to optics ; navigation and star-gazing (judicial astrology) to astronomy. To be sure, Aris­ totle contended that practitioners of the inferior sciences merely know the facts, while the mathematicians of the superior sciences have the demonstrations and the explanations (this is the classic distinction between arsltechne and scientia/ epistlme). The fact remains, however, that he himself created the conceptual tool in favor of which his commentators would eventually discard his principle of disciplinary autonomy, so as to turn the exceptions into the rule : the vast area of subaltern sciences developed rapidly in the fourteenth century and embraced more and more sciences as the century progressed. Aristotle's own hesitations, his virtual double-talk about autonomous yet subordinate sciences, attracted the attention of his commentators. Take for example Aristotle's ambiguous claim that it is not possible to apply the arith­ metical demonstration to the attributes of extended magnitudes, unless extended magnitudes are numbers. Did Aristotle mean that an extended magnitude is or is not, in the final analysis, reducible to a number? Since Aristotle did not leave a clear answer, his commentators were free to find one for themselves. In trans­ lations of Euclid's books philosophers observed the similarity between Euclid­ ean arithmetical and geometrical propositions. This anti-Aristotelian theme (the affinity between arithmetic and geometry) was developed from the twelfth century and throughout the later Middle Ages, but with particular vigor, persistence, and success by the fourteenth-century Oxford Calculators. Their enthusiasm for quantification and measurement was indeed outstanding. This theme, I will argue in the ensuing discussion, is central for decoding the historical meaning of Johannes de Muris' Notitia artts musicae. Among those Aristotelian observations that implied some kind of mathemati­ zation I will single out two incidents : in his Physics (book v11.5) Aristotle investigated the proportions involved in motion and declared that velocity is in direct proportion to the moving force and in inverse proportion to the resistance of the medium. This can be expressed with anachronistic notation as V = FIR. Aristotle's argument rests on the preconception that natural motion must be represented by simple mathematical relationships, a maxim that also held for the music of the Ars antiqua. But Aristotle's reasoning was inadequate : for one thing, V is greater than zero even if R is much greater than F, as if a fl y could move a mountain ; for another, the argument fails to account for how a runner, say, 17

For a comprehensive discussion of the unity and subalternation of the sciences, see Steven

J. Livesey; Theology and science in thefourteenth coztury: thru questions on the unity and subaltmzation of the sciencesfromjohn of Reading's Commentary on the Soztences (Leiden : Brill, 1 989), 20-76.

A R ISTOT E L I AN T H O U G HT IN T H E F O U RT E E NTH C E NTU RY

expends more force accelerating to his maximum velocity than maintaining it, though the resistance be the same. This error of Aristotle's engendered far­ reaching studies on the proportional relations between velocity and forces, whereby more complex proportions were proven to be more adequate for a mathematical description of motion. As we shall see later, this new type of proportion supported Johannes de Muris' innovative possibilities of imperfec­ tion by remote parts. In the Categories (ch. 8), Aristotle observed that qualities admit of variation of degree and that the same object may exhibit a quality in a greater degree than it did before. Yet, he continued, not all qualities admit such change, such as qualities that are dispositions, like the quality of knowing grammar or the quality of health. Here it is the object that varies in the degree that it possesses and not the quality itself. These remarks of Aristotle concern the very philo­ sophical base of measurement and initiated a heated debate over theological questions of how God's grace (caritas) is augmented. As I have already stated, repercussions of this debate, defined in the fourteenth century as the problem of "the intension and remission of forms", clearly resonate in Johannes de Muris' text, and provide an important key for the assessment of its innovative import.

Johannes de Muns and the Oxford Calculators It is worth while inquiring into the relationship between Muris' mathematical and astronomical works as well as his contribution to the articulation and repre­ sentation of musical time. Can we attribute his innovations in the field of men­ sural notation to his achievements as a mathematician and astronomer? Muris' accomplishments as an astronomer, especially his original contributions to the diffusion of the Alfonsine tables in his astronomical writings from 132 1 onwards, attest to his experience with concrete quantification. His astronomical works concern measurements of time and motion in a scientia-media discipline that was on the one hand part and parcel of the academic Qyadrivium and on the other hand associated with such practical needs as the accurate rendering of a perpetual calendar and devising means for telling time at different times of the year. 1 8 Less well known is Muris' treatise on �otion, De moventibus et motis, encapsu­ lated within the fourth book of his best-known mathematical work, the OJ!_adri­ partitum numerorum of 1343. 19 Here he studied in the abstract comparisons of 1 • Emmanuel Poulle, "John de Murs", DSB, vn : 1 28-33, emphasizes not only Muris' know­ ledge of trigonometry and geometry, but also his deep concern with both adjusting theoretical calculations to practical usage and recording astronomical observations, which were in general very rarely performed during the Middle Ages. 1 9 For a new edition and study of this source, see Lt OJ!_adripartitum numuorum dejean de Murs,

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different rectilinear and circular motions, sometimes commensurable, sometimes incommensurable, applying numbers and fractions to physical quantities as he did in his astronomical works. In these writings Muris clearly broke with the Aristotelian paradigm, but he made no reference to natural motions involving acceleration or deceleration or to motion against a resisting force. In fact, there is no reference in the whole of the treatise to these two key concepts of fourteenth-century Mertonian mathematical physics, namely acceleration and resistance. In her recent edition of Muris' OJ!,adripartitum, Ghislaine L'Huillier stressed the conservative tone of Muris' mechanics. Muris, she argued, clearly adhered to the simple Aristotelian theory of the relationship between velocity and force (V = FIR ) and omitted all mention of the complicated but widely accepted exponential formula proposed by the Mertonian Calculator Thomas Bradwardine in his Tractatus de proportionibus of 1328, which can be expressed as Fi!R2 = (Fi IR 1 ) V, 1 v._ 20 According to L'Huillier a goodly number of French mathematicians, including Jean Buridan, were reluctant at the time to adopt Bradwardine's law of dynamics. Although L'Huiller did not explain her finding, it appears that French mathematicians rejected Bradwardine's exponential ex­ pression as a legitimate interpretation of the Aristotelian relationship between velocity and force, 2 1 probably because it jarred with Aristotle's explicit warning against the application of excessively complicated mathematics to physics (such as the compound proportions that Bradwardine applied in his formula). 22 L'Huiller argued that in reaffirming the simple Aristotelian relation, Muris sided with Buridan, and she concluded that Muris cannot be counted among the pioneers of scientific research in the fourteenth century- at least with regard to physics proper. 23 Viewed against this background, the very presence of Calculatory methods ed. Ghislaine L'Huillier, Memoires et documents publics par la Societe de !'Ecole des Chartes, 32 (Geneva : Droz, 1990).

20 For Muris' analysis of motion see OJ!_adripartitum numuorum, 485 and 508-1 1. For Brad­ wardine's law of dynamics see Thomas of Bradwardine: his Tractatus de proportionibus; its signifi­ cancefor the development of mathematical physics, ed. and trans. H. Lamar Crosby, Jr, University of Wisconsin publications in medieval science, 2 (Madison : University of Wisconsin Press, 1955),

75-86.

Bradwardine corrected an inherent contradiction of Aristotle's rule, namely, that motion occurs even when resistance is greater than force (see above). 22 At most, according to Aristotle, mathematical fi gu res were useful for physics to represent a body at rest or in periodic motion, that is the regular, simple, and uniform celestial motions. Furthermore, representation of change in intensity, or acceleration was impossible for the very notion of a rate of change was impossible: "There cannot be motion of motion or becoming of becoming or in general change of change." (Aristotle, Physics, £2, 225b 1 5). 21 L'Huillier, Le O!!ftdripartitum Numuorum dejean de Murs, 29. But L'Huillier argued that, as far as pure mathematics is concerned, Muris was not only familiar with concurrent develop­ ments but actually inspired them and even to a certain extent anticipated them. See ibid., 23-9. 21

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and ideas in the musical discourse of Muris (including, among others, the compound proportion of proportion as involved in Muris' innovative idea of imperfecting by remote parts) is intriguing. Rereading Muris' Notitia artis musicae of 1321, this relationship to the Oxford Calculators raises several questions concerning the early history of the Mertonian school, its possible sources, its cultural resonance and social impact ; the position of Muris in the history of mathematics and mathematical physics, and finally the relationship among Muris' mathematical, astronomical, and musical works. From the perspective of music history the ensuing discussion will demonstrate that the new mathematics of the fourteenth century played a significant role in the gradual emergence of a new concept of musical esthetics that paved the way to the acceptance of musical imperfection and discord as constitutive parts of a new ideal of beauty in music. The boundaries of the argument

No direct evidence has yet been found to substantiate an assumption of per­ sonal links or even indirect contact between Muris and the Oxford Calculators. (This may be a consequence of the paucity of available biographical data on the individuals concerned.) My argument, therefore, rests entirely on textual analy­ sis. Most urgent, for the sake of the arguments below, is the question of the direction of dependency : was Muris' advanced musical thought nourished by Mertonian mathematical knowledge, or, conversely, should we attribute the mathematical breakthroughs of the Mertonians to contemporaneous musical wisdom ? Perhaps a third alternative exists : that both Muris' musical innova­ tions and the Calculators' new mathematics emerged independently as two distinct manifestations of a common source. To set the scene for probing these questions further, we need first to eliminate the possibility that the similarity between the Calculators' works and Muris' musical thought was merely acci­ dental, an insignificant coincidence of two separate and unrelated develop­ ments. Historians of science have repeatedly argued that their interest in the quantifi­ cation of qualities was by no means peculiar to the Mertonian Calculators. It was encouraged, if not actually initiated, by theological discussions of the increase of caritas; by medical-pharmaceutical quests for enhancing medical treatments by compounding drug effects ; and by studies of the intensification and propagation of light found both in optics and in debates concerned with divine illumination. In other words, these attempts to rationalize and account for processes such as motion, growth, intensification, or remission attest to a widespread, central, and general concern of the time. Musical and mathematical theories of variability must then be seen as part of this broader context of the fourteenth-century preoccupation with quantification and measurement. It would be misleading, 81

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certainly as a first hypothesis, to interpret the resemblance between the musical and mathematical discourse as mere coincidence. Indirect circumstantial evi­ dence may support the conjecture that the two activities stemmed from a com­ mon source. Marshall Clagett has noted the influence of the thirteenth-century French geometry of motions on the kinematical works of Thomas Bradwardine, the founder of the Mertonian Calculatory tradition. 2• Since Muris worked at the Sorbonne, where he was already a master of arts in 132 1 , we can assume that he was familiar with such French mathematical works. Indeed, his earliest math­ ematical work, the Canones tabule tabularum mentions a squaring of the circle quoted also in Gerardus of Brussels's Liber de motu (c. 1 2 50). 25 The possibility that Muris' ideas were inspired by the Mertonian mathematical revolution is weak­ ened by the fact, already mentioned, that he ignored the innovations of the Calculators in the essay on mechanics in book 1v of his OJ!,adripartitum nume­ rorum.

Finally, and conversely, no evidence can be adduced to support the hypo­ thesis that the Calculators absorbed Muris' notions and principles in their own works. There is one significant reference to a theory of music in Bradwardine's Tractatus de proportionibus of 1 3 2 8 . Bradwardine referred to the ancient Pythago­ rean theory of harmony as authoritative and corroborating evidence for measur­ ing continuous quantities (in this case musical strings) by discrete numbers. 26 Musical harmonies were expressed by simple ratios such as 2 : 1 or 3 : 2, which describe the proportional relation between the lengths of two strings. Brad­ wardine here admitted that the philosophical roots of his mathematical ap­ proach to physics rested in the Platonic-Pythagorean tradition, a tradition that was subdued or toned down in the realm of natural philosophy during the age of Scholasticism, but prevailed in the domain of music side by side with Scholastic musical thought throughout the Middle Ages. Bradwardine's notion of music, therefore, seems to have been anchored more in classical textbook expositions of the science of music than in the surrounding soundscape or in contemporaneous developments in musical theory. The affinity between musical and mathematical-physical discourse

It is of paramount significance that Johannes de Muris declared at the very beginning of his musica practica (the second book of the Notitia ) that sound is generated by motion, is measured by time, and is therefore transient by nature. As Ulrich Michels has shown, Muris was relying on Aristotle's De anima 11.7-8 Marshall Clagett, "Gerard of Brussels", DSB, v : 360. For further discussion on the relation between Gerardus' Lib(r de motu and Johannes de Muris' musical thought see below, pp. 98-9. 26 Bradwardine, Tractatus de proportionibus, 74-5. 24 2

'

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and o n Boethius' De institutione musica 1.3. 21 I t is not a mere coincidence that Muris quoted from Aristotle's De anima, where Aristotle discussed the empirical aspects of sound and the relation between sound and motion. Thus Muris prepared the ground for a discussion that subordinates music to natural philoso­ phy in general and to theories about motion in time in particular. The signifi­ cance of this development should not be overlooked : the traditional bond between music and pure arithmetic, with its universal validity, seems to be dissolved here and replaced by a more promising association of music with other natural phenomena, equally transient and moving in time. It is true that already in the Franconian phase music had become gradually subjected to natural philosophy. The Franconian application of conceptions, definitions, and principles drawn from Aristotle's Physics and Metaphysics marks a definite depar­ ture from the Pythagorean-Boethian tradition. But the Aristotelian notions were applied for a systematic classification of rhythmical entities according to essential qualities, and for ordering the discussion. The transitory character of musical sound and its relation to motion in time was de-emphasized not only by the qualitative discourse of the Franconians but also by the tyranny of perfec­ tion, whereby musical motion was reduced to a static periodic motion. This, as we observed earlier, was perfectly suitable for the thirteenth-century under­ standing of music as imago Dei. Granting this continuity of time, Muris saw the various note-values as different parts of a single continuum. For him, long and short belonged to one and the same whole in the same dimension : Temporis aliud maius aliud minus: maius, quod motum prolixiorem, minus, quod breviorem habet ceteris eisdem, secundum unam dimensio­ nem metitur. Haec autem specie non differunt, nam maius et minus speciem non variant. 28 One (part] of time is greater, the other lesser ; the greater, which has a longer motion, and the lesser, which has a shorter (other things being equal), are measured according to one · dimension. But these do not differ in species, for greater and lesser do not vary the species. It is clear that Muris argued here against the Franconian scheme of rhythmic classification according to essences, and the resulting division of rhythmic Muris, Notitia artis musicat, 49. Ibid., 66. Michels, Dit Musiktraktatt dtSjohanntS dt Muris, 73, suggests that Muris speaks in this passage about the general notion of time (as distinct from musical time) : "Auf den Zeitbegriff im allgemeinen, nicht im musikalisch speziellen Sinn bezieht sich die Erklarung zum ttmpus maius und minus bei Muris." I interpret the passage as referring to musical time. Therefore, the term specitS in the above quotation refers to "short" and "long" prolongations ("maius, quod motum prolixiorem, minus, quod breviorem habet ceteris eisdem") and not, as implied by Michels's commentary, to "perfect" and "imperfect" rhythmical values. 21

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motion into three distinct species (longa, breve, and semibreve). Muris inherited from his predecessors the conception of the rhythmic component - the prop­ erty of prolongation - as a quality attributed to the subject, which is the non­ measured sound. But unlike his predecessors he believed that qualities can be increased by addition in a manner similar to the addition of discrete objects. Muris probably knew the variety of opinions about the mechanism of intensifi­ cation, and he seems to have sided with Richard Middleton (d. c. 1360) and Duns Scotus (1265-1308), who argued that "more" or "less" of the same quality does not change the species, since in the case of increase in the degree of a quality there is a new reality added to the pre-existing one. In contrast to the" Aristote­ lian-Thomistic doctrine, Muris like Scotus posited that accidental qualities have a latitude (latitudo fomutrum ), which means that they are divisible into infinite "parts", that is, infinite possible degrees of intensity. If the degree is changed, the individual quality also changes. In other words, in this new ontology things are no longer individuated by their substantial form, but rather by all their accidental forms which are particu­ lar to each individual. Thus, when accidental qualities are contracted into an individual thing, they are represented as concrete instances at the given degrees of intensity. The range of a quality in its individual instances could then be interpreted as an increase in the number of parts of that quality. Such an account of intensification lends itself naturally to a geometrical or arithmetical represen­ tation : since it is the quality itself that has parts, variation in the intensity of a quality can be explained as the addition or subtraction of parts of that quality. 29 The very core of Muris' theory is the insistence on the continuous dimension of time in which all types of values (longer and shorter) participate. Thus, in the general framework of a decisive conceptual upheaval, Muris' rhythmic theory - which, on the face of it, can hardly be said to carry any metaphysical import- does nonetheless imply a potent philosophical shift. It is solidly tied to a novel quantitative approach to qualitative variation that emerged in the fourteenth century. The Oxford Calculators and the latitude of qualities The history, as well as the driving force, behind the Scotist revolution have been extensively studied and documented. 30 What concerns us here, however, is its 29 Edith Sylla, "Medieval concepts of the latitude of forms : the Oxford Calculators", Archives d'histoire doctrinale et littiraire du Moym Age, 40 (1973) : 223-83, at 2 51-7. See also Amos Funkenstein, Theology and the scimtific imagination from the Middle Ages to the stvmttmth cmtury

(Princeton: Princeton University Press, 1 986), 308-9. 0 For the history of metabasis in the Middle Ages, see Steven J. Livesey, "Metabasis: the ' interrelationship of sciences in antiquity and the Middle Ages" (PhD dissertation, University of California, Los Angeles, 1982). For the theological background of the dissolution of the

J O H A N N ES D E M U R I S A N D T H E O X F O R D C A LC U LATO RS

scientific application, the most important of which was the new fourteenth­ century mathematization of motion or other processes in time. In mathematics, Scotus' philosophical refinements found immediate repercussions in the works of the Mertonian Calculators. Their mathematical theories certainly penetrated Parisian circles, where they often underwent significant change and develop­ ment. In France as in England, doctrines of measurement were applied to theological problems, most notably in commentaries on the Sentences. The Oxford Calculators were a heterogeneous group of logicians and math­ ematicians who developed quite varied types of work that recent scholarship divides into two branches : the logical-sophistic branch associated with univer­ sity disputations, and the mathematical branch linked to natural philosophy or physics. 3 1 Furthermore, even when analysing the same problem, whether or not for the same purpose, they split into rival parties. Focussing on the problem of variability, for example, they completely disagreed over the proper mathematical model for explaining variability, how it should be applied, and how one understands the relationship between latitude and degrees. Some measured intensity by proximity to the maximal degree of the latitude, and remission by the distance from this maximal degree, 3 2 while others measured intensity by the distance from the zero degree of the latitude and remission by the proximity to the zero degree. 33 But either way their approach marked a revolutionary turn towards the mathematization of natural phenomena involving some kind of mutation. Johannes de Muris shared the premise with the core of the Oxford Calculators that there must exist a larger whole of the same quality or species, which allows comparison by quantification. Both understood the latitude of qualities to be a continuum. The latter, by definition, cannot be composed of things of different species.34 Among the Calculators John Dumbleton is of special interest for us, since his account of latitude and degrees seems to be very close to Muris' notion of the latitude of rhythmic duration. Furthermore, recent studies stress that Dumble­ ton's theory of physical variability was not merely theoretical or speculative, but tied to reality and claiming to describe reality. This new finding makes a com­ parison between his ideas and Muris' practical theory all the more interesting. 3 5 Aristotelian world-view and the resulting early experimentations with mixing genera or quanti­ fication of qualities, see Sylla, "Medieval concepts of the latitude of forms", and also eadem, "Medieval quantifications of qualities: the 'Merton school'", Archivt for the history of exact sciences, 8 (1971) : 9-39. Edith Sylla, "The Oxford Calculators", in CHIMP, 540-63. H Ibid., 140-52. Clagett, "Richard Swineshead and late medieval physics", 132-8. 14 Dumbleton's argument reads as follows : "Item non proprie fit distantia ex illis que sunt diversi speciei et non communicant" (quoted in Sylla, "Medieval concepts of the latitude of forms", 253 n. 79). 15 See Edith Sylla, "The Oxford Calculators and mathematical physics: John Dumbleton's 31

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Dumbleton stressed the distinction between essential, non-variable qualities and accidental, variable ones 36 and conceived degrees of intensity to be divisible into their constitutive parts. For him "latitudes" are physically identified as the intensity of a quality at given points. 37 In other words, he identified the latitude of a quality with the degree (gradus) of intensity as contracted into an individual in its specific instantiation. This enabled him to develop a one-dimensional co­ ordinate system to represent this latitude of forms. The increase of intensity was seen as analogous to a geometrical line, in which the parts, however minute, represent degrees of intensity. Intensification in this context entails passing through all the intensities between the original degree and the degree attained. Dumbleton's method made it possible to measure the variability of real natural processes by comparing different lines representing different intensities, that is, by adding or subtracting parts of intensities. 38 Of crucial significance is the fact that this notion of a latitude of qualities makes apparently opposite. qualities - such as hot and cold or fast and slow­ the two extremes or delimiting points of a continuum that extends from the minimum to the maximum degree of intensity. 3 9 This crucial elimination of the categorical opposition and contrast between the two extremes of the same qual­ ity is one of the most important contributions of fourteenth-century thought to the rise of early modern intellectual culture. A whole system based on hierarchy and compartmentalization began here to give way to increasingly daring gener­ alizations which led eventually to the modern notion of variety and diversity as simply the full unfolding of an essentially homogeneous formative principle. Summa logicae et philosophiae naturalis, Parts I I and I II", in Sabetai Unguru (ed.), Physics, cosmology

and astronomy, 1300-1700: tension and accommodation, Boston studies in the philosophy of

science, 1 2 6 (Dordrecht: Kluwer, 1 99 1), 129-6 1 .

36 Sylla, "Medieval concepts o f the latitude o f forms", 253 n. 79. A s already mentioned, this new approach diametrically opposed Thomas's distinction between the pair of concepts "small" and "large" ( parva vtl magna) on the one hand and the pair "more" and "less" (magis et minus) on the other. See above, p. 74. See also Clagett, "Richard Swineshead and late medieval physics", 37 Sylla, "Medieval concepts of the latitude of forms", 2 5 1-2. 132-4. 38 "Subiectum qualitatis intenditur et remittitur per adquisitionem et deperditionem realem qualitatum sicut quantitas maioratur et minoratur per appositionem partium et amotionem earundem" (Ibid., 255 n. 83). Attention should be called to one of the earlier pre-14th-c. developments, which may have influenced Dumbleton's theory. It is well known that one of the major sources of quantification of qualities was the Arabic and other medical literature. Galen's De simplicibus medicinis and Constantine's Liber graduum explained the range of variation in human health in terms of the latitude of the human temperament. This latitude, they argued, contains four degrees, each having three parts. Curiously, Muris likewise divided the latitude of duration into four grades, each subdivided into three parts. See below, pp. 89-90. 39 In the terminology of the Calculators, the minimal degree was often named non gradus, that is, zero degree of intensity. The maximum degree was usually refered to as gradus su-,,,mus, or gradus intmsissimus.

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But it was fourteenth-century philosophers and mathematicians who initiated this conceptual shift. And it had, already in the fourteenth century, important repercussions in esthetics in general, in musical esthetics in particular, and above all in the understanding of the relation between God and man. I shall discuss these developments later. Beyond Aristotle

Johannes de Muris and the Oxford Calculators were in many senses Aristote­ lians, while in another sense they clearly advanced far beyond the Aristotelian horizon. In line with Aristotle's world-view, they abstracted mathematical meas­ ures or properties from physical entities and studied them as if they were separate from real entities and deprived of existence. They also followed Aristo­ tle's discussion in the Categories of the property of qualities to admit variations. But they disregarded the restrictions that Aristotle imposed on a mathematical conceptualization of qualitative variability. They did away with the Aristotelian categorical distinction between mathematics and physics. Geometrical entities, such as lines or circles, were made useful not only for periodic motion but also for the representation of accelerated motion and of variability in general. In fact, they eroded the general Aristotelian injunction against transferring methods and mixing categories in their quantification of qualities through the application of arithmetical computations to continuous quantities. Significantly, not only the practice of doing physics was completely revolu­ tionized in comparison with the Aristotelian classificatory style, but also math­ ematics was now transformed : Dumbleton, as we saw, used geometrical lines to represent different latitudes. Other philosophers applied other geometrical fig­ ures, such as the various types of angles (horned angles, curvilinear angles) or concentric circles."° It is this idea of using geometrical figures for describing changing physical variables, or for representing and visualizing motions and mutations rather than static perfections, that reflects a nascent awareness of the formal symbolic nature of mathematical expression. In the discussion that follows, as elsewhere (especially in Chapters 6-8), I will discuss the conceptuali­ zation and representation of increasingly complicated rhythmical relations in the light of this shift in the mode of understanding mathematics and in its application to the study of change and variability. Here I contend that the new mathematical physics of the early fourteenth century and the new rhythmic theory of Muris reflect a similar mode of thought. The similarity that I have . in mind goes deeper than a common disposition towards quantification of qualitative changes. It concerns a common set of problems, communicated by similar vocabulary. In addition, it reflects a similar. 40

See above, p. 7 2 .

R H Y T H M M AT H E M AT I Z ED A ND D E M Y S T IF I ED

scientific attitude and a common methodology for working out these problems;" Further, we shall see in the theory of Johannes de Muris very subtle manifesta­ . tions - even if blurred by conflicting tendencies - of the transformation of mathematics into a mere tool, a language stripped of cultural-esthetic values. The affinity in question is not presented for its own sake as another exercise in historical analogy. Its true historical meaning goes so far as a claim that the fourteenth century saw the disenchantment and demythologization of both music and mathematics. This is reflected in the massive development of and experimentation with measurement performed with the hitherto illegitimate mathematics of the infinite, which was now applied everywhere : for measuring local motions, or qualitative mutations, or rhythmic relations, or relations among species within the latitude of their perfection. Whether in music or in physics, these measurements concerned the problem of relating finite values to infinitely divisible continua. The relaxation of the age-old opposition to musical imper­ fection in Muris' innovative musical thought can now be related to the rehabili­ tation of infinity in late-medieval philosophy. Infinity and imperfection in the thought ofMuris

Muris raised the problem of the infinity involved in the continuity of time and the finite nature of rhythmic values, and distinguished between the mathemati­ cal and the physical components that constitute a musical sound. Considered as a physical entity (Jonna naturalis), the duration of a musical sound has a specific latitude, limited by minimum and maximum. For in nature there are neither infinitely large nor infinitely small magnitudes ; rhythmical divisions are empiri­ cally limited by the nature of sound and of the human voice.42 �oniam ergo vox tempore mensurata unionem duarum formarum, naturalis scilicet et mathematicae, comprehendit, licet quod ratione alterius fractio non cessaret, tamen propter aliam vocis divisionem necessarium est alicubi terminari. Nam sicut omnium natura constan­ �ium positus est terminus et ratio magnitudinis et augmenti sic parvitatis et diminuti. Demonstrant enim naturales, quod natura ad maximum et minimum terminatur. · Vox autem est per se forma naturalis iuncta per accidens quantitati. Igitur oportet earn habere terminos fractionis, quorum latitudinem nulla 41 For a discussion of the use of these languages in philosophy and theology, see Murdoch, "From social into intellectual factors", 277-89. 42 Here we see Muris' sensitivity to the distinction between physics and mathematics. The fact that a sound can be measured by mathematics does not mean there is an identical structure between them. While in the abstract, mathematical measurement can be applied to a physical being, rhythmic divisions are empirically limited by the nature of sound and voice themselves.

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vox quantacumque frangibilis valeat praeterire. Hos autem terminos volumus comprehendere ratione:0 Therefore, since musical sound (vox) measured by time constitutes a union of two forms- the natural and the mathematical -even though according to the latter the division of the sound is endless, according to the former its division must end somewhere. Just as for all things existing by nature there is a limit to their magnitude and increase, so also to their smallness and decrease. Natural philosophers show that nature is limited by a maximum and minimum. Musical sound is in itself (per se) a natural form joined accidentally (per accidens) to quantity. It therefore ought to have limits to its division, the latitude of which cannot be exceeded by any sound, however divisible. We wish to understand these limits by reason. Johannes de Muris did not merely reflect the new mathematical trends of quantifying intensive magnitudes. He adopted them and spoke their analytical language.44 In the theory of the Calculator Dumbleton, the latitude of a given quality was abstracted and notated by a geometrical line. A homological model is evident in Muris' Notitia, where the rhythmic whole is represented by the maximal note-value, called in Muris' text the longissima. Neither the term nor the concept has a previous history, and the note had never before been seen in concrete music notation. Muris demonstrated the latitude and the limits (termini) of a musical sound by the following table, where the latitude of a prolonged sound is divided into four grades of perfection, and each grade is further subdivided into three parts in the relation of 3 : 2 : 1 . 45 First grade (Major mode)

Second grade (Mode)

.,

.,

,

Triplex longa Longissima Maxima

54 Duplex longa Longior Major

27 Simplex longa Longa Magna

27 Perfect longa Longa Perfecta

18 I mperfect longa Semilonga Imperfecta

Breve Breve Brevis

,

81

,

•

9

Muris, Notitia artis musicae, 69. In a philosophical vein, Muris grasped the revolutionary principle of quantifying physics, notwithstanding the structural incompatibility between mathematics and physics. His concep­ tual substratum does not necessitate a qualitative unity between the measure and the measured. Like Dumbleton, Muris gave separate reality to the latitude of given qualities and identified the latitude with the degree of intensity in a given instantiation. "' Table after W. Oliver Strunk, Source readings in music history (New York: W. W. Norton, 1 950), 177 ; I have italicized the note-names given in the table in Muris, Notitia artis musicae, 79. 43

44

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Third grade (Time)

•

9

Perfect breve Breve Brevis

Fourth grade (Prolation)

♦

3

•

6

♦

3

Imperfect breve Semi breve

Minor semibreve Minor

Brevior

Brevissima

♦

2

Perfect semibreve Minor

Imperfect semibreve Semiminor

Parva

Minor

♦

I

Minim Minim

Minima

The extreme low end of the musical continuum was now given the name of who explained in his De maximum speed for the motions of the heavens, but also for every action, for walking or for playing the lyre, there is a minimum time and a maximum speed for all such actions.46 Aristotle thus provided the very weapon with which his system was under­ mined, allowing the idea of a minimum quantitative dimension for the existence of natural substances side by side with his notion of the absolute continuity of matter. As terms belonging to the vocabulary of the Calculators' conceptual language of the intension and remission of forms, minimum and maximum were used abstractly, that is, secundum imaginationem, with a logical intent only. In this way the terms could denote acceleration or intensification to infinity and diminution to zero degree. 47 Dumbleton, however, insisted on the reality of the process of change and the reality of the latitude of the forms. Dealing with a musical sound as a concrete natural phenomenon, Muris had to determine a concrete, finite minimum and maximum. A physical minimum, unlike a math­ ematical minimum or a point, is indivisible in spite of having a definite length (perhaps a very short one). Thus, Muris argued in the passage quoted above, a sound, like all other natural phenomena, can have neither an infinitely large nor an infinitely small magnitude. Muris referred here to the Aristotelian notion of minima naturalia, but in the general anti-Aristotelian setting of his theory he turns the notion of minima naturalia in a different direction than atomism, towards concrete quantification. In Muris' quantified notion of rhythmic duration, all the longer degrees are measured by the rhythmic minima, and any degree representing a note-value is the sum of its constituent parts. 48 minima (sc. nota). The term itself came from Aristotle, caelo that not only is there a minimum time and a

46

Aristotle, De cat!o, B6, 288 b30-28914. For a discussion of the struggle with the notion of a rhythmic minima in the musical thought of the 14th c., see below, pp. 1 24-30. 47 The Calculators were not all of one voice regarding their understanding of the procedure involved in quantifying intension and remissions. For a panoramic view on their different concepts of measurement, see Clagett, "Richard Swineshead and late medieval physics". 48 Like Dumbleton, Muris gave separate reality to the latitude of given qualities and identified the latitude with the degree of intensity in a given instantiation. For Muris the

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Ab unitate igitur, quae tertia pars est ternarii qui perfectus est, usque ad 81, qui similiter est perfectus, dicuntur esse termini de maximo ad minimum cuiuslibet vocis, totaque eius longitudo inter hos terminos est inclusa. 49 So from unity, which is the third part of the number three, which is perfect, up to 81, which is similarly perfect, they are said to be the limits from the maximum to the minimum of any sound, and its entire length is included between these limits. The rhythmic latitude of sound is structurally identical with the latitudoformarum as described or symbolized by a geometrical line in Dumbleton's works. Muris' table of the four grades of perfection implies not only the idea of com­ mensurability of various lengths of time, but also previews a multitude of possi­ ble rhythmic combinations. Such combinations would make the texture of parts and whole more subtle. For example, Muris loosened the rigid and thrifty rhythmic combinations of his predecessors. He legitimized the procedure of imperfection not only within each of the four grades, but also by crossing the boundaries between the grades. Thus the rhythmic long, for example, could now be imperfected or diminished not only by its immediate part, the breve, but also by a remote part, namely the semibreve. Interestingly enough, in Muris' time, some of those rhythmic varieties did not yet have a musical counterpart. This is an important issue, which will be further elaborated in Chapter 6. The dense variability within the latitude, evident as a possibility in Muris' notion of musical time, also permeates the Calculators' discourse. The affinity between Muris' refinements of musical time and the Calculators' analyses of motion is founded on their shared attempt to introduce order into the disturbing problem of variability and disorder. Both mapped out all the possible types of different distributions of qualities throughout given segments of space or time. The Calculators emphasized the distinction between uniform (uniformis) and difform (differmis) distribution. In other words, they distinguished between subjects manifesting a quality in uniform intensity or a movement in uniform velocity (distribution of a quality in time) and subjects manifesting several intensities of a quality. The class of difform distributions was further divided into different types of non-regular distributions. In discussing velocity, the most common distinctions were between motus uniformis (uniform motion), motus dif­ formis (difform motion), motus uniformiter differmis (motion in which the velocity quality in question was the duration of a note-value. Being a physical entity (Jonna naturalis), a musical sound measured by time has its specific latitude, limited by minima and maxima. A sound, like all other natures, demonstrates that in nature there are neither infinitely large nor infinitely small magnitudes. 49

Muris, Notitia artis musicae, 72. 91

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increases uniformly) and motus uniformiter difformiter difformis (motion i n which the acceleration -increases uniformly). 50 These efforts towards rationalization by means of verbal arguments have a clear correspondence to Muris' Notitia artis musicae. Like the Calculators, Muris sorted out types of distributions throughout a definite extension in the dimen­ sion of time (the longa), with its axis on the field of rhythmic variability. His classifications are based on the distinction between perfect division (division into three equal parts) and imperfect division (division into two equal parts). In sorting out internal differences in the composition of the rhythmic longa Muris distinguished between : (1) longa perfecte perfecta -a perfect whole consisting of three perfect breves, which contain nine semibreves ; (2) longa perfecte imper­ fecta-one whole consisting of two perfect breves, altogether six semibreves ; (3) longa imperfecte perfecta-a longa diminished or imperfected by one or two of its nine constitutive semibreves, and (4) longa imperfecte imperfecta- a longa diminished or imperfected by one breve plus one semibreve. The terminological resemblance between the distinctions in mathematics and music is apparent, as is the resemblance of content. 5 1 Let us take a closer look into longa imperfecte perfecta and longa imperfecte imperfecta. Both, once again, represent a new option of diminution or imperfec­ tion by the parts of parts. In other words, Muris' rhythmic ideas imply the operation not only with simple proportions (the proportion between a longa and a breve - the immediate part of the longa), but also with compound proportion, or proportions of proportions (the proportion between a longa and a semibreve). 5 2 Composite proportions serve to relate the whole long to the parts of its parts. But according to the Pythagorean-Boethian tradition, propor­ tions that depart from simplicity and singleness (all proportions other than multiple or superparticular ones) are excluded from consonance and harmony and have no place in a well-ordered musical system. 5 3 Muris, however, seems to have gone against the grain of this fundamental rule and to have applied in his novel rhythmic system the hitherto inconceivable and cumbersome composite proportions. If so, are we not looking at the moment where music was no longer w Marshall Clagett, The scirna of mechanics in the Middle Ages, University of Wisconsin publications in medieval science, 4 (Madison: University of Wisconsin Press, 1 959), 247-8. " See Muris, Notitia artis musicae, 94-7. It could be the case that Muris or the Calculators had adapted Boethius' division of numbers into paritu par, part"tu t'mpar, t'mparitu par, impariter impar. The point is that such traditional schemes of classification were applied in the early 14th c. to the heretofore inconceivable realm of variability and contingency. 2 ' Muris, Notitia artis musicae, 93 : "Semibrevis est pars ternaria longae, nam est tertia pars brevis, et brevis est tertia pars longae, ergo semibrevis est tertia pars tertiae partis longae. Igitur imperficere potest longam." 3 ' Anicius Manlius Severinus Boethius, Fundamrntals of music, trans. Calvin M. Bower, Music theory translation series (New Haven: Yale University Press, 1 989), 14-15.

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expected to reify those mathematical formulae that the Pythagorean mathemati­ cal tradition labeled perfect, simple, or regular ? Here lies a neglected key t o the true innovative dimension o f Muris' a rs nova. It marks a departure from the Boethian legacy (or should we say tyranny?) ; it constitutes the beginning of a paradigm shift. To reinforce my claim, I suggest comparing Muris' turn to a parallel conscious and resolute attempt to introduce composite proportions in order to solve anomalies within the realm of late­ medieval physical theories of local motion. We have already noticed Aristotle's oversimplification in explaining the relation between force and velocity as a simple proportion, V= FIR. To remove the paradox that any mobile body could be moved by any mover, the Calculator Thomas Bradwardine devised in 1328 the following formula "in which both forces and resistances could vary and yet maintain a proportion of greater inequality and always produce a proportionate velocity in moving bodies": 54 Fi !R2 = (Fi /R1)v. 1 v.. Bradwardine's solution was aimed at a concrete physical question ; it targeted reality as much as Muris' theory did. Thus, physical pro­ cesses of change or motion are now accounted for by a new type of mathematics - less simple, less "beautiful", but more fruitful. 5 5 Muris' and Bradwardine's mathematical descriptions may be construed as a very preliminary acknow­ ledgement of the futility of the Pythagorean dogma. Both grasped the discrep­ ancy between the complexity of actual motio�s and the simplicity of the alleged mathematical relations. This understanding may become even more meaningful when we realize that the Pythagorean value system continued to have a power of tenacity in music, even while its conceptual base was being undermined. Muris did not appear to realize fully the revolutionary potential inherent in his new perspective. He still remained within the climate of Pythagorean ideas, which had asserted for generations a distinction between odd numbers, as associated with order and rationality, versus even numbers, which from the time of the Pythagoreans had symbolized boundlessness, disorder, and irrationality. Anchored to the Pythago­ rean hierarchy of good and evil numbers, Muris wrote : Cum igitur ternarius omnibus se ingerat quodammodo, hunc esse 4 F or the quotations see J ames A. Weisheipl, "The interpretation of Aristotle's Physics and ' the science of motion", in CHIMP, 52 1-36, at 535. " As already mentioned, Thomas Bradwardine's solution presents a mathematical alterna­ tive that eliminates the difficulty involved in Aristotle's theory; see above, p. 80. His solution is incorrect in physics, yet in some ways holds true for music: to double the velocity one raises the rhythmic values exponentially. Let me repeat at this point that although Bradwardine did not refer to contemporaneous music theorists, he gave the Pythagorean theory of harmony as an example of a legitimate use of quantitative methods for comparing qualitative d ifferences. See Bradwardine, Tractatus de proportionibus, 75-86.

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perfectum non debet amplius dubitari. Per cuius oppositum, cum ab ipso recedat binarius, relinquitur imperfectus, cum etiam binarius numerus sit infamis. 56 Since the number three enters into all things in one way or another, its perfection should not be doubted any more. Conversely, since the number two departs from it, it is left imperfect, for indeed the number two is ill-famed.

The theological value of the number three is as central to Muris as it was for Franco. It crosses throughout the whole chain of being and guarantees the world's perfection : �od autem in ternario quiescat omnis perfectio, patet ex multis veresi­ milibus coniecturis. In Deo enim, qui perfectissimus est, unitas est in substantia, trinitas in personis; est igitur trinus unus et unus trinus. Maxima ergo convenientia est unitatis ad trinitatem. In intelligentia post Deum esse et essentia et compositum ex hiis sub numero ternario reperitur. In primo corporum caelo: movens, mobile, tempus. Tria sunt in stellis et sole: calor, radius, splendor ; in elementis : actio, passio, materia; in individuis: generatio, corruptio, subiectum; in omni tempore finibili: principium, medium, finis; in omni morbo curabili : augmentum, status, declinatio. Tres operationes intellectus ; tres termini in syllogismo ; tres figurae arguendi ; tria principia intrinseca rerum naturalium ; tres potentiae entis non orbati; tres loci differentiae correlativae ; in toto univet'so tres lineae. 5 7 That all perfection rests in the number three is made clear by several likely conjectures. In God, the most perfect being, there is unity of substance, a Trinity of persons; he is thus three in one and one in three. There is therefore a maximal correspondence of unity to Trinity. After God, in the nou­ menal world, being and essence and that which compounded of them are found to be three in number. In the first heaven of bodies : the mover, the moved, time. There are three [qualities] in the stars and the sun: heat, ray, splendor ; in the elements: action, passion, matter ; in individuals : generation, corruption, underlying matter ; in every finite time: begin­ ning, middle, end; in every curable disease: increase, climax, decline. There are three operations of the understanding ; three terms in a syl­ logism; three figures of argument; three intrinsic principles of natural things; three powers of a being not deprived [of one of its senses] ; three '' Muris, Notitri:t artis musicae, 6 8-9.

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" Ibid., 67

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loci of correlative difference ; in the whole universe there are three lines [i.e., the geometric axes]. This paroxysm shows how Muris perpetuated his predecessors' analogical world-view. The symbolic deification of nature, so characteristic of sound theology and so much in line with the Thomistic interpretation of the universe as a sign of God, is promoted in the Notitia to the same extent as it was promoted in the musical discourse of the thirteenth century. Yet, as we shall see shortly, Johannes de Muris observed the dominant figures of thought of his predecessors while at the same time giving them a new perspective, transplant­ ing them into a new climate of thought. It is this "dualism", involving simultane­ ous reconfirmation and transformation of received musical ideals, that makes the Notitia a difficult text for historical interpretation in general and for a decisive evaluation of its innovative import in particular. Taking such dualism to be symptomatic of fourteenth-century amendments and improvements of the re­ ceived Scholastic world-view, I turn now to the much discussed problem of the status of imperfection in Johannes de Muris' Notitia artts musicae. Muris' initial claim about imperfection is enigmatic indeed : In omni vero tempore vocem mensurante quemdam modum perfectionis priores rationabiliter assignaverunt, illud tempus tale ponentes, quod per ternarium posset suscipere sectionem, opinantes in ternatio (sic ed. ; recte ternario] omnem esse perfectionem. Et propter hoc tempus perfectum pro mensura cantus cuiuslibet posuerunt, scientes quod in arte imperfe­ ctum non convenit reperiri, quamvis huius oppositum aliqui moderni, quod abest, se crediderunt invenisse ; quorum intentio clarius in sequen­ tibus exponetur. 58 To every time interval that measures sound, our predecessors reasonably assigned as it were a kind of perfection, supposing that time to be such as to sustain a division into three, for they believed all perfection to be in the number three. For this reason they set down perfect time for the measure of any music, knowing that it is improper for the imperfect to be found in art, although certain moderns believed that they had discovered the opposite of this, which is unsuitable. Their intention will be more clearly set forth in what follows. These lines conceal more than they actually disclose. What is unsuitable: the facts of imperfection as found in the music or the claim that those new musical facts are imperfect? Muris stated his intent to clarify what lies behind the supposed discovery of the modern,: but we are left unsure about his own position in the debate over the status of allegedly wayward innovations. While explicitly 58

Ibid., 66.

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accepting the paradigm of rhythmic perfection, Muris did not dismiss once and for all the possibility of musical imperfection. He urged his reader to wait and see : there was more in the modernls artistic intention than a simple violation of good old ideals, but this needed to be properly expounded since it could not be appreciated through superficial observations. It should be observed that the possibility of imperfection is bounded by a plea for the necessity of perfect time. This might mean that, whether or not imperfection was introduced into the art of music, it was not a question to be resolved by identifying actual instances of imperfection, but rather a question of interpretation. How far can the notion of perfection be stretched ? What is the outermost scope of rhythmic perfection ? As we shall see, it is this scope that was enlarged in the fourteenth century, effecting a transplantation of musical perfec­ tion into the new conceptual space, which cultivated tolerance towards imper­ fection and infinity. Thus, Muris reassured his readers, although in principle duple values are ill-famed (infamis), there is a class of duple values that can be considered perfect while still remaining in some respect imperfect : Cum igitur ternarius omnibus se ingerat quodammodo, hunc esse perfe­ ctum non debet amplius dubitari. Per cuius oppositum, cum ab ipso recedat binarius, relinquitur imperfectus, cum etiam binarius numerus sit infamis. Sed [ex his] compositus quilibet numerus (similitudine] convenientiaque, quam habet ad ternarium, perfectum potest merito reputari.59 Since the number three enters into all t�ings in one way or another, its perfection should not be doubted any more. Conversely, since the number two departs from it, it is left imperfect, for indeed the number two is ill-famed. But any composite number formed from these may properly be considered perfect through the similarity and correspond­ ence it has to the ternary. Muris intended here to demonstrate that rhythmic imperfection - or rhythmic disorder - could unexpectedly be considered as equivalent to rhythmic perfec­ tion. He observed that the three imperfect binary numbers in his four grades of perfection, namely the numbers 6, 18, 54, are also multiples of 3. Hence they resemble ternary numbers and participate in the Holy Trinity. He continued : Muris, Notitia artis musicae, 68-9. According to Leofranc Holford-Strevens (private com­ munication), the last two sentences of the paragraph (the last sentence of this quotation and all of the next) are corrupt both in Michels's edition and in GS 111 : 292-301, at 293. Strunk, Source readings, 173-4, wrests sense from the passage by silent emendation; I print Strunk's implied text with the variations from Michels in brackets (Michels and Gerbert also print "compositum sic quilibet"). Strunk's translation, based on GS, has been a helpful guide to my translation of the Notitia artis musicae throughout, though I have not followed it very closely. 9 '

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Est enim [tempus], cum sit continuum, non solum iterum in ternarios sed in infinitum partibil[e] incessanter. 60 For time, since it is continuous, is divisible not only again by three but also endlessly to infinity. Referring to the relation between the measure and the measured, Muris said that since that which is divided and measured by discrete units is a segment of a continuum, hence every division is arbitrary or contingent and not necessary. Thus a division by two can be "converted" to a division by three, or by any other number. Muris, like his predecessors, tries to save the ideal of rhythmic perfec­ tion, but here his reaffirmation does not come from the notion of divine timelessness, but rather from the ambiguity of time itself. Given the continuity of time, every division is an ad-hoc division implying an infinite number of other possible divisions. Thus, mediation between the new music and the music that it replaced consisted in the persistence of the traditional canon, which was redefined in line with the new sensibilities and dispositions of the fourteenth century. The ultimate scope of perfection was now reinterpreted : that is, relaxed and modified through the inclusion of numbers that have the number three and its powers for factors. There is an unprecedented tolerance towards the possibility of a conti­ nuity rather than dichotomy between perfection and imperfection. In the same vein imperfection may replace perfection in cases of motion in imperfect time (tempus imperfectum ) which is equal to motion in perfect time (tempus perfectum), for three binary values are equal to two ternary values in 6, 1 2 , 1 8 , 2 4 , etc. In fine huius opusculi notandum est, quod contingit fieri cantum ex perfectis notulis de tempore imperfecto ut tres breviores, et ex imperfe­ ctis de tempore perfecto ut duo breves. Adaequantur enim tres binarii et duo ternarii in 6, 1 2 , 18, 24 et sic addendo 6. Sunt autem tres binarii perfecti de (tempore) imperfecto, sed duo ternarii imperfecti de per­ fecto, et ad invicem revolvuntur et aequa proportione finaliter adae­ quantur. Et ex perfectis de perfecto et imperfectis de imperfecto sicut convenit decantatur.61 At the end of this work it should be noted that it is possible for a song to made up of perfect notes in imperfect time, such as three imperfect breves (breviores) , and with imperfect notes in perfect time, such as two [perfect] breves. Three twos are made equal to two threes in 6, 1 2 , 1 8 , 24, and any multiple of 6. But three twos are imperfectly perfect, while two threes are perfectly imperfect, and they are turned about one another, 60

Muris, Notitia artis musicae, 69. Ibid., 84. Michels's interpolated "tempore" (in angle brackets) is not only unnecessary but misleading, though Strunk similarly mistook the sense. 61

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and become equal in the end by equal proportion. And it can also be sung with the perfectly perfect and the imperfectly imperfect as appro­ priate. The notion of perfection is enriched rather then stained by its opposite. For Muris these contraries were no longer mutually exclusive, disjunctive concepts : they could be compared and equated. Such a modification in the understanding of fundamental categories is not self-understood, and we shall try to ground it in some very specific intellectual activities of the late thirteenth and early four­ teenth centuries. To understand Muris' approach to imperfection in music, we need to consider the broader context of vigorous attempts made in the fourteenth century to mediate between disorder and order, infinity and finiteness, unity and plurality, simplicity and complexity. Of special relevance to our case are such endeavors in astronomy and in the closely related field of the geometry of motion. The metaphysical and physical centrality of the circle and of circular motion induced both astronomers and mathematicians to develop a special interest in the movements of points on the surface of rotating spheres. Their main goal was to find a way of measuring the velocities of rotating spheres, so as to solve the difficulty arising from the fact that a rotating sphere is constituted from a multitude, indeed an infinity of point velocities. These cannot be grasped or measured, and therefore need to be represented by an alternative motion that is simpler and known, yet equal in some way to the complicated motion of the sphere. Since Muris was both an astronomer and mathematician, he was probably familiar with those geometrical works which study spherical or circular motions, and emphasized comparisons and equations between different types of motions. Although he did not expose his sources, he might have drawn the idea and procedure of equating different rhythmic motions from geometrical or astro­ nomical theories that had been developed at his own university - the university of Paris, around the mid thirteenth century-such as the Liber de motu of Gerardus of Brussels. Here Gerardus endeavored to reduce variability to uni­ formity by converting the varying curvilinear velocities of the points of geo­ metrical figures in rotation into a simple uniform rectilinear motion.62 The first part of his book is concerned with the translation of varying curvilinear velocities possessed by different lines to a uniform rectilinear velocity of points. Its second part translates surfaces in rotation to uniform motions, and the last deals with the reduction of solids in rotation to equivalent uniform linear motions. Gerardus drew heavily on the Archimedian geometry of movements and cites several Archimedian propositions that deal with the equation and the 62

See Clagett, "Gerard of Brussels", DSB, v: 360.

J O H A NN E S DE M UR IS A ND T H E OX F ORD C A L C U LATOR S

translation of one form to another. Thus Gerardus cites the proposition that equates a circle to a right triangle whose sides, including the right angle, are the radius and the circumference of the circle respectively. Another proposition is that which equates the lateral surface of a cone to a right triangle, one of whose sides adjacent to the right angle is equal to the circumference of the base of the cone and the other of which is equal to the slant height of the cone. 6 3 Since Johannes de Muris used the Moerbeke translations of Archimedes and referred specifically to Archimedes' De mensura circuli in his De arte mensurandi of 1 344, it is logical to assume that his ideas about the equation between composite binary numbers and ternary numbers, or between three perfect binaries in imperfect time and two imperfect ternaries in perfect time, could have origi­ nated in the astronomical tradition of equating and comparing different mo­ tions. 64 Another important source that nourished the works of both Gerardus and Muris is Aristotle's comments in his Physics (v11.4) on the equality of different motions and on the condition for comparing motions. Aristotle, let us recall, restricted comparisons of velocities to cases in which there is a specific identity of figures in the space traversed. Thus, discussing the possibility of equating the velocity of a rectilinear and circular motion, Aristotle outlined the comparison in detail and then explained its inadequacy on the ground that equality in velocity takes place only when the distance or space traversed is of the same species, that is; follows the same contour or figure. These passages in the Physics concern the conceptual core of most of Gerard us' propositions -the comparison of linear and circular motions. Gerardus disregarded Aristotle's qualifications and used Aristotle's authority to validate and develop further cases of equality between linear and circular motions. Following Gerardus' fundamen­ tal approach, the Oxford Calculators saw as their main objective the reduction of variations in velocities to a simple uniform velocity. The key notion in the new measurements and quantitative analyses of dif­ ferent difform or complex motions was the notion of equivalence ; the reduction of complexity to simplicity through the detecmination of a simple uniform state or motion that is equivalent to the initial difform and irregular state or motion. The central fourteenth-century theme of eliminating ontological oppositions manifests itself here again. By quantifying the relationship between uniform motion and difform motion, these mathematicians demonstrated a continuity between opposites, between the finite and the infinite, between the uniform and the difform, and between complexity and simplicity. This understanding, let me repeat, is crucial to the understanding of the role of late-medieval thought in the

63

Ibid. See Marshall Clagett, "The use of the Moerbeke translations of Archimedes in the works of Johannes de Muris", Isis, 43 (1952): 236-42. 64

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early modern process of demystifying the physical world and the mathematical language that describes it. Perhaps the most famous result of these endeavors to mediate between opposites was the formulation of the rule of "mean speed". This rule translated a uniformly difform motion into a simple uniform motion : a body moving uniformly difformly (in uniform acceleration) will in a given time cover the same distance it would while moving uniformly with its mean speed. In the termino­ logy of the Calculator Richard Swineshead, the rule reads: Cum motus localis uniformiter difformis correspondet quo ad effectum suo medio gradui, sic patet quod tantum per idem tempus ponitur pertransiri per medium gradum sicut per ilium motum localem uniformi­ ter difformem. Unde quiscunque motus difformis, et etiam quecunque alia qualitas, alicui gradui correspondet . . . Since local motion uniformly difform corresponds to its mean degree (of velocity) in regard to effect, so it is evident that in the same time so much is traversed by means of (a uniform movement) at the mean degree as by means of the uniformly difform movement. Furthermore, any difform motion - or any other quality - corresponds to some degree. 65 This basic _rule engendered a list of variations, such as : If a subject has a given degree at one end and increases rapidly in temperature at first as one moves away from this end of the subject, and then, increases more and more slowly with distance until the degree of the other end is reached, it will correspond to a greater degree than a subject whose temperature increases uniformly from one end to the other. 66 Other equations could involve two different types of motions: circular and linear. The following rule concerns the problem of rotation in which the speed of each point of the rotating body is in direct proportion to the distance of the point from the center of the body: "In a uniform motion of a rotating wheel each point of which is moving with different velocity, the velocity of the wheel as a whole is measured by the linear path traversed by the point which is in most rapid motion."67 Here again, a motion composed of infinite velocities, repre­ sented by the infinite number of points on the radius of a rotating wheel, is translated into a simple uniform motion. It is in this context of the geometry of motion per se and as applied to astronomical and natural motions that Muris' theory of legitimate imperfections 65 Clagett, Thescience of mechanics in the Middle Ages, 244-6. For different versions of this rule, 66 see ibid., 255-329. Sylla, "Medieval concepts of the latitude of forms", 257. 67 Clagett, The science of mechanics in the Middle Ages, 235-7.

1 00

J O H A N N E S DE M U R I S A N D T H E OX F O RD C A L C U L A TO R S

i s to b e situated. Here was founded a common infrastructure for the rhythmical developments throughout the fourteenth century and the guiding premise of the early modern study of the spatial and temporal dimension of motions. By the end of the fourteenth century the mathematical techniques of dealing with simultaneous variable rates would account for the extended use of rhythmic proportions in the Ars subtilior, and for the general procedure of shifting mensu­ ration, of augmentation and diminution. Can these rhythmical devices be construed as musical analogies of the "mean speed" theorem ? Do they not presuppose a regulative common value, or measure, or velocity (equivalent to the uniform point velocity of the Calculators) that serves as the converting mean point of two different motions ? Later, in the fifteenth century, music theorists would conceptualize this common unit and name it the integer valor. By allowing controlled imperfection and by focusing on transitions from one kind of measurement into another, Muris paved the way towards the new music of the mid and late fourteenth century. This new music allowed the sensation of real time -the network of expectations and anxieties that constitutes human time -to be articulated through the projection of changes in rhythms and meters, which would be increasingly emphasized as the century went on . Music was now just starting to abandon the thirteenth-century uniform periodicity that participated in an unbroken inner communion with the eternal. To put it differently, time, as a factor of human existence, became an aspect about which music now could, and now began to, say something. Composers, as well as listeners, paid heed to time as an object of human concern in the same way as philosophers. Through a close reading of Muris' Notitia, we can see a new concept of music unfolding, one that eventually allowed rhythms to imitate worldly processes or motions and to manifest earthly variability through a variety of intensities. But this musical worldliness does not seem to reflect a real, or rather misinterpreted, decline in religious life. Its interconnections with the peculiar fourteenth­ century theological sensitivities and with the contemporaneous mathematical and philosophical developments, suggest a new style of thinking about music ­ music demathematized while mathematized ; music dedivinized while attending to the new religious and theological trends of the period.

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4 M U S I C A L T I M E A N D T I M E I N G E N E RA L T H E C H A L L E N G E F RO M M AT H E M AT I C S

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Infinity and continuity

I considered Johannes de Muris' Notitia artis musicae in relation to the Calculators' theory of the composition and quantification of continuous qualities. In this chapter, I will survey other mathematical and other musical theories. As with all surveys, in undertaking this one I run the risk of superficiality. My aim is not to provide an exhaustive analysis. Rather, I hope to achieve the following: to highlight some basic theoretical issues common to musicians and mathematicians ; to examine the relationships between conflicting musical schools in the light of contemporane­ ous conflicts between mathematical schools ; to suggest tools and criteria for analysing music theories ; to scrutinize concepts whose understanding depended on .certain logical and mathematical subtleties ; and finally, to evaluate whether the significance of these subtleties surpassed the appreciation of individual theorists. What actually is a continuum ? How do different points on the continuum relate to each other, and to the whole continuum ? But if a continuum contains such distinct points, how can it be absolutely continuous? Moreover, what is the meaning of a minimum, or of the first and last points of a continuum, if a continuum, if indeed continuous, has no such points ? If it has no such points, can nothing ever begin or terminate ? These questions describe what we know as Zeno's paradox, which originated with the fifth-century-Be philosopher Zeno of Elea. But the paradoxical nature of the infinitely divisible challenged and baffled the minds of fourteenth-century thinkers, whether they were phiN T HE PRE C E D ING C H A P TER ,

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IN F IN ITY A ND C ON T I N U ITY

losophers or mathematicians, theologians or musical theorists. The principal guide to these perplexities was Aristotle's discussion of the nature of the infinite and the continuous. Aristotle's theory of the infinite was based on the rejection of the actual existence of an infinitely large or infinitely small entity. Aristotle regarded as possible potential infinity only. 1 Furthermore, Aristotle rejected any possible mediation or relation between an indivisible unit and a continuum. This was because his definition of continuity required that the components of a con­ tinuum unite to become "one". Accordingly, Aristotle concluded that nothing that is continuous can be composed of indivisibles. A line cannot be composed of points, the line being continuous and the points indivisible, for the extremities of two points can neither be one, since of an indivisible there can be no extremity as distinct from some other part, nor together, since that which has no parts can have no extremity, the extremity and the thing of which it is the extremity being distinct. 2 In other words, according to Aristotle, the idea of continuity requires that be­ tween any two points on a given continuum - no matter how close they are ­ there is always a line, which is infinitely divisible into increasingly smaller lines. As already noted in the preceding chapter, Aristotle's arguments against atomism must be seen in the larger context of his prohibition against the mixing of disciplines and against the mixing of two branches or species of the same discipline. The continuous and the discrete, we recall, represented two different species of mathematical entities. Each, according to Aristotle, has its own mathematical field. Continuous magnitudes are the domain of geometry, whereas discrete quantities are the matter of arithmetic. 3 In general, late-medieval philosophers accepted Aristotle's arguments against atomism. But a small minority did not, reviving the Greek concept of atomism. This was not, however, a simple and unanimous alternative.'' Although most of these "indivisibilists" conceived continua to be composed of sizeless atoms, Nicholas Bonet (for example) revived Democritus' notion that atoms have a very small extension. Walter Chatton followed the Pythagorean theory of a finite number of indivisibles, while the majority of the indivisibilists held that atoms were infinite in number. Either atomists accepted the implications of atomism, namely the discontinuity of local motion or other continuous processes such as 2 Aristotle, Physics, C4-8. Ibid., C6, 231 124-30. Aristotle's prohibition against the mixture of species hindered the development of quanti­ tative physics, for it prevented the mixture of qualities with quantities. For a discussion of this aspect of Aristotle's philosophy, ·see above, pp. 27-8. 4 On the rise of atomism in the 14th c., see John E. Murdoch, "Infinity and continuity", in CHIMP, 564-92, at 575-84. See also John E. Murdoch and Edward Synan, "Two questions on­ the continuum : Walter Chatton, O.F.M. and Adam Wodeham, O.F.M.", Franciscan studits, 26 (1966) : 2 1 2-88. 1

3

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augmentation and diminution, o r they attempted to reconcile indivisibilism and the continuity of motion and change.' Among the Oxford Calculators there was a minority who accepted atomism. Walter Burley (c.1275-1344/5) believed that continua were composed of an infi n ite number of mediate points or indivisible particles. When the continuum represented an increase in intensity, he argued that the originally given intensity was destroyed (tota destructa) and the body successively acquired new degrees of intensity. Each new higher degree, however, did not contain the previous degree as a whole contains its parts, for each degree was indivisible. 6 The core group within the Mertonian Calculators accepted Aristotle's theory of continuity but broke, as we saw in Chapter 3, with the Aristotelian prohibition against the mixture of the continuous and the discrete. They developed a revolutionary "additive theory", or method of quantifying the relationship between intensities, and laid the foundation for a mathematical analysis of physical events.' Philosophers and mathematicians spared no effort in refuting opposing views. Indivisibilists worked out theories of "touching indivisibles", which would save the phenomenon of continuity. The Aristotelians argued that indivi­ sibilism was absolutely incompatible with geometry, an idea that was in turn fruitful for theology. The infinite excess between any rectilinear angle and a curvilinear angle, for example, was seen as a perfect model for the infinite distance between God and the world, or between individuals of distinct species. 8 Problems of infinity and continuity were also investigated from the purely logical point of view. The relation of this logical literature to music will be discussed later. Here it should be noted that, in logic, it was not infinity itself that was scrutinized, but rather propositions involving the term "infinity" and other related terms. Surprisingly, it is in logic that the medieval analysis of continuity and infinity reached its culmination. This was reached when the traditional physical concerns with ascribing limits (minima and maxima or first and last instances) to continuous processes or motions was mixed with the search for the limits of logical inferences, namely, in "sophisms". There, proposi­ tions about processes in time and their confounding limits (beginning and termination) were phrased in an odd and awkward way and called logical subtleties or sophisms.9 I_ will relate these logical subtleties to the equally intricate rhythmic style of the Ars subtilior in Chapter 7.

' Murdoch, "Infinity and continuity", 575-80. Sylla, "Medieval concepts of the latitude of forms", 231-57. 7 See above, pp. 84-7. • John E. Murdoch, "Subtilit4tes anglicanae in fourteenth-century Paris: John of Mirecourt and Peter Ceffons", in Madeleine P. Cosman and Bruce Chandler (eds.), Machaut's world: scimce and art in thefourteenth century (New York: New York Academy of Sciences, 1978), 51-86, at 61-4. 9 Norman Kretzmann, "Incipitldesinit", in Peter Machamer and Robert Turnbull (eds.), Motion and time, space and matter (Columbus : Ohio State University Press, 1976), 101-36, at 1056

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The idea o f measure (mmsura) embraced not only all the quantitative predi­ cates and relations that could belong to a subject in widely varied circumstances, but also all the logical predicates that such a subject could acquire. It united the otherwise distinct disciplines through a common methodology and a common language of analysis. 10 We have already encountered some of its vocabulary and appreciated the similarity between the discourse of fourteenth-century music and those of theology, philosophy, and mathematics. The rich vocabulary of mmsura includes terms like divisiones, divisibilitas, partes proportionates, excessus, pars, totum, mediate, immediate, ante, post, latitudo, gradus, uniformis, differmis, minima, maxima, incipit, and desinit. These terms must sound familiar to musicologists engaged in the study of medieval music. Within the discourse of music and elsewhere, these terms concern either the limits of continua (maximum/minimum; incipitIdesinit) or the inner composition, order, number, position, properties, and relations among the parts of such continua. 1 1 I n attempting to analyse rhythmic notions, theoreticians struggled to under­ stand the properties of musical time and their relation to general time. Debates over the composition of continua divided musical theorists as well as mathema­ ticians and philosophers into rival parties. None of the rhythmic theorists active during the fourteenth century adopted an atomistic theory, which regarded general time as an aggregate of an infinite number of indivisible units (each being a "now"). Although such a theory would have meshed more easily with the measurement of time in music, theorists adhered to the Aristotelian tradi­ tion, which conceived of time as absolutely continuous. Yet they differed in their perception of the conceptual base of measured music. 1 2 Given the postulated continuity of time, some theorists felt they had to account for the discrete values represented by rhythmic figures, for "numbered sound" apparently conflicted with, if it did not contradict, the continuity of time. The question of the composition of the time signified by a musical figure was fundamental. Is this time continuous and therefore susceptible to all con­ ceivable divisions? Or is musical time an aggregate of discrete units, divisible only into such parts as are multiples of the given unit? In other words, theorists asked whether it was necessary for the measure to be similar in structure to that which was measured. The composition of musical time was a crucial determi­ nant - in theory- of the types of rhythmic values, as well as of the limits of their possible subdivisions. This theoretical account of time determined whether 11. Also see John E. Murdoch, "Propositional analysis in fourteenth-century natural philo­ sophy: a case study", Synthes,, 40 (1979) : 117-46, at 124-8. These logical subtleties will be discussed in relation to the Ars subtilior; see below, pp. 2 13-29. 11 10 Murdoch, "from social into intellectual factors", 280-87. Ibid., 283--5. 12 In this sense, music theorists followed the trend that prevailed among contemporaneous philosophers. 105

M U S IC A L TIM E A ND T I M E IN G E N E R A L

there could be a correlation between the theory and the practice of the fourteenth century, and whether theorists could discuss time in terms that could account for the variety of rhythms actually used in fourteenth-century music. These rhythms included not only multiples of basic units, but also subdivisions into parts that were not always commensurable with one another. Closely related to the problem of the composition of time, and no less confusing, were difficulties involved in determining the existence and properties of the minimal time-interval u ed in music. The fact that the minimum could (and in practice did) shift from one style to another sharpened the problem of conceptualizing the notion of a minimum, and pointed out the complexity involved in the relation between the concept, its conventional sign, and what was actually signified. Finally, there were difficulties in explaining the relation between shorter and longer note-values : theoreticians had to decide for each value whether it was a whole or a part of other values. This problem entailed the question of the conceptualization in theory, and the possibility in practice, of causing imperfection by remote parts. For us to understand and evaluate the theoretical writings of the fourteenth century, it is necessary in the case of each individual theory to reconstruct its underlying assumptions regarding the nature and the composition of musical time, its relation to time in general, the order and properties of the parts that constitute the continuum, and finally the relation between the rhythmic signs and what they signify. This is so important that the discourse about rhythmic issues at this time cannot be understood without first coming to understand the philosophical and mathematical presuppositions that undergirded the different possible answers to these basic questions. It is in the context of these basic assumptions that the consistency, value, and historical importance of each theory should be examined. The main sources recording disagreements over these issues are the seventh book of Jacobus of Liege's Speculum musicae, Johannes de Muris' Notitia artis musicae, the Anonymus OP's Tractatus de musica, the fourth book of John Tewkes­ bury's OJ!,atuor principalia musicae (which appears also as Anonymus 1, De musica antiqua et nova), Anonymus v1, Tractatus defiguris sive de notis, and the anonymous De musica mensurabili previously attributed to Theodoricus de Campo. 1 3

The composition of musical time Jacobus of Liege regarded the issue of time as a major point of dissension between antiqui and moderni. The difference between the two opposing schools 1 3 The discussion of these sources will proceed by topics or problems and not by chrono­ logy. When available, recent editions are used as listed in the notes below.

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centers o n a subtle and interesting issue: the question o f the relationship between the facts that rhythmic signs signify a numbered quantity of units and that "what is numbered" may be continuous. As Jacobus saw it, the two rival parties held completely opposite opinions. The moderni (whose chief representa­ tive, according to Jacobus' editor Roger Bragard, was Johannes de Muris) conceived musical time as absolutely continuous, whereas the antiqui (with whom Jacobus concurred) considered time to be composed of discrete and numbered units. Jacobus quotes the moderni as claiming that the traditional definition of perfect time Insufficiens est, ut aiunt, quia tempus perfectum, sicut est divisibile in partes duas inaequales vel in tres aequales, ita aeque bene divisibile est in duas partes aequales, eodem valore temporis nihilominus remanente. 1 4 is insufficient, they say, for just as a perfect time is divisible into two unequal or three equal parts, so it is just as well divisible into two equal parts, the time-value nevertheless remaining the same. These divisions were possible, the moderni argued, owing to the continuity of musical time : Corpus quantum continuum, antequam actu fuerit divisum, aeque divisi­ bile est in partes duas aequales, sicut in tres, vel in duas inaequales. Ergo similiter erit de tempore, cum sit species quantitatis continuae. 1 5 A body, insofar as it is continuous, before it has actually been divided, is equally divisible into two equal parts as it is into three, or into two unequal ones. Therefore the same will hold for time, since it is a species of continuous quantity. The two statements quoted by Jacobus are not directly derived from any of Muris' treatises, and their authorship remains questionable. Yet they do reflect Muris' view, since it is implied that the time denoted by a rhythmic figure is continuous and not discrete. It also implies that although a figure is determined within a metric context and thus indicates for the performer a particular number of time-units, this is irrelevant to the inner composition of the time embraced within a given rhythmic value. As we saw in Chapter 3, this view recalls the innovative "additive theory" of the intension and remission of form. Both Muris and the "additivist Calculators" conceived arithmetic as not only a mere tool for counting a set or an aggregate of discrete objects, but also a language capable of expressing differences in size, length, or intensity. In refuting the moderni's argument, Jacobus displayed subtlety and great 14

73),

Jacobus of Liege, Speculum musicat, 7 vols., ed. Roger Bragard, CSM 3 (Rome : AIM, 1 955., Ibid., VII : 27-8. 27.

VII :

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insight into logical problems. Jacobus' controversy with the moderni in general, and his methodology and its relation to contemporaneous Ockhamist logic and science in particular, will be the focus of the next chapter. It is worth noting here that Jacobus (through metalinguistic analyses of propositions, which I will discuss later in Chapter 5) demonstrated the inadequacy of his opponents' theory. First, he pointed out that the moderni erred in applying the rule according to which any division of a continuum is theoretically possible before an actual division is made. They were mistaken because once a certain segment of time is represented by a rhythmic figure it ceases to be an indefinite segment of time and becomes a definite and discrete rhythmic value represented by a specific figure. This figure in turn signifies a certain species of note-value or a certain number of time-units, organized in a certain mensuration. Otherwise, not only would the distinction between perfect division and imperfect division be mean­ ingless, but so would the distinctions between various types of figures. Applying a basic logical rule, Jacobus argued that if two objects have identical descriptions or definitions, then the two are identical. Thus, if a perfect time can be divided into two equal parts and if an imperfect time can be divided into three equal parts, then perfect time is identical to imperfect time and imperfect time is identical to perfect time. But this is patently absurd. The modern theory, Jacobus concluded, entailed that the "objects" of musica mensurabilis (the res musicales them-selves or the terms used for describing them) were all confused and that they had no distinct identity. Videtur igitur ex dicendinepugnare tempori perfecto, ut ab imperfecto distinguitur, divisibile esse in duas partes aequales. Primo probatur hoc ex dictis istorum quia repugnat divisioni quam fecerant de tempore, scilicet quod aliud sit perfectum, aliud imperfe­ ctum. Membra enim illius divisionis, si vera sunt quae hie dicuntur, coincidunt, et hoc est quod supra declarare promiseram. Si enim tempori perfecto, ut est membrum illius divisionis distinctum a tempore imperfecto, competit in duas dividi partes aequales, cum hoc competat tempori imperfecto, imo secundum istos in sua ponatur descri­ ptione, similiter et si tempori imperfecto competat in tres dividi partes, ut hi asserunt, quid consequens est ad hoc, nisi quod tempus perfectum tempus sit imperfectum vel e converso ? Illa enim sunt eadem quorum definitiones sunt eaedem. 16 It will be seen from what is to be said that perfect time, as distinct from imperfect time, resists being divisible into two equal parts. First, this is proven by their own statement, because it resists the division they had made regarding time, namely that the one is perfect, 16

Jacobus, Speculum musicae, vn : 28.

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the other imperfect. I f the things said here are true, the members o f that division coincide, and this is what I promised to clarify above. If perfect time, being a member of this division distinct from imper­ fect time, is capable of being divided into two equal parts -since imperfect time is capable of this ; indeed according to them that is part of its definition - and similarly, if imperfect time is capable of being divided into three parts as they assert, what is the consequence of this if not that perfect time is imperfect or the converse ? Those things are the same whose definitions are the same. In reply the moderni argued : quod non appellatur tempus imperfectum praecise quia in (partes) duas aequales sit divisibile vel divisum, cum etiam tempus perfectum aequali­ ter in duas divisum veraciter imperfectum vocaretur; nee etiam dicitur tempus imperfectum quia divisibile sit in partes tres aequales, quia etiam tune imperfectum perfectum diceretur, sed ob hoc imperfectum nomina­ tur qu�a non attingit perfectum perfecti temporis valorem sed ipsius tantummodo continet duas partes. 1 7 that imperfect time is not so called strictly because it is divisible or is divided into two equal parts, since perfect time too, when divided into two, would be correctly called imperfect. Nor is it called imperfect time because it is divisible into three equal parts, since then the imperfect would be said to be perfect. But the reason it is named imperfect is that it does not reach the full value of perfect time, but contains only two thirds of it. Neither perfect nor imperfect time is so called because of the possibility of its division into three or two ; either divided into the other number of parts would take the other name. Both are continuous, and one is a rational subset of the other. Jacobus, however, invoked the authority of Aristotle to show that this argument is logically deficient. Non valet ista responsio quia, qui sic dicunt, materialiter et non formali­ ter tempore perfectum et imperfectum inter se distinguunt, cum tamen materia principium distinctionis non sit, quia in fundamento naturae, idest in materia, nihil est distinctum, secundum Commentatorem. Sed actus est qui distinguit et separat, ut habetur (sexto) Metaphysicae. 1 8 This answer is not valid, for those who say this distinguish perfect and imperfect time from one another materially and not formally, even though matter is not the principle of distinction ; for according to the 17

Ibid.,

VII : 29. 109

18

Ibid.

M USIC AL TIME AND TIME IN GENERAL

Commentator [Averroes!, in the basis of nature (that is, in matter) nothing is distinguished. But it is action that distinguishes and sepa­ rates, as stated in the sixth book of the Metaphysics.

Since perfect time and imperfect time are both equally divisible into two or three parts, they must share their essential definition. Therefore, perfect and imperfect time differ not in their form (essence) but in their matter. But accord­ ing to Aristotle, matter is not the principle of individuation. Thus, Jacobus argued, there is no real distinction between perfect and imperfect time-values. The problem with this argument is that the issue at stake is the need to distinguish not between two species (say a table and a dog, each of which, according to Aristotle, is defined exclusively through its unique essential form) but between two members of the same species, (say between two men or between two longae). Now, according to Aristotle, matter does individualize one individual from another particular of the same species, not the form or essence as Jacobus mistakenly argued. 1 9 Jacobus considered the problem of time once again while refuting Muris' theory of time as summarized in his ninth conclusion, 20 according to which "time can be divided into as many equal parts as one wishes (tempus possit dividi in quotlibet partes aequales)". 2 1 Jacobus countered that there was a dis­ tinction between time per se, . which he regarded as an indefinite matter, a mere potentiality, and the specific time denoted by rhythmic figures. Unlike continu­ ous time as such, rhythmic figures are not indefinite but represent a distinct rhythmic concept and a certain number of time-units. Therefore, musical time or time signified by a rhythmic figure is numbered and discrete. Dicendum quod, licet tempus materialiter et absolute sumptum et ut continuum dividi possit in quot volueris partes aequales ut in duas, tres, quattuor, sic ceteris, non tamen ut per notulas significatur musicas, ut saepe dictum est. Alicer enim est divisibile ut per longam signatur perfectam, aliter ut per imperfectam. Important enim notulae quaelibet determinatas temporis morulas et in hoc inter se distinguuntur, licet in hoc generaliter conveniant quod tempus important ad modum quo annus, mensis, dies, quadrans, hora, momentum, uncia, atomus. Item notulae musicae non videntur tempus pure continuum importare

19 My interpretation follows Franco's explanation, according to which the species of the longa has three varieties -perfect, imperfect, and duplex. Thus, the perfect longa and the imperfect one are not two distinct species. See above, p. 36, where I argue that theorists regarded the property of imperfection as an accidental quality of an essentially perfect figure. 20 For analysis of the methodology that underlies Muris' conclusions and on the methodo­ logical and formal resemblance between Muris' Notitia artis musicae and other mathematical treatises, see below, pp. 186-7. For Jacobus' arguments against the first eight conclusions, see 21 below, pp. 1 69-78. Muris, Notitia artis musicae, 87.

I IO

THE CO MPOS IT ION OF MUS IC AL T I ME

sed discretum et numeratum ad determinatas partes applicabile vel applicatum, ut supra declaratum est, et per Modernos confirmatum, etiam per ilium qui has novem ponit conclusiones. Advertenda autem sunt dicta eius in hac nona conclusione : Omne, inquit, continuum divisibile est in quotlibet pctrtes eiusdem proportionis, sicut in duobus vel in tribus vel in quctttuor, et cetera. Tempus est de numero continuorum. Ergo potest dividi in quotlibet pctrtes ctequctles. Fiet ergo cctntus ex ductbus, tribus, quctttuor, quinque, sex, septem, octo, novem semibrevibus ctequctlibus eiusdem figurcte, et cetera. 11 It must be said that although time, taken materially and absolutely and as a continuum, can be divided into as many equal parts as you wish (such as into two, three, four, and so forth), nevertheless it cannot in the way signified by musical notes, as has often been said. For it is divisible in one way when signified by a perfect, in another way when by an imperfect long. Notes refer to any determinate units (lit. : "little spaces") of time and are distinguished in this respect from one another, though in general it is appropriate for them to refer to time in the same way as a year, a month, a day, a quadrant, an hour, a moment, an ounce, an atom. Also, musical notes do not seem to refer to purely continuous but to discrete time, numbered by determinate parts either applicable or ap­ plied, as stated above and confirmed by the moderm; especially by him [Muris] who drew these nine conclusions. But one must take note of his words in the ninth conclusion : "Any continuum can be divided into as many parts of the same proportion as one wishes, into two or into three or into four, etc. Time is numbered among the continua. It can therefore be divided into as many equal parts as you like. A song may therefore be made out of two, three, four, five, six, seven, eight, nine equal semibreves of the same shape."

Jacobus seems to have interpreted Muris' ninth conclusion to refer to the continuity of the time signified by a breve. Michels understood Muris' ninth conclusion in a similar way : " Tempus here is not time in general . . . but the term for the time-content of a breve."n Yet a close reading of Muris' text leads to the opposite view, that the ninth conclusion does refer to time per se rather than to any time embodied within a musical note. Though Muris made this the last in 22 Jacobus, Speculum musicae, v11 : 85-6. For the precise relationship among the time-words at the end of the first paragraph, see Johannes Vetulus de Anagnia, Liber de musica, ed. Frederick Hammond, CSM 27 (AIM, 1 977), 2 8 : a quadrant = 6 hours, a moment = 'f40 hour or 90 seconds, an ounce = 1/1 2 moment or 7 ½ seconds, an atom = •;�• ounce or %6 second. For the quotation from Muris, cf. Notitia artis musicae, 104 (the only significantly discrepant reading is "Tempus est de genere continuorum"). H Michels, Die Musiktraktate desJohannes de Muns, 106.

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M U S I C A L T I M E A N D T I M E I N G E N E RA L

his list o f conclusions, this particular conclusion seems to b e unconnected to the previous eight. These constitute a unified group describing eight conditions under which imperfection could take place. In the ninth conclusion the subject is changed to the apparently unrelated notion of time and its properties. Muris' various possibilities of imperfection will be discussed in Chapter 6 and ex­ plained there in terms of the exhaustion of the different possible distributions of smaller rhythmic values throughout a definite extension of time. For the mo­ ment it is sufficient to note that the ninth conclusion provided the rationale for the general possibility of dividing time into any number up to nine equal parts. The so-called "Petronian divisions" of the single tempus brevis, upheld by Jacobus, is one example of such a division but not the only one. It is unclear why Jacobus -a most pedantic and precise theorist - understood the term tempus of the ninth conclusion in reference to the breve and not to time in general. Could Jacobus have failed to see that while the first eight conclusions use specific rhythmic terms (longa, brevis, semibrevis, etc.), the ninth conclusion uses the abstract term tempus rather than the well defined brevis, and thus compromises the uniformity of his conclusions? Jacobus' misinterpretation of Muris' theory is particularly puzzling in the light of his own admission that Muris was one of the moderni who held that musical notes signified discrete and numbered time, as Muris had indeed indicated : Est enim notula figura quadrilatera soni numerati tempore mensurati ad placitum significativa. 24 A note is a four-sided figure that arbitrarily signifies a numbered sound, measured by time. Muris, like Jacobus, distinguished between general time and the specific time denoted by a rhythmic figure, as is confirmed by a didactic dialogue extracted from Muris' Compendium musicae practicae: �id est tempus in generali ? Mensura vocis prolatae sub uno motu continuo. �id est tempus longu m ? �od in tres partes et partium partes usque ad vocis minimum separatur. 25 What is time in general? The measure of a voice prolonged through one continuous motion. What is the time of a longa (lit. : "a long time") ? That which is sepa24

Muris, Notitia artis musicae, 9 1 ; quoted by Jacobus, Speculum musicae, vu : 44. Johannes de Muris, Compendium musicae practicae, in Johannis de Muris Notitia artis musicae et Comprndi�m musicae practicae; Petri de Sancto Dionysio Tractatus de musica, ed. Ulrich Michels, CSM 17 (AIM, 197 2), 119-45, at 119. 25

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rated into three parts and parts o f parts down to the smallest unit of the voice. Jacobus misread this passage as self-contradictory. He argued that it implied that the property of continuity, which was predicated of the whole genus of time, was deprived of its subordinating species - the time of the long. Non