Generalized musical intervals and transformations [Revised ed.] 9780199890194, 9780195317138, 9780199759941

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Generalized Musical Intervals and Transformations

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Generalized Musical Intervals and Transformations

David Lewin

OXFORD 2007

UNIVERSITY PRESS

Oxford University Press, Inc., publishes works that further Oxford University’s objective of excellence in research, scholarship, and education.

Oxford New York

Auckland CapeTown Dares Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in

Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Copyright © 2007 by Oxford University Press, Inc. Originally published 1987 by Yale University Press Published by Oxford University Press, Inc. 198 Madison Avenue, New York, New York 10016 www.oup.com Oxford is a registered trademark of Oxford University Press All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data Lewin, David, 1933-2003. Generalized musical intervals and transformations / David Lewin.

p. cm, Originally published: New Haven: Yale University Press, c1987. Includes bibliographical references and index. ISBN 978-0-19-531713-8

1. Music intervals and scales, 2. Music theory. 3. Title.

ML3809.L39 2007 781.2’37—de22 2006051121

13579 8 6 4 2 Printed in the United States of America on acid-free paper

For June and Alex ,

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Contents

Foreword by Edward Gollin 1X

Preface XH Acknowledgments XXVli Introduction XX1X

1. Mathematical Preliminaries ] 2. Generalized Interval Systems (1):

Preliminary Examples and Definition 16 3. Generalized Interval Systems (2): Formal Features 31 4. Generaltzed Interval Systems (3): A Non-Commutative GIS; Some Timbral GIS models 60 5. Generalized Set Theory (1): Interval Functions; Canonical Groups and

Canonical Equivalence; Embedding Functions 88 6. Generalized Set Theory (2): The Injection Function 123 7. Transformation Graphs and Networks (1): Intervals and Transpositions 157 8. Transformation Graphs and Networks (2): Non-Intervallic Transformations 175

9, Transformation Graphs and Networks (3): Formalities 193 10. Transformation Graphs and Networks (4): Some Further Analyses 220 11. Appendix A: Melodic and Harmonic GIS Structures;

Some Notes on the History of Tonal Theory 245 12. Appendix B: Non-Commutative Octatonic GIS Structures;

More on Simply Transitive Groups 251

Index 22)

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Foreword to the Oxford Edition Edward Gollin

It has been nearly twenty years since the initial publication of David Lewin’s Generalized Musical Intervals and Transformations (GMIT), and the work has aged well. This is due in part to the foundational nature of the book’s subject matter. The work, a methodical examination of the concept of a musical interval, explores how the familiar notion of interval as “a distance extended between pitches in a Cartesian space” is merely one specific case of a more general idea, one that can embrace different kinds of musical objects (durations, meters, Klangs, timbres, and so on), different (i.e. non-Euclidean) geometries, and different orientational perspectives (interval as action or gesture rather than as simply measurement of distance between things). Along the way, the work recasts set theory, the concepts

of transposition and inversion, and notions of musical time in this generalized image. But the work has maintained its relevance and importance as well because of the brilliance and musicality of its author. David had a gift for finding musically

significant examples for his sometimes abstract concepts, and a gifted musical imagination that delighted in finding new ways to hear and understand familiar musical passages. While GMIT does not offer the extended musical analyses of his later books, Musical Form and Transformation or Studies in Music with Text, the work is nonetheless rich with smaller analytical gems. To be sure, transformational theory has evolved in the years since GMIT first appeared—the analytical use of Klumpenhouwer networks, the development of neo-Riemannian theory, and the resurgence of spatial] methodologies and metaphors

in analysis all postdate David’s seminal study. But each of these subsequent developments can find its basis in the framework David sets forth in GMIT: Klumpenhouwer networks apply the Generalized Interval System (GIS) concept recursively to create networks of networks; neo-Riemannian theory, which emerged from

explorations begun in chapter 8 of GMIT, takes families of contextual transforma- i&

Foreword to the Oxford Edition

tions to be the formal intervals between the familiar set of harmonic triads or seventh chords; spatial methodologies simply extend the idea of transformational networks to create graphs that embrace all members of a family of objects (pitches, pitch sets, rhythmic durations, and so on) related by certain contextually significant intervals. !

One notable new feature of this edition is an author’s addendum (the preface), drawn from a previously unpublished typescript titled “Updating GMIT,” which presents, in a sometimes synoptic form, concepts or musical examples David had planned for a future edition of GMIT. The document was likely written in the summer of 1987 and was used as the handout for a talk given at the Eastman School of Music in the fall of that same year. It should not be surprising to those who knew David’s incredible industry and the speed with which he could read and suggest revisions to others’ work that David would have been drafting plans for a new edition of GMIT so soon after its publication—for David, it was often difficult to stop thinking about a project, or tinkering with its ideas, once begun, and the document clearly represents David’s residual energy following the writing of GMIT. The examples explored in the addendum are diverse, although certain themes recur. For one, David seems to have been particularly concerned with examples that involve non-commutative groups of operations, no doubt because such groups often defy our accustomed and familiar intuitions about the way intervals work. For another, David seems to have been interested in finding examples that do not simply involve individual pitch classes (transformations of melodies, of Lagen in triple counterpoint, of ordered hexachords), again because these are less familiar, and often reveal less intuitive aspects of interval. Although the document is perfectly intelligible, some sections of “Updating GMIT” deserve additional comment. 1. The error in figure 8.2 (g minor instead of g# minor) that prompted David’s commentary in section I has been corrected in this edition. The first section of David’s notes was expanded to become his article “Some Notes on Analyzing Wagner: The Ring and Parsifal” (19th-Century Music 16.1, 1992, reprinted in David Lewin, Studies in Music with Text [Oxford University Press, 2006]). 2. David developed and expanded section IV into a pair of unpublished exercises for his math and music course at Harvard University. Exercise 5 (2 pages) directs the student to discover the elements of the Q—X group acting on the augmented triads of sc (014589) and then find transformations of the “rapture of the 1, David has written articles on each of these topics subsequent to the publication of GMIT. Klumpenhouwer networks are the topic of two articles: “Kiumpenhouwer Networks and Some Isographies that Involve Them,” Music Theory Spectrum 12.1 (1990): 83-120, and “A Tutorial on Klumpenhouwer Networks, Using the Chorale in Schoenberg’s op. 11, no. 2,” Journal of Music Theory 38.1 (1994): 79-101. David’s most significant post-GMIT contribution to neo-Riemannian theory is the article “Cohn Functions,” Journal of Music Theory 40.2 (1996): 181-216. Two of David’s contributions to graphical methods of analysis are “The D-major Fugue Subject from WTCII: Spatial Saturation?” Music Theory Online 4.4 (1998), and “Notes on the Opening of the F# Minor Fugue from

x WTC I” Journal of Music Theory 42.2 (1998): 235-239.

Foreword to the Oxford Edition

strife” figure under Q4, Q8, and X5 in Schoenberg’s Ode to Napoleon (as in David’s example 2 from the addendum). An optional part of that exercise encourages students to explore transformations of characteristic tetrachords in Schoenberg’s Ode using the members of the same Q—X group. Exercise 8 (3 pages) ex-

plores the simple transitivity of the Q—X group and has the student find the (interval-preserving) elements of the commuting group, {T,, T,, T,, I,, [,, I}. David’s article “Generalized Interval Systems for Babbitt’s Lists, and for Schoenberg’s String Trio” (Music Theory Spectrum 17.1 [1995]: 81-118), in particular “Part 5: Background on Non-Commutative GISs,” explores the relationship between non-commutative GISs and their commuting groups. 3. David similarly developed and extended section V into an exercise for his math and music course (exercise 9, 4 pages). The Daniel Harrison article to which David refers was published as “Some Group Properties of Triple Counterpoint and Their Influence on Compositions by J. S. Bach” (Journal of Music Theory 32.1 [1988]: 23-49). David inserted a manuscript page into the “Updating GMIT” typescript that presents a TPERM and VPERM analysis of Bach’s c-minor fugue from the Well-Tempered Clavier, Book I. The manuscript notes that the diagram is modeled after Schenker’s “Table of Voices” from “Das Organische der Fuge” in Das Meisterwerk in der Musik, Band I, p. 59, and further observes that the Lagen symbol “‘A’ can mean ‘Subject,’ ‘B’ can mean ‘Countersubject’ and ‘C’ can mean ‘any third part of roughly characteristic rhnythm’” (emphasis Lewin’s), suggesting that the methodology is not bound to works in strict triple counterpoint. David’s diagram, however, has not been incorporated into the author’s addendum of this volume because David wrote no accompanying text for it—creating new text would have adversely disrupted David’s prose in the rest of the section. David, however,

did use the c-minor fugue analysis as part of exercise 9 in his math and music course, which I present below for interested readers to explore if they wish (terminology has been adapted to conform to the text of “Updating GMIT”): PART I OF EXERCISE 9: (a) Complete the partially-filled diagram below, which pertains to the c-minor fugue in Book I:

Meas. _ Stufe Lage TPERM interval VPERM interval

7 i

11 jaat 15 Vv

20 i

26.5 i

(b) Discuss features of the construction which you find revealed by the double intervallic analysis. For instance, does the use of 3-cycles

bring out any aspect of the structure? Do the TPERM and VPERM xi

Foreword to the Oxford Edition

analyses coincide as they did [in the A-major Prelude]? What aspects of the piece are bound together by repetition of TPERM intervals? By repetition of VPERM intervals? 4. Section VI considers the GIS structure of a family of 12-tone-row transfor-

mations that David first explored in his article “On Certain Techniques of ReOrdering in Serial Music” (Journal of Music Theory 10.2 [1966]: 276-287). David refers in the section to “an excellent work, as yet unpublished” by Andrew Mead. That work was published in two parts as “Some Implications of the PitchClass/Order-Number Isomorphism Inherent 1n the Twelve-Tone System: Part One” (Perspectives of New Music 26.2 [1988]: 96-163) and, more pertinent to Lewin’s

addendum, “Some Implications of the Pitch-Class/Order-Number Isomorphism Inherent in the Twelve-Tone System Part Two: The Mallalieu Complex: Its Extensions and Related Rows” (Perspectives of New Music 27.1[1989]: 180-233). David, of course, never created a second edition of GMI7T, an undertaking that, he wrote, would have involved “[fixing] a lot of errata & corrigenda; some major rewrites here and there; a reasonable amount of bibliographic updating.”* This edition of GMIT, while retaining the text of the original, does incorporate the corrections indicated by David’s errata list. Moreover, while it does not attempt to identify

or alter passages that David felt needed rewriting, the articles cited in this foreword give a picture of David’s evolving ideas about transformational theory. And while David may have wanted a new edition of GMIT, rather than a second printing, he was also eager to make GMIT available to students and scholars. In these respects, this Oxford edition fulfills David’s wishes—that his ideas be available to all who seek them, so that they may grow, evolve and multiply. 2. 1995 e-mail correspondence, recipient unknown.

Xil

Preface

a)“Tr S Er b)fkSfF

I. The following figures redo those of figure 8.2 on p. 179. Music examples la and b present scores of the relevant passages.

eo & > D cat Bh ——> F

:gi ——> L —, —, ' L — + — [E] B (Db --) Gb ——+ [b)] f (-Ab) Tarnhelm Valhalla

L = LEITTONWECHSEL; +— = MAJOR-MINOR; S = “BECOMES SUBDOMINANT OF”.

> a . a a,~° PP

a) Tarnhelm, Rheingold III, 37ff.

1) 1 CS Aree OY”Bes CE ee 9 Yo” A 1=e haAWiuOA eesSDOe

(7 ee 2) eee |) Oe a ees) _6

EXAMPLE la xi (muted horns)

Preface b) Modulating section of Valhalla, Rheingold I, Sff.

P a . p Ly

ap = a|.

ll i a =~ 7 es ee 2 or LF i t—“(tis‘“‘i a AN = Pie : 3 me

= 2 rd ia L.a |i220 iad ew Tei hUel eh UL UUme Oe st CO OO stsi‘(‘(‘(‘(‘ v —— > Y

hte Aree NW ee. ee. eee eee Oe Oe: ee. ee 4

a A s e a 2 it e ea. Ty, 2. te of oT. te eee rs “* y¥ s ve Se

Sj ___~__1a¥ po li 1 pe p__|_} } xml TY i Ht ff y ———

Foil hh Et OM OL LLC OR Oe

PL ee OO a ae et aCeOl Oe L.A = Ce ie lie? ‘BeOY ee Le * 2ee dd 21

A) ee + _—___§ } eee

A G4 a beSeBeEE 8 eo Burg fi. CEee) ieee(Die kt ae. eeeist : ganz

Sp nn sichtlich geworden.) pf*an4% Ls we Pp

EXAMPLE 1b

The analysis is better than that in the book. It brings out a clear isography between the passages. Figure 8.2a in the book is not a well-formed “graph” by the later defi-

nition. (SUBM is not = LT SUBD on major as well as minor Klangs: (C,+) SUBM = (e,—) but (C,+)LT SUBD = (e,—)SUBD = (b,—).) The symbol “(G,—)”

on figure 8.2a is a misprint for (Gt,—).' The discussion of section 8.1.2, pages 179-180, still applies: a group that contains L, S, and +— operations on Klangs will not be simply transitive in equal temperament. (For instance, (C,+)SSSS =(E,+), but (C,+)L +— also = (E,+).) xiv 1. See item 1 in the foreword, p. x.

Preface

Later in the Ring, Wagner develops the relationship of Valhalla and Tarnhelm themes very ambitiously. Figures c) through f) below analyze a transformation that occurs at the climax of Walkiire II,2: Woton, coming to realize the full implications of Valhallagate, ironically gives his blessing to Hagen (“So nimm meinen Segen, Niblungen Sohn!”). Music examples 1c through le are coordinated with the figures.

c)iAb —-> Ab} d)i gh gt

Lis ©€-N gs | govt e) gt —— 'E ¢! or im (*

| + | re On 2 a

c) d) | en a ae oun fr et ttt tthe E “cross-meter” is heard as one of several contending temporal patterns by Edward T. Cone, “Analysis Today,” Problems of Modern Music, ed. Paul Henry Lang (New York: W. W. Norton, 1962), p. 44. Elsewhere, we can read that the first three notes we hear form a rhythmic group which “is clearly 3; the motion of the two quarter notes into the next downbeat defines the meter precisely.” This is the hearing of James Rives Jones, “Some Aspects of Rhythm and Meter in Webern’s Opus 27,” Perspectives of New Music vol. 7, no. 1 (Fall-Winter 1968), p. 103. Later,

still on page 103, he refers to “the 3 meter” as “already ... established” by the cited criterion. 39

3.31 Generalized Interval System (2)

at the attack of the opening E}, then the D and the G on figure 3.1, with the rests that follow them, each fill one such span. I do not know what to make of this. The idea of quintuple “perfections” seems a better metaphor for my hearing this aspect of the music than does the idea of quintuple “meter.” Figure 3.1 also shows the recurrent GIS,-interval (2, 7), interlocking the mensural function of “7 beats later” with the pitch-class interval 2. The first (2, 7) recurrence links the two notes of the B—B} accompaniment figure to the corresponding two notes of the C#—C figure, 7 beats later. The second recurrence (third occurrence) of (2, 7) links the attack of the cadential G to the beginning of the accompaniment figure A—G# 7 beats later. In sum, the recurrent GIS,-interval (2, 7) links aspects of the recurrent accompaniment figure with other events of the music.

pitch-class _ recurrent compositional temporal

intervals GIS3-intervals functions intervals (11, 1)» defines the beat; spans ——> 1

a the accompaniment figure 11

wn 5)-————» Pc interval 11, via GIS3interval (11, 1), gives a special mensural function to “5S beats later”? ————_ 5 “5S beats later” is

also a maximal wait between 3——» (3, 2)—————> engages the notated meter; attacks; that

ee ;

spans the approaches to follows the the cadences -——-— cadences

2 ——» (2, 7) relates the accompaniment figures to other events 7 beats later or earlier. ——?» 7 (Cf. GIS3-interval (11, 1).) FIGURE 3.2

Figure 3.2 summarizes the discussion of figure 3.1 so far, in the form of a table. It shows how the recurrence of GIS,-intervals gives special meanings 40 to the pitch-class intervals 11, 3, and 2, as those interrelate with the temporal

Generalized Interval Systems (2) 3.3.1 intervals 1, 5,2, and 7, allin connection with various compositional features of the music.

S3-elements GIS3-interval vectors of S3 sets (Eb, 0)

+ (B, 3) (8, 3) + (Bb, 4) ne 4) (11,1)

+ (D, 5) +(11,5) (3,2) (4,1) +(C§, 10) +(10,10) (2,7) (3, 6) (11, 5)

+(C, 11) +(9,11) (1,8) (2, 7) (10, 6) (11, 1) FIGURE 3.3

Figure 3.3 applies to the opening of the passage a theoretical construction suggested by the temporal aspect of GIS,. The left-hand column of the figure lists the first six members of §, in their order of appearance during the passage.

First comes pitch-class Eb at time-point 0, instancing element (Eb, 0) of S;. Next comes pitch-class B at time-point 3, instancing element (B, 3) of S,. At this time (i.e. just after time-point 3), we become aware of a 2-element S,-set, that is, the set ((Ep, 0), (B, 3)). The elements of this set form one GIS,-interval

with a positive time-component, that ts, the interval (8,3) from (Eb,Q) to (B, 3). The GIS,-interval (8, 3) is listed in the second row of the second column in figure 3.3; that interval is the sole constituent within the interval vector of the 2-element S,-set.

Now the pitch class Bp occurs at time-point 4, providing the new S;, element (Bb,4) for the third row of column 1 on figure 3.3. At this time (i.e. Just after time-point 4), we become aware of a 3-element S,-set, ((Eb, 0), (B, 3), (Bb, 4)). Besides the GIS,-interval of (8,3) already listed in row 2, the elements of the 3-member set produce new GIS,-intervals of (7, 4) (= int,((Eb, 0), (Bb, 4))) and (11, 1)(= int, ((B, 3), (Bb, 4))). These new constituents for the interval vector of the expanded 3-element S,-set are listed in

the third row of column 2 on figure 3.3; the old interval vector expands to include the occurrences of the two new intervals. When pitch class D enters at time-point 5, the 3-element $,-set expands to a 4-element S,-set ((Eb, 0), (B, 3), (Bb, 4), (D, 5)), and the interval vector of the 3-element set expands to adjoin new occurrences of the GIS,-intervals

(11, 5)(= int;((Ep, 0), (D,5))), (3, 2)(= int,((B, 3), (D, 5))), and = (4,1) (= int,((Bb, 4), (D, 5))). The new intervals are listed in the fourth row of column 2, on figure 3.3. At this time (i.e. just after time-point 5), we become = 4]

3.3.1 Generalized Interval Systems (2) aware for the first time that some pitch-class intervals are predominating over others, and that some temporal intervals are predominating over others, as we note the various intervals going by. In earlier work I have suggested that our becoming aware of such predominances is associated with our marking such a time-point as an “ictus.” * The present analysis supports that theoretical idea, since time-point 5 is both a notated barline, indeed the first notated barline,

and also audible to some extent as marking the attack of an intuitively “strong” quarter. These considerations lice behind my bracketing the first four entries of column | on figure 3.3, as belonging together in a special way. According to the theory just sketched, it 1s only when the 4-element S,-set has been completely exposed, 1.e. itis at and only at a moment just after time-point 5, as we listen along, that we first become sensitive to any regular mensural structuring in the passage. Just after time-point 5 we become aware that temporal interval 1 is predominating over other temporal intervals; we can then (and only then)

hear temporal interval | as a beat with which to measure other temporal intervals. In that sense the GIS,-structure of figures 3.1—3.3 really ““begins”’ for a listener at (and only at) time-point 5, the first written barline and the first perceived ictus; any GIS,-structure preceding time-point 5, according to this theory, is reconstructed by a listener at (or following) time-point 5. Not only ts

the beat established at time-point 5, the pitch-class interval 11 is also established at the same time, as a predominating pitch-class interval. Pitch-class

interval 11 is thus bound psychologically to the establishment of mensural structure in the piece, as part of one and the same Gestaltist experience that a listener will be having just after time-point 5.

pc interval 11 11 (NB § beats later)

recurs @ time-point 5: = 1 : +

temporal interval 1] \/ \/ Now! recurs @ time-point 5: FIGURE 3.4 Figure 3.4 symbolically collates the ideas discussed just above. The figure also suggests how the temporal interval of “5S beats later’ has already acquired a special significance at the moment the listener hears time-point 5. "5 beats

later,” namely, spans the temporal distance from the opening attack to the first ictus. Figure 3.4 shows how that temporal distance is already clearly associated with the structural function of the recurrent pitch-class interval 11, 4. David Lewin, “Some Investigations into Foreground Rhythmic and Metric Patterning,” Music Theory, Special Topics, ed. Richmond Browne (New York: Academic Press, 1981),

42 101-37.

Generalized Interval Systems (2) 3.3.1

the pitch-class interval between the opening Eb and the D of the first ictus, 5 beats later. These considerations enable us to analyze with greater precision how the idea of “being in 3”’ might arise, and how it would become associated with the compositionally thematic Ep—D gesture.°

Let us now inspect the fifth row of figure 3.3, investigating how our impressions develop when C# is attacked at time-point 10, introducing new manifestations of the GIS,-intervals (10, 10), (2, 7), (3, 6), and (11, 5). We now hear a second pitch-class interval of 3, but simultaneously we also hear a third pitch-class interval of 11. The latter interval, by recurring yet again, continues

to predominate over other pitch-class intervals. Indeed, its predomination itself recurs. Furthermore, we now (i.e. just after time-point 10) hear for the first time the recurrence of a GIS,-interval (not just of a temporal interval or pitch-class interval). The recurring GIS,-interval is (11, 5), recently discussed

in connection with the possibility of asserting a “thematic ? meter’ when listening at time-point 5. The sensations prompting such a possible assertion are thereby intensified at time-point 10. Time-point 10 ts experienced as an

“ictus” in the formal sense of the theory mentioned earlier. Dynamic and registral accents at time-point 10, the loud low C# attack, support the possible hearing of a “‘strong beat’’ there, should one want to assert “3 meter” beyond purely mensural considerations. We can then expand figure 3.4 to figure 3.5, which portrays a provisional impression one might have while listening just after time-point 10.

Ll Cn ‘ LN, J p “XT \/ te Now? (ictus?) f Pot (11, (GIS 3 interval (11, 5) 5) (11, 5) recurs @ time-point 10) FIGURE 3.5

Now let us turn to the sixth row of figure 3.3, investigating our impressions when C is attacked at time-point 11, introducing new manifestations of the GIS,-intervals (9, 11), (1, 8), (2, 7), (10, 6), and (11, 1). (11, 1) here is a recurring GIS,-interval; it confirms our already developed sensations about the beat-defining and other mensural functions of the pitch-class interval 11. The GIS,-interval (2, 7) also recurs at time-point 11. This builds up another

183-216. 43

5. My analysis of listener psychology just after time-point 5 partly elaborates, partly qualifies, and partly takes issue with the thought-provoking approach to this passage by Christopher Hasty, in his important methodological and analytic study, “Rhythm in Post-Tonal Music: Preliminary Questions of Duration and Motion,” Journal of Music Theory vol. 25, no. 2 (Fall 1981),

3.3.1 Generalized Interval Systems (2) mensural matrix that tries to expropriate (11, 1) for its own purposes, trying to put an ictus at time-point 11, rather than time-point 10. Figure 3.6 sketches this notion.

tt 3 Now? (ictus?)

Lt

2.7) : recur @ time-point 11.) (2, 7)

FIGURE 3.6

The ictus on Cin figure 3.6 corresponds to a written barline; this was not the case with the conflicting ictus on Cf in figure 3.5. That C# was 5 beats after the D-ictus of figure 3.4; the C of figure 3.6, which picks up the Hauptstimme D in register, is 6 beats after the D-ictus. The mensural conflict of “5 units after’? (D-ictus to C#-ictus) and “6 units after” (D-ictus to C-ictus) is highly thematic in op. 27 as a whole.® The mensural reading of figure 3.6 tries to associate the C#—C event in the music with the “accompaniment figure,” the figure that projected the B—Bpb

event. In contrast, the mensural reading of figure 3.5 tried to associate the D—C# descent with the thematic Ep—D descent. Figure 3.5, of course, “did not

know about” the forte and staccato C natural coming up right after the forte and staccato C#. We have already discussed how the recurrent GIS,-interval (2, 7) interacts with the accompaniment figure more generally. This completes the exegesis of figure 3.3. The theoretical notion of an “unfolding interval vector,” made abstractly available by the temporal aspect of GIS,, was analytically useful for examining our impressions of figure 3.1 as those developed note-by-note, and for discussing to a significant extent our impressions of the music beyond that. GIS, was particularly useful because it enabled us to consider pitch-class structure and mensural rhythmic structure in conjunction with each other, rather than as independent features of the passage. That is the essence of GIS, in its capacity as what we shall soon call the direct product of GIS, and GIS,. 6. Aconflict between mensural distances of 5 and 6 units figures in the relation of the rhythmic ostinato to the written meter at the opening of the first movement. A reprise of this rhythmic conflict occurs at the opening of the coda in the last movement (mm. 56—62). The written meters are functional, in ways too complex to indicate here. Special accents attach to the loud and dense trichord-pairs of the middle movement. The lower chords of the trichord-pairs attack at the barlines of measures 4, 9, 4 bis (6 measures after 9), 9 bis, 15, 20, 15 bis (6 measures after 20), and 20 bis. The resulting pattern projects alternating spans of 5 and 6 written measures. Indeed this may well be the strongest mensural function for the written measure as a temporal span in the

44 movement.

Generalized Interval Systems (2) 3.3.3

3.3.2 ExampLe: Let GIS, be the GIS involving time-points, that just figured

as “GIS,” in the preceding example. For the present example, let GIS, be a GIS of durations as in 2.2.3 earlier: S, is a certain family of “durations” x, y,... related by certain stipulated proportions; [VLS, is the multiplicative group of such proportions; int,(x, y) is the quotient y/x. We construct a new GIS, GIS;, as follows. S, is the Cartesian product S, x S,, that is, the family of pairs (s, x), where s is a time-point and x is a duration. We can conceive (s, x) as modeling an event that begins at time-point s and extends for a duration of x (units) thereafter. IVLS,, is the direct-product

group IVLS, @ IVLS,. Each member of IVLS, is a pair (1,,1,), where i, is a member of IVLS, (representing a number of beats between time-points) and i, is a member of IVLS, (representing a quotient of durations). (i,, 1)

and (j,,j2) combine in the direct-product group IVLS, according to the rule (i,,12)(j1,.J2) = (iy + J1,12J2). ints((s, x), (t, y)) is defined as (int, (s, 0), int, (x, y)), that 1s, loosely speaking, as (t — s, y/x). To put it another way, if int,((s, x), (t, y)) = (i,,1,), then event (t, y) begins 1, units of time after event (s, x) and extends for i, times the extent of event (s, x). The reader can take it on faith that GIS, as constructed above is indeed a GIS. It will not be necessary to produce an analytic example, I think, in order to convince the reader that this GIS is a useful theoretical tool. It combines

two aspects of our mensural rhythmic intuition, as they impinge upon us conjointly, into one compound structure. We are still and again presuming a fixed unit of time here, by which we measure durations and distances-betweentime-points. In both example 3.3.1 and example 3.3.2 we combined a given GIS, witha

given GIS, in a certain manner, obtaining a new GIS that modeled the conjoint action of the two given GIS structures. We can make our procedure formal and general, as a means of combining any given GIS, and any given GIS, into a “direct-product GIS.” The following definition gives the procedure. 3.3.3. DEFINITION: Given GIS, = (S,,IVLS,,int,) and GIS, = (S,, IVLS,, int,), the direct product of GIS, and GIS,, denoted GIS, ® GIS.,, is that GIS, = (83, IVLS;, int,) which is constructed as follows. 53isS, x S,, the Cartesian product of S, and S.. That is, the elements of S3 are pairs (S;,8,), where s, and s, are elements of S, and S, respectively.

IVLS, is IVLS, @IVLS,, the direct-product group of IVLS, and IVLS,. That is, the members of IVLS, are pairs (i,,i,), where i, and i, are members of IVLS, and IVLS,; further, the members (i,,i,) and (j,,j,) of IVLS, combine (to form a group) under the rule (i,,1.)(j,,j2) = (ij,,i2j2). The function int;, from S, x §, into IVLS,, is given by the rule

int,((s,,52),(t,,tz)) = (int, (s;, t,), int,(s., t,)). 45

3.4.1 Generalized Interval Systems (2)

It 1s straightforward to verify that GIS,, as defined above, is indeed a GIS, i.e. that GIS, satisfies Conditions (A) and (B) of Definition 2.3.1. This finishes our investigation into methods of constructing new GIS structures from old in a general abstract setting. Now we shall see how the notions of “transposing” and “inverting” elements of a space S arise naturally

in any GIS; we shall see how operations of transposition and inversion interrelate characteristically with intervallic structure, and we shall explore how the operations combine among themselves. We shall also explore other characteristic operations, the “interval-preserving operations”; these may or may not be the same as the transposition operations, depending upon whether the group of intervals is or is not commutative. ’ 3.4.1 DEFINITION: Given a GIS; given an interval i of IVLS; then transposition by i, denoted T,, is defined as a transformation on S as follows. Given a sample element s of S, the i-transpose of s, T,(s), is that unique member of S which lies the interval1 from s. That is, T;(s) is well defined by the equation int(s, T;(s)) = 1.

This definition conforms to our abstract intuition that the i-transpose ofa given element s should lie the interval 1 from s. The definition also conforms to the way in which we already use the word “‘transposition” in connection with some GIS structures, namely those involving pitches and pitch classes. T,(s) is indeed well defined by the equation within the definition. Condition (B) of 2.3.1 assures us that given i and s, there exists a unique t such that int(s, t) = 1; it is precisely this t which we are now calling “the i-transpose of s,” T;(s). 3.4.2. THEOREM: Each T;, is an operation; that is, it is 1-to-1 and onto as a transformation on S. The transposition operations form a group of operations on S. That group is anti-isomorphic to the group of intervals. Specifically, let us consider the map f, defined from IVLS onto the family of transpositions by the formula f(i) = T,; then fis an anti-isomorphism. That is, fis 1-to-1 as well

as onto; and TT; = T;;. Proof (optional): We shall] prove the assertions of the theorem in an order different from that in which the theorem states them. First we show that f1s an

anti-homomorphism, i.e, that T;T, = Tj. Given intervals i and j, given a sample sin S, then we write int(s, T,(T;(s))) as the group product of the two intervals int(s, T;(s)) and int(T,(s), T;(T;(s))), using Condition (A) of 2.3.1. Now int(s, T,(s)) = j, by the defining equation of

Definition 3.4.1. And by the same equation, int(T,(s), T,(T,(s))) = 1. Hence 7. All the groups in our specific examples of GIS structure so far have been commutative.

46 Later on, in chapter 4, we shall study a non-commutative GJS of musical interest.

Generalized Interval Systems (2) 3.4.4 int(s, T,(T,(s))) is expressed as the group product of the two intervals j and 1. That is, int(s, T,(T,(S))) = ji. Thus T;(T;(s)) lies the interval ji from s. So it is

equal to T;,(s). We have shown: For any sample s, T;T;(s) = Tj,(s). So the transformation T,T, has the same effect on S as does the transformation T,,; the functional equation T,T, = T), is true, as claimed. So the map f is an anti-homomorphism. We show now that it Is an antiisomorphism. We have only to show that fis a 1-to-1 map. Supposing that the functional equation T; = T; is true, we wish to prove that i and j must be the same interval. Fix any s. Since T; = T; by assumption, T,(s) = T,(s). Then, by Definition 3.4.1, i = int(s, T;(s)) = int(s, T;(s)) = j, which is what we had to show. It remains only to prove that each T, is an operation, and that the family of transpositions 1s a group of operations. To prove all this, it suffices to show

that the family of transpositions satisfies Conditions (A) and (B) of 1.3.4 earlier, namely (A) that the family is closed and (B) that for each T, there is a

T; satisfying T;T,; = T;T, = 1. Condition (A) is evident from the formula T;T,; = T;,: The composition of two transpositions is a transposition. In order to prove Condition (B), we shall prove a lemma: T, is the identity operation 1 on S. That is true since, given sin S, T,(s) is the member of S which lies the identity

interval from s; hence T,(s) = s. Or T,(s) = 1(s); that being the case for any samples, T, = 1 as asserted by the lemma. Now we can prove Condition (B). Given any interval i, take j = i-*. Then ji = ij =e. T, is the transformation demanded by Condition (B): T,T, = T,, = T, = 1; likewise T;T,; = 1. q.e.d. 3.4.3. THEOREM: Fix some referential member ref of 8; then LABEL(T,(s)) =

LABEL(s):i. Proof: LABEL(T;(s)) = int(ref, T,(s)) by definition of LABEL in 3.1.1,

= int(ref,s)int(s,T;(s)) by 2.3.1(A), = LABEL(s)int(s, T,(s)) by 3.1.1, = LABEL(s)-i_ by 3.4.1. Theorem 3.4.3 tells us that no matter what ref is chosen for LABELing purposes, the label for the i-transpose of s is the label of s, right-multiplied by iin IVLS. The natural question arises: What happens when we left-multiply ref-LABELs by 1? We shall explore that right now. 3.4.4 DEFINITION: Fix some referential member ref of S. Given any interval i, the transformation P, (more exactly P!*)is defined on S by the formula

LABEL(P,(s)) = i: LABEL(s), which is to say the relation

int(ref, P;(s)) = 1: int(ref, s). 47

3.4.5 Generalized Interval Systems (2)

Given ref, i, and s, Condition (B) of 2.3.1 tells us that a unique member of S, call it t, satisfies the relation int(ref, t) = i- int(ref, s). Definition 3.4.4 takes that unique t, given ref, i, and s, and calls it P(s). Note that the specific value of P.(s) depends on ref as well as on i and s. In contrast, the value of T,(s) was well defined in 3.4.1 independent of any choice of ref. 3.4.5 THEOREM: The transformations P. form a group of operations isomor-

phic to IVLS under the map fi) = P.. In particular, the formula PP. = P. is valid.

Proof (optional): LABEL(P(p,(s))) = j° LABEL(P((s)) = i-(j- LABEL(s))

= (ij): LABEL(s) = LABEL(P;,(s)). The elements P(P;(s)) and P.(s) thus

JJ a

have the same LABEL (lie the same interval from ref). So they are the same:

P.P(s) = P.(s). This being the case for any sample s, the functional equation PP. = P.. is true.

The map f of the Theorem is thereby a homomorphism of IVLS onto the family of transformations P.. To prove that f is an isomorphism, it remains

only to show that f is 1-to-1. Suppose P, = P; we are to infer that 1 = j. Fix any s; then i- LABEL(s) = LABEL(p(s)) = LABEL(P(s)) by supposition; and that = j- LABEL(s). In sum, i- LABEL(s) = j - LABEL(s). Hence, multiplying that equation through on the right by the inverse of LABEL(s),

i = j as desired. We can now use arguments just like those we used in the proof of 3.4.2, to show that the family of P. is a group of operations. The operations P. have a special property. They are what we shall call the “interval-preserving” transformations. 3.4.6 and 3.4.7 develop the formalities. 3.4.6 DEFINITION: Given a GIS (S, IVLS, int), a transformation X on S will be called “interval-preserving” if X has this property: For each s and each t,

int(X(s), X(t)) = int(s, t). 3.4.7 THEOREM: No matter what ref is chosen, the interval-preserving transformations on S are precisely the P.. Proof (optional): We show first that P. preserves intervals. We can write int(P.(s), P.(t))

=LABEL(P,(s))~ 'LABEL(P,(t)) by 3.1.2, =(i- LABELS(s))~'i- LABEL(t)) ~~ by 3.4.4, =(LABEL(s)7! - i7')(Gi - LABEL(t)) =LABEL(s)~ 'LABEL(t)

=int(s, t) by 3.1.2. 48 Thus P. preserves intervals. Now suppose X is any interval-preserving

Generalized Interval Systems (2) 3.4.8 transformation on S. We shall show that there exists some interval 1 such that

X = P.. The i we want here is LABEL(X(ref)) = int(ref, X(ref)}). Given any s, we can then write LABEL(X(s)) = int(ref, X(s))

= int(ref, X(ref))int(X(ref), X(s)) via 2.3.1(A), = 1-int(X(ref), X(s)) by construction of 1 here, = :'int(ref,s) because X is interval-preserving, =i:LABEL(s) _ by the definition of LABEL, = LABEL(P;(s)) via 3.4.4. In sum, we have LABEL(X(s)) = LABEL(P,(s)). Since X(s) and P,(s) have the same LABEL, we infer that X(s) = P,(s). Since s here is an arbitrary sample member of S, X = P, as a transformation on S. Thus our intervalpreserving X is in fact this particular P,. q.e.d. Because of 3.4.7 we can conceive the transformations P; as somewhat less dependent on the choice of ref. It is true that a transformation labeled “P,” by one choice of ref might be labeled “*P,”” by another choice of ref. However the interval-preserving transformations en masse, literally “‘as a group,’ remain

the same family of transformations en masse, no matter what ref 1s chosen. The interval-preserving property does not depend on the choice of ref, and that property is sufficient to define the family of transformations as a group. 3.4.8 THEOREM: Given an interval 1, Conditions (A) through (D) below are all logically equivalent: If any one of them is true, then they are all true. (A): T; preserves intervals.

(B): For some choice of ref, T; = P,. (C): For any choice of ref, T; = P;. (D): iis centralin IVLS. That is, icommutes with every j in [IVLS (1.8.2).

Proof (optional): Suppose (A) is true; we prove that (C) follows. Fix any ref. Since T; preserves intervals by assumption, 3.4.7 tells us that

T; =P; for some j. For any s, LABEL(s):1 = LABEL(T;(s)) by 3.4.3; that = LABEL(P;(s)) since T; = P,; and that = j}-LABEL(s) by 3.4.4. In sum, LABEL(s)-1 = }-LABEL(s) for any s. Consider s = ref in particular: LABEL(ref)-i = j: LABEL(ref), But LABEL(ref) = int(ref, ref) = e. Hence

i= Jj. T; = P, as desired. ,

Clearly the truth of (C) entails the truth of (B). Now suppose (B) is true; we prove that (D) follows. We are supposing T; = P; for some ref. Then for every s, LABEL(T;(s)) = LABEL(P,(s)). Then for every s, LABEL(s)-i = 1: LABEL(s) (3.4.3 and 3.4.4). It follows that i commutes with every j, which is Condition (D) as desired. For given any j, find the s which lies the interval jfrom ref; then LABEL(s) = }; substituting j for LABEL(s) in the most recent

equation involving i, we obtain ji = ij; i commutes with the given j. 49

3.4.9 Generalized Interval Systems (2) Now we close the logical chain by showing that the truth of (D) entails the

truth of (A). When we have done this, we shall have shown that (A) implies (C), (C) implies (B), (B) implies (D), and (D) implies (A); hence the truth of any

one entails the truth of all the others. Supposing (D) is true, then, we show that (A) will be true. Fix any ref. Given any s and any t, we write int(T,(s), T;(t))

= LABEL(T,(s))"'LABEL(T,(t)) — by 3.1.2

= (LABEL(s):i)"}(LABEL(t)-i) by 3.4.3 =(i-*- LABEL(s)~')(LABEL(t) - i) =j~!- (LABEL(s) *LABEL(t)):i

=i7!-int(s, t):i by 3.1.2 = int(s, t) by the assumption of Condition (D), that 1 is central. In sum, assuming Condition (D), then int(T;(s), T;(t)) = int(s, t) for every s and t. Or, assuming Condition (D), T; will be interval-preserving. Or: (D) implies (A) as desired. q.e.d.

3.4.9 Coro.varigs: (A): In a commutative GIS (a GIS whose group of intervals is commutative), the transposition operations are precisely the interval-preserving operations. (B): In a non-commutative GIS, there exist transposi-

tions that do not preserve intervals, and there exist interval-preserving operations that are not transpositions. 3.4.10 THEOREM: Any transposition operation commutes with any intervalpreserving operation.

Proof: Fixing any ref, consider T; and P;. We apply 3.4.3 and 3.4.4 to various LABELS. Taking any sample s, we have LABEL(P,(T;(s))) =] LABEL(T,(s)) = j- (LABEL(s) - 1) = (j- LABEL(s)) - i = LABEL(P,(s))-1 = LABEL(T,(P;(s))). Thus P;T;(s) and T;P;(s) have the same LABEL, they lie the same interval from ref; so P;T;(s) = T,P;(s). This being the case for any sample s, the functional equation P,T, = T;P; is true: T; commutes with

P;. q.e.d. We are now ready to define and study “inversion operations” on an abstract GIS. For each u and each v in S (v may possibly equal u), we shall define an operation I‘, which we shall call “‘u/v inversion.”’ Figure 3.7 helps us visualize an appropriate definition for the operation, conforming to our spatial intuitions. The figure shows how we conceive any sample s and its inversion I(s) (short for I¥(s) here) as balanced about the given u and v in a certain intervallic

proportion. I(s) bears to v an intervallic relation which is the inverse of the 50 relation that s bears to u. The inverse proportion is symbolized by the mirror

°e

Generalized Interval Systems (2) 3.5.2

§

int (s, u) . u

v

yy (1(s), v)

I(s) FIGURE 3.7 relation of the two arrows on figure 3.7. The interval from v to I(s), which is the inverse of int(I(s), v), will then be the same as the interval froms tou. Thatis, we

intuit int(v, I(s)) = int(s, u). We can use this equation to define I‘ formally in any GIS. 3.5.1. DEFINITION: Given any u in S and any v in S, the operation I} of u/v inversion is defined by the equation

int(v, I¥(s)) = int(s,u) for alls. The operation is well defined by the equation: Given any s, set i= int(s, u) and find the unique t which lies the interval i from v. That t, which satisfies the equation int(v, t) = int(s, u), is precisely the value for I*¥(s). We have been referring to the “operation” IX prematurely; so far we have

constructed a transformation, but we have not verified that the transformation is indeed an operation, t.e. onto and 1-to-1. The reader may verify, as an exercise, that IY as defined is onto and 1-to-1. (Given t, find an s such that I¥(s) = t. Prove that if I¥(s’) = IX(s), then s’ = s.) 3.5.2 THEOREM: Fix a referential element ref of S. Set 1 = LABEL(v) and j = LABEL(u). Then

LABEL(I‘(s)) = i: LABEL(s)7! °j. Proof: To save space, we write “I” for “I}” here.

int(v, I(s)) = int(s,u) (3.5.1). So int(v, ref )int(ref, I(s)) = int(s, ref)int(ref,u) (2.3.1(A)). Thence,

LABEL(v)"*LABEL(I(s)) = LABEL(s)!LABEL(u) (3.1.1; 2.3.2)

Or: i7'- LABEL(I(s)) = LABEL(s)™! -j

Or: LABEL(I(s)) = i: LABEL(s)7* :j q.e.d. 51

3.5.3 | Generalized Interval Systems (2)

The formula of this theorem is very useful despite the dependence of all the LABELs (including 1 and j) on a possibly arbitrary ref. We shall use the formula to analyze this question: When are IY and IY the same operation on S? In the familiar GIS of twelve chromatic pitch classes, for instance IC = If} = [D'= JA = IC'and so on. That is, C/C inversion has the same effect on any sample pitch class as does F#/F% inversion, or A/D? inversion, or D#/A inversion, or B/C# inversion, and so on. In this special GIS one can intuit that I< and I¥ will be the same operation if and only if the pitch class w is the C/C inversion of the pitch class x. One might thereby conjecture that I‘ and IY will be the same operation, in a general GIS, if and only if the thing w is the u/v inversion of the thing x. This conjecture is valid if the GIS is commutative. When the GIS is not commutative, we must be content with the broader view provided by the following theorem. 3.5.3. THEOREM: I* = IY as an operation on S if and only if w = I*(x) and the interval int(x, u) is central. Proof (optional): Imagine ref fixed. Let i, j, k, and m be the respective LABELS for v, u, w, and x. Then, via 3.5.2, given any s,

LABEL(I\(s)) = i LABEL(s)~! j, while LABEL(I%(s)) = k LABEL(s)~' m. Hence the condition that IY be the same operation as I¥ is equivalent to the condition that

i LABEL(s)~! j = k LABEL(s)“! m_ for afl s in S. Now as s runs through the various members of S, the inverse of its LABEL runs through all the various intervals in IVLS. Hence the condition under discussion is equivalent to the condition that

in} = knm_ for every n in IVLS. And that condition is equivalent, via group theory, to Condition (A) below.

(A): (k7!i)n = n(mj7') for every n in IVLS. In sum, IY = I” as an operation if and only if Condition (A) above is satisfied.

Now we shall prove that Condition (A) is satisfied if and only if w = I¥(x) and int(x, u) is central. That will prove the theorem as stated. Suppose, then, that Condition (A) is true; we shall show that w = I'(x) and int(x, u) is central. Condition (A) being true by supposition, it is true for n = e; therefore k~'i = mj~!. Let us call this interval c. Condition (A) then tells us that cn = nc for every n in IVLS, so c is central. c was taken equal to mj7!. It turns out that c also equals j~'m. To see that,

we write mj~! = c; thence m = cj; thence, since c is central, m = jc; thence

52. jim = c as asserted.

Generalized Interval Systems (2) 3.5.5 Let us review: Assuming Condition (A), we have so far shown that k7'i =

mj! = j-'m = c is central interval. Now k~'i = LABEL(w)"'LABEL(v) = int(v, w), via 3.1.2. Likewise j~!m = int(x, u). Thus we have shown: int(v, w) = int (x, u) and int(x, u) = c 1s central. Now the relation int(v, w) = int(x, u) is exactly the relation which tells us that w = I*(x) (Definition 3.5.1). Thus, assuming Condition (A), we have proved that w = I*(x) and int(x, u) is central as desired.

Now we prove the converse half of the theorem: Supposing that w = I¥(x) and that int(x, u) is central, we prove that Condition (A) is satisfied.

w = I(x) (by supposition). So

int(v, Ww) = int(x, u) (3.5.1). Or: LABEL(w)7'LABEL(v) = LABEL(u)7'LABEL(x) (3.1.2). Or:

k~'i = j-'m (meaning of i, j, k, m).

Furthermore, we have supposed that j~'m, which is LABEL(u)~ 'LABEL(x),

which is int(x, u), is a central interval. Let us call this central interval c.

j-'m = c; thence m = jc; thence m = cj; thence mj! = c. So k~'i = mj~! = c, a central interval. c being central, cn = nc for every n in IVLS. Substituting k7'i = mj7! for c, we obtain

(k~'i)n = n(mj7') for every n in IVLS. And that is precisely Condition (A). q.e.d. We used ref and LABEL in our proof of Theorem 3.5.3. Note, however, that the statement of the theorem does not depend on a choice of ref. 3.5.4 CORROLARY: I! = IY if and only if int(v,u) is central. Proof: Applying the formula of Definition 3.5.1 to the algebraic truism int(v, u) = int(v, u), we infer that u = I*(v). Then, by the logic of Theorem 3.5.3 just proved, IY = I° if and only if int(v, u) is central. The corollary tells us that in a general GIS, v/u inversion may well not be the same operation as u/v inversion, despite the relations v = I*(u) = I*(u);

u = I(v) = Iv). The operations I* and I’ both transform u to v, and v to u. But there may be some other s, other than u or v, such that I'(s) is not the same element as I°(s). Indeed the corollary assures us there will be some such s if int(v, u) is not central. These considerations indicate how carefully and rigorously we must proceed hereabouts; intuitions based on our familiarity with a number of commutative GIS structures will not always be reliable.

IY if and only if w = I'(x). 53 3.5.5 COROLLARIES: In a commutative GIS, I¥ always = I'; generally, IY =

3.5.3 Generalized Interval Systems (2) In a non-commutative GIS, there will always be some inversion operation I\ which is not the same as I’. (For there will always be some int(v, u) which is not central.)

Now we shall see how inversion operations IY combine with transpositions T, and interval-preserving operations P. 3.5.6 THEOREM: For any transposition T, and any inversion I’,

(A): T,1i = I, where x = T,(u). (B): IVT, = I¥ where w = T_!(v). (C): T, commutes with I* if and only if n is central and nn = e. Proof (optional): We shall fix some ref and use LABELs to help our computations. Let 1 = LABEL(yv); let j} = LABEL(u). Then

LABEL(T I*(s)) = LABEL(I*(s))-n (3.4.3) = 1+ LABEL(s)7!-j-n (3.5.2) = LABEL(I%(s)),

where x is the member of S whose LABEL is jn (3.5.2). Since LABEL(x) =

jn and LABEL(u) = j, x = T,(u) (3.4.3). Now TI'(s) I*(s) have the same LABELs, via the chain of computations above. Hence T,I‘(s) = I}(s). (Both lie the same interval from ref.) Since s was an arbitrary sample member of 8, TI’ = IY as an operation. This proves (A) of the theorem. To prove (B), we start with a similar chain of computations.

LABEL(I* T (s)) = i(LABEL(T,(s)))~'j (3.5.2)

= i(LABEL(s) - n)~}j (3.4.3) = in~'LABEL(s)~}j = LABEL(I"(s)),

where w is the member of S whose LABEL is in ~! (3.5.2). Since LABEL(w)

= in~' and LABEL(v) = i, w is the n~ transpose of v (3.4.3). Since T7' = T,', we can write w = T7'(v). (The map of n to T, is an anti-isomorphism; n~' maps to T7'.) We go on to infer the operational equality of the operations I‘ T, and I*, exactly in the way we inferred an analogus equation when proving (A) of the theorem. This proves (B) of the theorem. Using the formulas of (A) and (B) that we have now established, we see

that T,I¥ = IVT, if and only if Iy = I, x and w being as in (A) and (B) of the theorem. By Theorem 3.5.3, this will be the case if and only if w = [X(u) and the interval int(u, x) is central. That will be so, according to 3.5.1, if and only if int(v, w) = int(u, x) and int(u, x) is central. But int(u, x) = n, since

x = T,(u), and int(v, w) = n“', since w = T_'(v). So, in sum, T, will commute with IY if and only if n = n7! and n is central. This proves (C).

54 q.e.d.

Generalized Interval Systems (2) 3.5.7 (C) of the theorem shows that, given any interval n in a general GIS, either T,, commutes with every inversion operation or T, commutes with no inversion operation. In the familiar GIS of twelve chromatic pitch-classes, T, commutes with every inversion operation: If you invert and then transpose by a tritone, the net result is the same as if you transpose by a tritone and then invert (about the same center or axis). In that GIS, no other interval of transposition has this property, save for the trivial interval of zero. (Tg is the identity operation.) In fact, no other T,, in that GIS will Commute with any inversion operation. (C) of the theorem shows us that this situation is related to the fact that6 + 6 = Omod 12, while n + n does not = 0 mod 12 for any other non-zero interval mod 12. Theorem 3.5.6 gave us insight into how inversions combine with transpositions. An analogous theorem will give us analogous insight into how inversions combine with the interval-preserving operations P.

IX, | 3.5.7 THEOREM: For any interval-preserving operation P and any inversion

(A): PI* = I” where w = P(v). (B): IXP = Ik where x = P“'(u). (C): P commutes with I\ if and only if P = T, for some transposition T, such that c is central and cc = e.

Proof (optional): We fix a referential element ref. Setting n= int(ref, P(ref)}) = LABEL(P(ref)), we write P = P, in the manner of 3.4.4 earlier. We can then manipulate pertinent LABELs to prove (A) and (B) exactly as we proved (A) and (B) for Theorem 3.5.6 above. To prove (C), we begin in a manner similar to that by which we proved 3.5.6 (C). Using (A) and (B), we note that PI¥ = I¥P if and only if IY = I, where w and x are as 1n (A) and (B). Via 3.5.3, this will be the case if and only if

int(v, w) = int(u, x) and int(u, x) is central.

Now LABEL(u) =j and, since x = P,!(u) = P,-:(u), LABEL(x) = n4-LABEL(u)=n74j (3.4.4). Therefore LABEL(x)"!LABEL(v) = (n7~!j)*j = j-' nj. And int(u, x) is precisely LABEL(x)~! LABEL(u) (3.1.2). So int(u, x) = j~'nj. We have now showed: P = P, commutes with I’ if and only if int(v, W) = int(u, x) = j- ‘nj and the interval j~'nj is central.

A little group theory provides the proof for the following lemma: The element j ‘nj of a group is central if and only if n is central. Thus either n is central, in which case j~/njis of course simply n; or else nis not central, in which

case j ‘nj is not central. We have now proved: P = P, and I’ commute if and only ifint(v, w) = int(u, x) = nand nis central. The rest follows from 3.4.8 and 3.5.6(C). q.e.d.

We have now seen how inversion operations combine with intervalpreserving operations, Earlier we saw how inversions combined with transpositions (3.5.6), and how transpositions commuted with inversion-preserving

operations (3.4.10). Earlier still, we noted that the transpositions formed a 55

3.5.8 Generalized Interval Systems (2) group of operations among themselves (3.4.2), combining according to a certain formula; the interval-preserving operations also formed a group among themselves (3.4.5) combining according to another formula. It remains to explore how the inversion operations combine with each other, and we proceed

to do so. 3.5.8 THEOREM: Fix ref, and let the LABELs of v, u, w, and x be respectively

i, j, k, and m. Then

ry = Pal ye Proof: Given any s, then 3.5.2 tells us that

LABEL(I*(s)) = i: LABEL(s)"!- j while LABEL(IM(s)) = k: LABEL(s)"!-m. Hence LABEL(I‘I“(s)) = i(LABEL(I%(s)))7'j = i(k -LABEL(s)~!+m)7}j = i(m~'LABEL(s)k~')j = (im7!')LABEL(s)(k7 }))

= LABEL(P;'T>'\s)) (3.4.3; 3.4.4). In sum, I‘I%(s) and P-'T>(s) have the same LABELs, and are therefore the same element of S. Since s was an arbitrary sample member of S, the functional equation of the theorem is true. q.e.d. 3.5.9 COROLLARY: I" is the inverse operation to I’.

Proof: Given u and v, take x = v and w = u in the formula of Theorem 3.5.8 above. Then m = 1 and k = j, as those intervals are defined in that theorem. So im! and k~'j are both equal to e; the formula of the theorem tells us in this special case that I‘ IY = P.T,, which is of course the identity operation. By the symmetry of the situation (reversing the roles of u and v), II’ is also the identity operation. q.e.d. 3.5.10 COROLLARIES: Let T and I be any transposition operation and any inversion operation in a commutative GIS. Then

(A); Ti =T and (B): IT = TUR. Proofs: (A) follows at once from 3.5.9 and the first remark of 3.5.5. To prove (B), set J = IT. Via 3.5.6(B) we know that J is an inversion operation. Then, invoking (A) just proved above, we infer that J = J~!. It

follows: IT = J = J"! = Ty"! = T"'I"! = T“'; the last step is again a consequence of (A) just proved. q.e.d.

We have now explored how various types of operations on the space of 56 an abstract GIS compose among themselves and with each other. Standing back

Generalized Interval Systems (2) 3.5.11 from the niceties of the specific formulas involved, we can get a useful global picture. 3.5.11 THEOREM: Let PETEY be the family of all operations on S that can be expressed as (functionally equal to) something of form PT, where P is some interval-preserving operation and T is some transposition. Let PETINV be the family PETEY plus the family INVS of all inversion operations. Then

(A) PETEY is a group of operations and (B) PETINV is also a group of operations. Proof (optional): We already know that PSVS, the family of mterval-

preserving operations, and TNSPS, the family of transpositions, are each groups of operations (3.4.5; 3.4.2). Let PT and P’T’ be any two members of PETEY. Set P” = PP’ and T” = TT’. Since PSVS and T'NSPS are groups, P” is interval-preserving and T” is a transposition; then P”T” is a member of PETEY as PETEY was defined. Furthermore, the composition of the given PT with the given P’T” is precisely P’T”, this member of PETEY. For (PT)(P’T’) = P(TP’)T’ = P(P’T)T’ (3.4.10) = P’T”. So PETEY is a closed family of transformations. To prove PETEY a group, it suffices via 1.3.4 to show that PETEY-asdefined contains the inverse of each of its member operations. Given PT in PETEY, then P™* and T™ are respectively members of PSVS and TNSPS, so that P~’T~ is a member of PETEY-as-defined. And P-!T“* is the inverse of

the given PT: PT = TP (3.4.10), so PT(P'T"*) = TP(P *T™') = 1 and (P-'T~')PT = (P''T~')TP = 1. So (A) of the theorem is proved. We use the criteria of 1.3.4 again to prove (B) of the theorem. We know that PETEY, since it is a group, contains the inverse of each of its members; we also know via 3.5.9 that INVS contains the inverse of each of its members. Hence PETINV, the set-theoretic union of the two families PETEY and INVS,

contains the inverse of each of its members. It remains only to prove that PETINV 1s a closed family of operations. Suppose that X and Y are members of PETINY; we have to show that XY 1s (Operationally equal to) a member of PETINV. We can distinguish four possible cases, which we take up one at a time below. Case 1: X and Y are both members of PETEY. Then XY, being a member of PETEY, is a member of PETINV. Case 2: X 1s a member of PETEY and Y is a member of INVS. Say X =

PT and Y = I. Now TI is some inversion-operation J (3.5.6(A)). And PJ is some inversion-operation K (3.5.7(A)). Then XY = PTI = PJ = K isa member of INVS, and therefore a member of PETINV as desired. Case 3: X is a member of INVS and Y is a member of PETEY. By an argument analogous to that of Case 2, now using 3.5.6(B) and 3.5.7(B), we conclude that XY is a member of INVS, and hence of PETINV as desired. Case 4: X and Y are both members of INVS. Then XY is a member of

PETEY (3.5.8). So XY is a member of PETINV as desired. q.e.d. 57

3.6.1 Generalized Interval Systems (2) We have seen that transpositions are naturally related in a number of ways to interval-preserving operations. We might conjecture that inversions should be naturally related to “‘interval-reversing” transformations, in the sense of the following definition. 3.6.1 DEFINITION: A transformation Y on the space S of a GIS will be called interval-reversing if

int(Y(s), Y(t)) = int(t, s)

for alls and all tin S. There is something to our conjecture above. Specifically, if the GIS is commutative, then the inversions are precisely the interval-reversing operations on S. But if the GIS is non-commutative, then there will not be any intervalreversing transformations at all! We shall now prove these facts, starting witha lemma. 3.6.2 LEMMA (optional): Let Y be an interval-reversing transformation; let ref

be fixed; then there is an interval i such that

LABEL(Y(t)) =i: (LABEL(t))“! for every tin S. Proof: int(Y(s), Y(t)) = int(t,s) by supposition. So

LABEL(Y (t)) ‘LABEL(Y(s)) = LABEL(s)~? LABEL(t) (3.1.2).

Take s=ref in the above equation, so that LABEL(s)=e. Set i= LABEL(Y (ref)) = LABEL(Y(s)) in the above equation. Then

LABEL(Y (t))* -i = LABEL(t). So LABEL(Y(t))? = LABEL(t)-i7* and LABEL(Y(t)) =i:LABEL(t)™, for any t. 3.6.3. THEOREM (optional): If IVLS is commutative, then the inversion operations reverse intervals, and every interval-reversing transformation is some inversion-operation. Proof: Fix some ref. The result of Lemma 3.6.2, in conjunction with the formula of Theorem 3.5.2, tells us that any interval-reversing transformation must be some inversion, specifically some I¥,,. It remains to show that any I} reverses intervals. Setting 1 = LABEL(v) and } = LABEL(u), we can write

int(1%(s), I¥(t)) = LABEL(I*(t))"* LABEL(I¥(s)) (3.1.2) = (iILABEL(t)7!j)7+ (G(LABEL(s)“*j) (3.5.2) = j- +LABEL(t)i7iLABEL(s)~*j

58 = LABEL(s)"'LABEL(t),

Generalized Interval Systems (2) 3.6.4 since IVLS is commutative! And that

=int(t,s) via3.1.2. q.e.d. 3.6.4 THEOREM (optional): [f IVLS 1s non-commutative, then there exists no interval-reversing transformation on S. Proof: We suppose that Y is an interval-reversing transformation and arrive at a contradiction.

For alls and ail t, LABEL(Y (t))~* LABEL(Y(s)) = LABEL(s)"! LABEL(t),

as in the proof for Lemma 3.6.2. Having the formula of that lemma at our disposal now, we can substitute for the LABELs of Y(s) and Y(t) in the above equation, using the special interval i of the lemma. We get the new equation

(i(LABEL(t))~')' -iLABEL(s)"' = LABEL(s)"*LABEL(t). Or: (LABEL(t)i“' )iLABEL(s)"' = LABEL(s)"'LABEL(t). Or: LABEL(t)LABEL(s)7! = LABEL(s)™! LABEL(t). But that equation, holding for a//s and t, says that IVLS is a commutative

group. And that contradicts the premise of the theorem. q.e.d.

59

4 Generalized Interval Systems

(3): A Non-Commutative GIS; Some Timbral GIS Models

During the discussion of transpositions, interval-preserving operations, and inversions, the reader may have been puzzled by the care with which noncommutative GIS structures were separated from commutative. After all, we have so far not encountered any specimen GIS which is non-commutative. Why should we be concerned at all about the non-commutative case? Why not save ourselves some trouble, and just stipulate in the definition of a GIS that the group IVLS should be commutative? The work of the present chapter will respond to these concerns by exploring a musically significant noncommutative GIS. I have already presented some of the work elsewhere, but it will have a special impact in the present context.’ 4.1.1 DEFINITION: By a time span, we will understand an ordered pair (a, x),

where a is any real number and x is any positive real number. The pair of numbers is understood to model our sense of location and extension about a musical event that “begins at time a” and “‘extends x units of time” thereafter. The family of all time spans will be denoted TMSPS.

We have already encountered one rhythmic GIS whose objects were certain time spans; that was example 3.3.2. There, we restricted the values for the numbers a to integers, and we restricted the values for the numbers x to

certain proportions. Using the same direct-product construction, we could also construct a GIS for time spans in which the number a could assume any rational value, and the number x any positive rational value. Using the direct1. David Lewin, “On Formal Intervals between Time-Spans,” Music Perception vol. 1,

60 no. 4 (Summer 1984), 414-23.

Generalized Interval Systems (3) 4.1.2

product construction, we can also construct a GIS for all time spans in the manner of 4.1.2 following.

4.1.2 Examp.e: Take S = TMSPS. Take IVLS to be the direct-product group of the real-numbers-under-addition by the positive-reals-undermultiplication. Define the function int, from S x Sinto IVLS, by the formula int((a, x), (b, y)) = (b — a, y/x).

Then (TMSPS,IVLS, int) is a GIS. It is commutative. The interval (b — a, y/x) measures our presumed sense that time span (b, y) begins b — a units after time span (a, x) and lasts y/x times as long.

This commutative GIS is useful and relatively simple, but it is not adequate as a model for the way we perceive time spans interacting under all circumstances. We shall now investigate why that is so. First we shall examine the family of time spans as a conceptual space independent of any particular

compositional context; then we shall examine how time spans behave in various specific musical contexts. To begin, then, let us focus on the time span (a, x) as a conceptual object

in a conceptual space, modeling our sense that something “begins at time point a” and “extends for x time-units”’ thereafter. We can ask, what is this absolute conceptual time-unit? In practice, we often proceed as if it were the minute. We do so, that is, when we write metronome marks which reduce various contextual units, in various pieces or passages of music, to fractions of a minute. The minute is not commensurate with our sense of a “‘beat,”” but we can use the second for that purpose if we wish, dividing all the metronome numbers by 60. Neither the minute nor the second, though, is very satisfactory as a would-be absolute conceptual time unit; both are derived from certain relative periodic motions of the earth, the sun, and the moon. Scientists today find these motions so erratic and irregular that they use other conceptual units of time for precise measurements. But even those units, deriving from certain sub-atomic motions, are clearly contextual. And that does not even begin to engage other technical problems involving Relativity and quantum mechanics in connection with such sub-atomic “‘fixed”’ units of time.

Jn short, if we declare any one time-unit to have absolute conceptual priority, that is a matter of computational convenience, or of scientific, sociological, or religious convention, rather then manifest musical reality. Abandoning this approach, we can make our absolute time-unit a matter of notational structure: We can call it “the brevis’ or “the perfection” or ‘‘the whole note” or “‘the notated beat,” for instance. But then we are throwing the whole problem back onto some notational convention that is highly restricted

socio-historically, a convention that indeed already presupposes a highly structured theory of measuring time by some pre-existing absolute unit. And

that will not help us in our inquiry. 61

4.1.2 Generalized Interval Systems (3)

We might try to assert that, though we cannot conceive an absolute time-

unit clearly, there will be some clear contextual time-unit, which we can identify and use for theoretical purposes, in any music that we may want to analyze. Such an assertion is reasonable in connection with a large body of music, and not only European music of the Classic-Romantic period. But the assertion is still not valid as a universal proposition about music, unless one is willing to restrict the use of the word “music” circularly, i.e. in precisely this

way. Making that restriction would involve at least a broad aesthetic contention. An even then, many critics who might feel no qualms about excluding as “music” say certain improvisations of John Coltrane, or Stockhausen’s Aus den sieben Tagen, would feel less comfortable excluding as music certain Tibetan chants, or sections of Elliott Carter’s String Quartet no. 1. Figure 4.1 (pp. 64-65) reproduces measures 22—35 of the Carter score. What could one assert as “the” (one clear contextual) time-unit for measures 22-327?

Figure 4.1 shows that our philosophical musings above do not simply come down to a matter of mensural versus non-mensural perceptions. Measures 22-32 of Carter’s piece have a very strong mensural character, despite our difficulty in pinning down “the” beat. The mensural profile of the passage

is especially—one might even say unusually—strong within each of the individual instrumental parts. For example, the first violin’s A in measure 26 and G in measure 28 each last precisely half as long as every other note in the first violin from measure 22 to measure 30; a player who does not hear this

mensural relation will not project the passage effectively. For another example, the B—F#-D—C#-D# of the cello at measure 32 and following is heard not only in syncopation against the otherwise regular half-note beat of measure 33 and following (half-note = MM90Q), but also as a rubato of the earlier cello melody C—G—E}—D-E, appearing eleven semitones lower in measures 2729. The earlier melody is presented in notes of constant duration whose beat, every five written eighths, is at MM48, not MM90. A cellist who does not hear the rhythmic relation of the transposed melodies will not project measures 32-35 completely effectively. In sum, the notion of “an” abstract conceptual time-unit, a unit by which we measure the number x of the formal time span (a, x), 1s a notion fraught with methodological problems. The number a of the time span (a, x), as well as the number x, is measured by our conceptual time-unit. For when we say that we perceive something that “begins at time-point a,” we mean implicitly that it 2. I am grateful to Jonathan W. Bernard for engaging my interest in this passage through his lecture, “The Evolution of Elliott Carter’s Rhythmic Practice,” delivered to the meeting at Yale of the Society for Music Theory on November 11, 1983. Bernard’s observations engaged many of the rhythmic relations I shall be discussing. The uses to which I shall put those observations,

62 as I expound my GlIS-theories, are of course my own responsibility.

Generalized Interval Systems (3) 4.1.2

begins just that number of conceptual time-units after some referential timepoint, “time-point zero.” Having noted this, we see that we must discuss not

only the conceptual time-unit in this connection, but also the conceptual “time-point zero.” What is this abstractly privileged moment that contributes toward measuring the number a of the time span (a, x)? Is it the moment of the Big Bang, or of the Biblical Creation? Is it a completely arbitrary moment very long ago? (And if so, why should we select an arbitrary moment to play a uniquely referential role?) Should we select “time-point zero” by a notational convention, e.g. as the first vertical line on the score of whatever piece we are

analyzing at the moment? Or should we presume to assert, explicitly or implicitly, that there must always be some one uniquely privileged moment, in the score or the performance of any passage we want to discuss, which we can

unequivocally label as a contextual zero time-point for the occasion? These methodological expedients involve difficulties similar to those discussed above in connection with the referential time-unit.

In one way at least, the choice of a zero time-point is less problematic than the choice of a temporal unit: The former choice does not affect the numbers attached to intervals in the GIS of 4.1.2, while the latter choice does affect those numbers. To see this, first suppose that we move our referential zero time-point back by m units into the past. Then the percept that was formerly manifest over the time span labeled (a, x) in the old scheme will now

be manifest over the time span labled (a + m, x) in the new scheme: What used to begin a units after (old) time-point zero will now begin a + m units after (new) time-point zero. Similarly, the time span labeled (b, y) in the old scheme will correspond to the time span labeled (b + m, y) in the new scheme. In the GIS of 4.1.2, the interval between the old labels is int((a, x), (b, y)) =

(b — a, y/x). In the same GIS, the interval between the new labels is int((a + m, x), (b + m, y)) = (b + m — (a + m), y/x) = (b — a, y/x). So, in transforming each old time span (a, x) to the new time span (a + m, x), we have not transformed the intervals involved: The interval between a pair of transformed spans is exactly the same as the interval between the corresponding pair of spans prior to transformation. That is, int((a + m, x), (b + m, y))

= int((a, x), (b, y)). Now let us suppose we keep the same referential time-point zero but change the unit of measurement so that what was x old units becomes xu new units, the factor u corresponding to the change of scale in measurement. Then percepts formerly corresponding to the time span (a, x) and (b, y) in the old

scheme will now correspond to the time spans (au, xu) and (bu, yu) in the new scheme: What used to begin a old units after time zero and extend x old units therefrom will now begin au new units after time zero and extend xu new units therefrom. We can see that this transformation does change the numbers

attached to intervals in GIS 4.1.2: int((a, x), (b, y)) = (b — a, y/x), while 63

4.1.2 Generalized Interval Systems (3)

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64

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Generalized Interval Systems (3) 4.1.2

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FIGURE 4.1 (continued) 65

4.1.2 Generalized Interval Systems (3)

int((au, xu), (bu, yu)) = (bu — au, yu/(xu)) = ((b — a)u, y/x). Intuitively, the

second percept we are discussing used to begin b — a units after the first percept; now it begins (b — a)u units after the first percept, the “unit’’ having changed. Of course the GIS of 4.1.2 knows nothing of “‘percepts”’ or “‘units”’; it simply knows that int((a, x), (b, y)) = (b — a, y/x) is not the same pair of numbers as ((b — a)u, y/x) = int((au, xu), (bu, yu)).

Lest the host of methodological problems under discussion appear insuperable, we should recall that we can finesse them all by restricting our attention to music in which we can identify and assert a referential time-unit and a referential zero time-point contextually. Then we can use the GIS of 4.1.2 without problems. There are plenty of pieces and passages for which we

can sensibly take this tack. On the other hand, there are also pieces and passages in which we cannot identify such referential entities contextually, music which we would agree nonetheless to consider highly structured mensurally, music within which it seems analytically valid—even necessary—to articulate time spans engaged in mensural interrelationships. We have already begun to explore the Carter example in this connection; we shall continue that analysis soon. Another example is provided by Stockhausen’s Klaviersttick XT. Stockhausen tells the pianist to look at the sheet of music and begin with any group of notes from among nineteen such groups dispersed over the score, “‘the first that catches his eye; this he plays, choosing for himself tempo ..., dynamic level and type of attack. At the end of the first group he reads the tempo, dynamic and attack indications that follow, and looks at random to any other group, which he then plays in accordance with the latter indications,” and so on and on. ““When a group is arrived at for the third time, one possible realization of the piece is completed.” * Each of the nineteen groups is notated quite traditionally as regards pitches and internal rhythmic proportions. But each group might be played at any of six tempi, ranging from very slow to very fast. Indeed, even during one performed realization, any group might be played two different times at two different tempi. The tempo of each performed group (after the first) depends on the instructions which appear at the end of the group just played, which might itself occur at any of the six tempi. In this context it makes no musical sense to speak of “‘the’’ referential

time-unit, beyond the interior of each performed group at its performed tempo. And yet, mensural relations among time spans from different groups

(especially consecutive groups) are highly audible, and therefore at least perceptually functional in any given realization. We have already mentioned, in footnote 3 of chapter 2, other examples of highly mensural music without a fixed time-unit. Nancarrow’s Studies for

Player Piano contain many pitch-canons involving elaborately changing 3. The cited text is from the Performing Directions by the composer on the score (Universal!

66 Edition no. 12654 LW, 3d ed., 1967).

Generalized Interval Systems (3) 4.1.2 tempo proportions.* Ligeti’s Poéme symphonique is performed by winding up

100 clockwork metronomes to varying degrees of tension, setting them at a variety of tempi, and releasing them, allowing them to run down over the course of eighteen to twenty minutes.° The effect includes an ironic poetic commentary, among other things, upon the very issue of the “‘contextual time-

unit.” To some extent in all this cited literature, and to a great extent in much of it, any time span has the potential for becoming a Jocal contextual timeunit, setting a local tempo. By “‘local,’’ I mean here not only over a certain temporally connected section of the total texture, but possibly also within a certain part, instrument, or instrumental group. For example, there is a more than clear mensural structure within the part played by any single metronome within Ligeti’s piece. For some less extreme examples let us return once more to figure 4.1, the Carter passage discussed before. The viola moves in notes of constant duration from its entrance at measure 25 up to the middle of measure 35; these local time-units “‘beat”’ the tempo of MM 180. The first violin “beats”’

constant local time-units at MM36 over measures 22—30, except for the A of measure 26 and the G of measure 28 which, as observed earlier, are each half the local time-unit in duration. At measure 33 the first violin starts to project a new constant local time-unit, that beats at MM90. The cello beats constant local time-units at MM120 over measures 22—26; then over measures 27—31 it

beats new constant local time-units at MM48. Finally at measure 32 and following, it stops moving in notes of constant duration as it plays a pitchvariation on measures 27ff., where MM48 began. This variation was discussed

earlier. The second violin beats its own constant local time-unit at MM96 from measure 22 through measure 26. Then over measures 27-30 it runs quickly through a number of local time-units at MM120, MM160 (m. 274), MM96 (m. 284), MM80 (to be discussed later), and MM60 (m. 30). Finally it

settles into a more stable local time unit in measure 31, beating at MM90. 4. Some of Nancarrow’s recent work involves irrational proportions like x. For the reader who may at first think such an idea 1s too bizarre to have any musical meaning, I append a brief exercise in conducting the tempo relation of 7. Imagine a horizontal line segment at chest height, in front of you and somewhat to your right. (I am supposing that you conduct right-handed.) Move your hand (arm) back and forth along the line at a constant speed, beating a horizontal 2 allegro vivo. Now imagine the line segment as a radius of a circle whose center is at the leftmost point of your beat. As you reach the rightmost point of your beat, start swinging your hand (arm) around and around the circumference of that circle counterclockwise, taking care always to keep your hand moving at the same constant speed. The amount of time it takes you to swing once around the circle is z times the duration of your earlier horizontal 2? measure. 5. The composer specifies very precisely how this is to be done. He also specifies that the piece is to be played after an intermission, so that the returning audience finds the metronomes already underway, with no persons on stage. The metronomes are to be arranged on risers, like a chorus; the slow beaters are at the lowest level and the fast beaters at the highest. I am indebted to Martin Bresnick for supplying me with this report, based on a personal communication from the

composer. 67

412 Generalized Interval Systems (3) Figure 4.2a collates and tabulates these various metronome marks, that reflect

the various tempi beat by the various local time-units in the individual instruments over the passage.

meas: 22 25 27 300 «31S 33 35

Vi. 1: 36 (72) |] 90 VI. 2: 96 //120/160/96/80/60// 90 Via. : 180

Ve. : 120 // 48 // (variable) FIGURE 4.2a

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a 90

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, 96 be. 60 90 VI. 2 Oo 160

120 96 80

re 180

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120

"48 ye be Ob FIGURE 4.2b

Figure 4.2b takes the numbers of figure 4.2a and represents them as pitches. This device will help clarify for musicians the numerical ratios 1nvolved among those numbers. The number 180, which labels the fastest beat on figure 4.2a, is represented on figure 4.2b by the highest pitch, high C. Slower tempi, in their numerical ratios to MM180, are represented on figure 4.2b by lower pitches in the corresponding frequency ratios to high C. For instance, on figure 4.2a the opening tempo of the cello is MM120, 2/3 of the 68 tempo MM180 coming up in the viola. On figure 4.2b the ttmpo MM120 is

Generalized Interval Systems (3) 4.1.2 represented by the opening pitch FS in the cello, a pitch whose fundamental is

2/3 the frequency of the high C coming up in the viola, the high C which represents the tempo MM180. The euphony of figure 4.2 is striking. It makes very clear the network of “numerical consonances’’ displayed by the tempo relations of figure 4.2a, which are also the proportional relations of the various local contextual timeunits. If we imagine a quartet actually playing figure 4.2b, we can get an even clearer idea of how this numerical network affects interactions among the performers. Here are some, and only some, of the relations the players will heed. (1) The 48 of the cello at measure 27 will lie a good numerical octave below the opening 96 of the second violin. (2) The 120 of the second violin at measure 27 will match the opening 120 of the cello. (3) The 120-to-160 relation in the second violin during measure 27 will match in its ratio the 36-to-48 relation between the first violin and the cello thereabouts. (4) The 80-to-60 relation in the second violin over measures 29-30 will retrograde, an octave lower, the 120-to-160 relation discussed in (3) above. (5) The 60-to-90 relation in the second violin, measures 30—31, will match an octave lower the earlier 120-to-180 relation between the cello at the opening of the passage and the viola entrance. (6) The 90 of the first violin at measure 33 will match the 90 of the second violin at measure 31, which in turn will match, an octave lower, the persistent 180 of the viola. (7) The 36-to-90 profile of the first violin part as a whole will match in transposed retrograde (or inversion) the 120-to-48 profile of the cello part up to measure 33. Exactly these numerical relations, and others of the same sort, must be projected to make the rhythmic structure of Carter’s passage come to life and communicate, not only between the players and their listeners, but also among the four players themselves. We shall discuss the above seven performance notes some more later on. On figure 4.2b, it is curtous how the Dp4 of the cello can be heard as a root whose major harmony is elaborated by the symbolic pitches of the figure over measures 22—30, along with a major seventh and an added sixth. To be

sure, it is doubtful that our tonal perceptions of roots, triads, harmonic sevenths, and added sixths can be used to assert analogous functions in the realm of tempo relations. And yet there is a certain suggestiveness in the idea that the cello part of measure 27 and following has a rhythmically “‘grounding’ function somehow analogous to the tonal root function of the symbolic Dp4 on figure 4.2b. This suggestion is useful for the cellist who wants the M M48 tempo at measure 27 and following to feel stable and referential, rather than syncopated and intrusive. The suggestion is also useful for understanding why just this tempo of MM48 (symbolized by the pitch Dp4 on figure 4.2b) is permitted to launch into a wide and free rubato at measure 32 and following

in the cello (symbolized by the “‘cadenza”’ on figure 4.2b), just at the time the 69

4.1.2 Generalized Interval Systems (3) other three instruments are all finally agreeing upon the new referential tempo MM90-or-180 (symbolized by the prominent pitch class C at m. 32ff. on figure 4.2b). Over the second-violin part of figure 4.2b as a whole, D} moves clearly to C. But the cello is still refusing to change its earlier D} for C as we leave the passage, though its D> has been abandoned. All of these remarks amplify the idea suggested above, that the cellist should refer the rhythm of the part over measures 32-35 not just to the beats of the upper instruments at MM90 but also to the earlier beat of the cello itself, MM48 at measure 27 and following. Figure 4.2b shows by analogy how the symbolic pitches of the “cadenza” in the cello at measures 32—35 are heard not only in relation to the Cs of the upper instruments there, but also in relation to the earlier D} of the cello itself at measure 27 following. However interesting it may be to think of MM48 as a numerical “ground”

organizing the tempi of measures 22-31, it is still clear that the players of the three upper instruments will not treat the cello part of measures 27-31 as a succession of referential time-units. That is, they will not adjust their own beats to conform in proper proportion with the cello beat of measures

27-31. Nor will the cello and the two violins treat the constant beat at MM180 in the viola from measure 25 on as a succession of referential timeunits for the entire passage; at least they will not do so until after measure 30. During measures 22—24, in particular, the MM36, MM96, and MM120 of the two violins and the cello will not be referred to a beat of MM180 in the viola, for the viola has not yet entered there. Once the viola is in, at measure 25 and thereafter, its constant MM180 will provide a useful check for the ensemble rhythm without necessarily establishing itself as referential, just as the sustained high C on figure 4.2b provides a useful check for the ensemble’s intonation without necessarily establishing itself as a root. Of course MM180 does eventually become much more referential during measures 33-35, along with its lower octave MM90; just so, on figure 4.2b, the pitch class C comes to dominate the tonal texture.

The players may decide on purely notational grounds to use MM120 (changing to MM180 by measure 33) or MM60 (changing to MM90 by mea-

sure 33) as a referential tempo for the entire passage. This makes a certain practical sense for early rehearsals, but it can hardly be recommended for an effective performance. We have all heard and seen players fighting their way through slow lyric lines, supposedly tranguillo like that of the first violin in measures 22—32 or sostenuto e cantabile like that of the cello in measures 27-32, all the while jerking their feet up and down spastically in an erratic approximation of some distantly related notational “beat.” These lyric lines are not syncopated, as such a method of production makes them sound to both player and listener. Rather, each line has its own autonomous local time-

unit, with respect to which it should project an essentially “first-species”

70 character.

Generalized Interval Systems (3) 4.1.2 Indeed, the search for any one overriding referential time-unit, to govern all of measures 22-31, is bound to fail. It must fail because it misconceives the

nature of the temporal space at hand. That space comprises a multitude of locally referential time-units, in various more-or-less consonant numerical relationships among themselves. Only by recognizing that character of the space can one hear the music progress over the passage, and not just “be.” What I mean will be amplified by the following review of the earlier performance notes, (1) through (7). (1) The MM48 of the cello at measure 27 will lie a good (numerical) octave below the opening MM96 of the second violin. That is, at measure 27 the cellist should hear the cello line moving at half the rate the second violin has been moving so far. This would be easy to hear if the cello came in a sixteenth-note later. In fact there is no problem within measure 27, getting the attack of the pitch C2 at the right time-point: The player need only continue beating MM120 up to that point. But once the new melody has entered, its tempo may sound arbitrary and its character “syncopated” unless the player hears the melody taking over, albeit out of phase, from the preceding tempo of the second violin, projecting its own local referential time-unit.

(2) The MM120 of the second violin at measure 27 will match the opening MM120 of the cello. MM120 1s the notated pulse. Still, the second violin should not simply be playing that pulse (with spastic foot-tapping or the like). What makes measure 27 come alive and communicate in this connection is an exchange of local referential beats, between the two mstruments. The second violin takes over the preceding beat of the cello, while the cello—as we

noted in (1) above—is about to take over the preceding local beat of the second violin, a rhythmic octave lower (and out of phase). The cellist and the second violinist should hear this ““voice-exchange with octave transfer,” to

hear themselves conversing with each other in a quite familiar tradition of chamber music. Figure 4.2b, showing the exchange of the symbolic pitch classes Dp and F in the two instruments at measure 27, serves as a guide to that tradition here. Just as the players would match those pitch classes if playing figure 4.2b, so they should match their exchange of tempi at measure 27 when playing the Carter passage. (3) The MM120-to-160 relation in the second violin, later in measure 27, will match the MM 36-to-48 relation between first violin and cello hereabouts. Assuming that everything else has gone right so far,this will happen automatically if the second violin plays the dotted eighth (beating MM160) as a precise 3/4 of the quarter note (beating MM120). That is not so easy to do as it is to say, but the ability should be available to a well-trained player of twentiethcentury music. Here, the two quarter notes of the second violin in measure 27 will function as locally referential timespans for the player. (4) The MM80-to-60 relation in the second violin, measures 29—30, will

retrograde the MM120-to-160 relation of (3) above, one rhythmic octave 71

4.].2 Generalized Interval Systems (3) lower. Here “MM80” is projected only by the Ep-triad-event in measure 29 of the score; the tempo is not beat by any recurrent durational unit. Nevertheless a conceptual MM80 is useful to the player in a manner indicated by figure 4.3.

(96) 120 160 96 80 60

LP MOLT SLIT Su ee

FIGURE 4.3

The figure transcribes the rhythm of the second violin from the middle of measure 26 through measure 30 into a new notation, using MM160 as a new

notational tempo of reference, that is, the tempo at which the new notated quarter beats. This transcription brings out clearly how the rhythmic peripatetics of the second violin are structured by the indicated relationship, that is MM80-to-60 answering MM120-to-160. The transcription “modulates” our

rhythmic hearing exactly as the second through sixth notes of the second violin on figure 4.26 would sound “‘modulated”’ if we listened to them in Bb

minor, rather than Db major. Just as the Fs of figure 4.2b would sound primarily as fourths below the adjacent locally referential Bhs, in our modulated pitch-hearing, just so the tempi of MM120 and MM60 on figure 4.3 sound in rhythmic proportions 3 : 4 and 4: 3 to the adjacent locally referential tempi of MM160 and MM80, that hie alongside them. Figure 4.3 demonstrates a logical internal structure for the second-violin passage as a rhythmic entity in itself; this structure will surely not emerge if the player adjusts each individual tempo of figure 4.3 only to the beats of the viola at MM180 hereabouts, or to the notated beat of the score at MM120. It 1s in order to bring out the quasi-palindromic structure of figure 4.3 that the “tempo” of MM80 is represented, exceptionally, by only one time span. (5) The MM60-to-90 relation in the second violin, measures 30-31, will match, a rhythmic octave lower, the earlier MM120-to-180 relation between the cello at the opening of the passage and the viola entrance. This needs no further discussion; the relation of the relations will emerge without special attention if the players are otherwise temporally “in tune.” (6) The MM90 of the first violin at measure 33 will match the MM90 of the second violin at measure 31. Obviously. Here MM 90 is a referential local time-unit. Likewise the MM90 of the second violin at measure 31 will have matched the MM 180 of the viola so far, using MM180 as a referential time unit. (7) The MM36-to-90 profile of the first-violin part as a whole over the 72 passage will match in transposed retrograde (or inversion) the MM120-to-48

Generalized Interval Systems (3) 4.1.2 of the cello part as a whole up to measure 33. [tis much harder to hear the large rhythmic proportion here than it is to hear the corresponding symbolic pitch

proportion on figure 4.2b. Nevertheless, it will aid communication between cello and first violin, as well as projection between ensemble and audience, if the first violinist hears the MM90 entrance at measure 33 speeding up the earlier MM36 of the instrument in exactly the same ratio as the MM48 of the cello at measure 27 slowed down that instrument’s earlier MM120. The proportion can be sensed when the two instruments rehearse the pertinent music by themselves. Via this proportion, the MM90 of the first violin at measure 33 engages and completes a large mensural structure functioning over measures 22—35; it is not simply a surrender of the first violin passively to the beats of the second violin and viola at measure 31 and following. Of course the first violin will use those beats to find its new tempo at measure 33.

To sum up: When performers confront the score of the Carter passage and the numerical network of local tempi or time-units displayed in figure 4.2a, they should not concern themselves with the question, ‘““Which one of these is the overall unifying referential tempo?” That question, a rhythmic analog to the sorts of questions asked about pitch structures by Rameau, Riemann, and Hindemith among others, has no definite answer here. Even if we try to force an answer by selecting MM48, or MM180, or MM60-then-90 as a “‘root’”’ tempo on the basis of this or that criterion, we shall still not be engaging thereby the temporal relationships that make this music progress and communicate. Those temporal! relationships, some of which were discussed in performance notes (1)—(7) above, involve patterns of local tempo “‘consonances,” patterns in which many different tempi can assume locally referential

roles. This attitude toward the numerical network of figure 4.2a, and the symbolic pitch network of figure 4.2b, is more in the spirit of Zarlino: It asks

not for one overriding referential unity, but rather for a splendid variety of consonant ratios among the entities involved, as they underlay and succeed one another, projecting a logical compositional idea. In this way of hearing the rhythmic space through which the passage moves, any time span has the potential for becoming locally referential, or behaving as if it were. For example, let time span r be the span covered by the first note of the cello in measure 22 of the score; let s be that time span covered by the F¢ of the first violin in measures 25—26; let t be that time span covered by the A of the first violin in measure 26. We can say if we wish that t begins 164 r-spans after the beginning of r, and lasts 14 times the duration of r. But this way of listening corresponds to the “‘foot-tapping”’ performance of the first violin’s melody. We can also say that t begins | s-span after the beginning of s, and lasts I/2 the duration of s. And that way of listening corresponds to a much more musical shaping of the melody. Taking s’ as the time span covered by the opening D of the first violin in measures 22—23, we could also say that t begins 4 s’-spans after the beginning of's’, and lasts 1/2 the duration of s’. This 73

4.1.2 Generalized Interval Systems (3) way of listening corresponds to an even more musical shaping of the melody.

Adopting the above attitude to Carter’s rhythmic space, we implicitly deny the relevance of GIS 4.1.2 as a model. Given abstract time spans s and t, we want to be able to conceive t as beginning a certain number of s-beats after s, rather than a certain number of possibly irrelevant “referential units” after s. If s is the span (a, x) and t is the span (b, y), the old GIS of 4.1.2 assigned int(s, t) = (b — a, y/x): t begins b — a referential units after s, and lasts y/x times as long. We want to replace that old notion of time-span interval by a new function: int(s, t) = ((b — a)/x, y/x). The new interval tells us that t begins

S; t,

(b — a)/x x-lengths (s-beats) after s, and lasts y/x times as long. The new

May | Ib, | Xy | y1 interval uses s itself as a measuring rod, to tell us how much later t begins.

Vv \ \ \/ ‘

fi \O ~~, ran oy ra NO aaa DA NR NY |f2 X2 ¥2 82jbo ts FIGURE 4.4

Figure 4.4 shows how our new “interval” works. On the figure, four numerical time-spans are indicated: s,; = (a,,X,), t; = (0,,Y,), 85 = (a5,X>), and t, = (b,, y,). We shall see later that 1t does not matter at all, for our new model, what the formal numerical time-point zero is, or what the formal numerical time-unit is. That is, it does not matter to what percept we attach the numerical time-span label (0, 1). On the figure, we can imagine an “upper instrument”’ projecting s, and t, at a slow tempo, and a “lower instrument”’ projecting s, and t, at a fast tempo. The dotted slurs arching above the upper instrument mark off x,-lengths, durations that mark a contextual (potential)

$,-beat. The dotted slurs arching below the lower instrument mark off x,lengths, durations that mark a contextual (potential) s,-beat. Using our new interval construct, we write int(s,,t,) = (4,2): t, begins 4 s,-beats after s,, 74 ~~ and lasts twice as long. Arithmetically, (b, — a,)/x, = 4andy,/x, = 2. Using

Generalized Interval Systems (3} 4.1.3.2

the new interval construct, we also write int(s,,t,) = (4,2): t, begins 4 s,beats after s,, and lasts twice as long. Arithmetically, (b, — a,)/x, = 4 and Y¥2/X2 = 2.

Note that our new “interval from s, to t,” is the same as our new “interval from s, to t,’’: int(s,,t,) = int(s,,t,) = (4,2). We shall discuss the implications of this a lot more later on. Note particularly that s, precedes s, on figure 4.4 in the obvious sense, while t,, the (4, 2)-transpose of s,, follows t,, the (4, 2)-transpose of s,. One sees that our intuitions about formal “‘transposition”’ will not be completely reliable in our new non-commutative GIS. (Our intuitions about interval-preserving operations will be trustworthy.) We are of course still far from having constructed a formal GIS in which our new notion of “interval” is to work. It is high time to do so now. 4.1.3.1. Lemma: Let IVLS be the family of pairs (1, p), where 1 is a real num-

ber and p is a positive real number. Then IVLS forms a group under the composition (i, P)(j,q) = G@ + pj, pq).

In this group, the identity is (0, 1) and the inverse of the element (1, p) is the element (—1/p, 1/p). The group 1s non-commutative.

The proof of the Lemma will be left as an exercise for the interested reader. Do not forget to show that the defined composition is associative: ((i, p) GJ, 4)){K, r) = Gi, p) (GQ. q) (k, £)).

4.1.3.2. THEOREM: Let int be the function that maps TMSPS x TMSPS into the group IVLS of Lemma 4.1.3.1 according to the formula int((a, x), (b, y)) = ((b — a)/x, y/x).

Then (TMSPS, IVLS, int) is a GIS. Proof (optional): We must show that Conditions (A) and (B) of Definition 2.3.1 obtain.

(A): Given time spans (a,x), (b,y), and (c,z), we are to show that int((a, x), (b, y))int((b, y), (c, z)) = int((a, x), (c,z)). We write int((a, x), (b, y))int((b, y), (c, z)) = ((b — a)/x, y/x)((c — b)/y, z/y) by the formula defining int in the theorem. This = (((b — a)/x) + (y/x)(e — b)/y, (y/x) (Z/y)) by the group composition in IVLS. Canceling factors of y in the

numerators and denominators, we see that this = ((b —a +c — b)/x, z/x) which = ((c — a)/x, z/x).

And that number pair its indeed int((a, x), (c, z)). 75

(B): Given the time span s = (a, x) and the interval (i, p), we are to find a unique time span t = (b, y) which lies the interval (i, p) from the time span s = (a,x). If any such b and y exist, they must satisfy the relation

int((a, x), (b, y))} = (i, p), Or

((b — a)/x, y/x) = (i, p), or

(b—a)/x=i and y/x=p, or

b=ix+a and y= xp.

So there can be at most one time span t in the desired relation to s and (i,p): That is the time span t = (b, y) = (x + a,xp). And in fact this particular t 7s in the desired relation: int(s, t) = int((a, x), (ix + a, xp)); this = (((ix + a) — a)/x, xp/x) by the formula defining int; that = (1x/x, p), which is

indeed (i, p) as desired. .e.d. At long last, we have before us a non-commutative GIS of musical interest. The GIS has important formal properties, which we shall now study.

4.1.4 THeorem: GIS 4.1.3 has properties (A) and (B) below. (A): For any real number h, the interval from time span (a + h, x) to time span (b + h, y) is the same as the interval from (a, x) to (b, y). (B): For any positive real number u, the interval from time span (au, xu) to time span (bu, yu) is the same as the interval from (a, x) to (b, y). Proof: (A): int((a + h,x), (b + h, y))

= (((b + h) — (a + h))/x, y/x) (4.1.3.2)

= ((b — a)/x, y/x) (algebra) = int((a, x), (b, y)) (4.1.3.2). (B): int((au, xu), (bu, yu))

= ((bu — au)/xu, yu/xu) (4.1.3.2)

= ((b — a)/x, y/x) (algebra) = int((a, x), (b, y)) (4.1.3.2). q.e.d.

Some commentary on this theorem is in order. The time spans (a, x), (b, y), and so on still rely numerically on an implied referential time-unit and an implied time point zero: (a, x) begins the number a of referential units after the referential zero time-point, and lasts the number x of referential units. The essence of Properties (A) and (B) in the theorem above is that the numerical function int for the GIS under present discussion does not depend at all on the choice of time point zero, or on the choice of referential time-unit.

To see this, suppose first that we move the referential zero time-point back h units into the past (= forward (—h) units into the future). An event originally associated with the time span (a, x) will now be associated with the

76 ‘time span (a + h,x): The event will begin a + h units later than the new zero

Generalized Interval Systems (3) 4.1.5

time-point. Similarly, another event originally associated with the time span (b, y) will now become associated with the time span (b + h, y). Property (A) of the theorem says that in GIS 4.1.3, the formal interval between the time spans associated with the two events is not affected by this transformation. Even though the time spans themselves, as number-pairs, change from (a, x) to (a + h, x) and so on, the interval between transformed spans is the same as the interval between the original spans. Now suppose we change the referential unit of numerical time, so that the old unit is u times the new unit. A duration of x old units is then the same as a duration of xu new units. And the number a of old units after time-point zero is the same as the number au of new units after time-point zero. Hence the events that were associated with time spans (a, x) and (b, y) 1n the old system will be associated with time spans (au, xu) and (bu, yu) in the new system. Property (B) of the theorem says that in GIS 4.1.3, the formal interval between the time spans associated with the two events is not affected by this transformation. Thus, in the GIS of 4.1.3 the function int(s, t) will always deliver one and the same pair of numbers (1, p), no matter what the referential time-unit and the referential zero time-point by which we calculate numerical durations and distances from time-point zero. To put this intuitively: Given event 1 and event 2 in a piece, we can play the music whenever we want and at any tempo we want, without affecting at all the pair of numbers (i, p) which GIS 4.1.3 will

deliver to us as the formal interval between the numerical time spans associated with the two events for any particular analysis.

The same can not be said for the commutative GIS of 4.1.2, studied earlier. In that GIS the interval between time spans (a,x) and (b,y) was (b — a, y/x); accordingly, if we replace the referential time-unit so that events once associated with those time spans now become associated with the new spans (au, xu) and (bu, yu), then the interval between the new spans is different. It is not (b — a, y/x), but rather (bu — au, y/x). We noted this earlier. In our present terminology, we can say that GIS 4.1.2 does not enjoy Property (B) of Theorem 4.1.4. In fact, a remarkable theorem is true. Not only does GIS 4.1.3 enjoy the two Properties of Theorem 4.1.4, but it is also essentially the only possible GIS involving time spans as objects that enjoys those two Properties. The meaning of the word “essentially” in the above sentence is made clear by Theorem 4.1.5 following.

4.1.5 THeorem: Let GIS’ = (TMSPS,IVLS’, int’) be any GIS with time spans for its objects that also enjoys Properties (A) and (B) of Theorem 4.1.4. Then the group IVLS of GIS 4.1.3 and the group IVLS’ of the given GIS’ are isomorphic via a map f such that, for all time spans s and t,

int'(s, t) = f(int(s, t)). 77

4.1.6.1 Generalized Interval Systems (3) Some commentary is in order before we launch into a proof. The idea of isomorphism between (semi)groups was discussed in 1.11.1 and 1.11.2 earlier. To review here: If G and G’ are abstract groups, a function f from G into G’ is ‘an isomorphism of G with G’”’ if fis 1-to-1, onto, anda homomorphism. fisa homomorphism if f(mn) = f(m)f(n) for every m and every n in G. Supposing f an isomorphism of the abstract groups G and G’, then the two abstract groups will have exactly the same algebraic structure under the identification of m in G with its image f(m) in G’. So the first thing Theorem 4.1.5 saysis that IVLS and IVLS’ have essentially the same algebraic structure, when we identify the member m of IVLS with its image f(m) in IVLS’. Second, Theorem 4.1.5 says that if we take the member m of IVLS to be int(s, t) in particular, then the function f whose existence is asserted makes the image f(m), a member of IVLS’, equal precisely to int’(s, t). Thus the function int’ is,

so to speak, naught but the isomorphic image of the function int under the isomorphism f whose existence the theorem asserts. In this sense, the given GIS’ is “‘essentially” the same as GIS 4.1.3. The (optional) proof of Theorem 4.1.5 1s lengthy. To help break it into manageable sections, we shall prove two lemmas. The lemmas appear below as 4.1.6.1 and 4.1.6.2. After that, we shall go on to the proof of the Theorem proper.

4.1.6.1 Lemma (optional): Let G and G’ be abstract groups. Let f be a function from G into G’ such that for all m and all nin G, f(m)"'f(n) = f(m™'n). Then fis a homomorphism. Proof of (optional) Lemma; Set m = n = e in the given formula; we get f(e)~*f(e) = f(e). It follows that f(e)~' is the identity in G’; hence f(e) is the identity in G’, e’.

Now set n =e and let m vary in the formula of the Lemma. We get f(m)~1f(e) = f(m™). Since f(e) = e’, we have f(m)™* = f(m™) for all m. Then we can rewrite the formula of the Lemma as

f(m~!)f(n) = f(m7'n) for all m and all n. Asm runs through the various members of G, m™ = oruns through the various members of G. Substitute o for m~ in the rewritten formula; we then obtain the formula

f(o)f(n) =f(on) for allo and all ninG. And thus f is a homomorphism, as claimed.

4.1.6.2 Lemma (optional): Within the group IVLS of 4.1.3.1, (i, p)*(j, 4) = (G — D/p, q/p).

78 Proof of (optional) Lemma: We already verified in 4.1.3.1 that @, p)~ in

Generalized Interval Systems (3) 4.1.6.2 this group was the element (—i/p, 1/p). Then (i, p)""G, q) = (—i/p, 1/p)G,q) = ((—i/p) + (1/p)i, (1/p)q) = (G — d/p. a/p) as asserted. Now we are ready for the (optional) Proof of Theorem 4.1.5. We take the time span (0, 1) as a referential object within the space of GIS’ for purposes of LABELing. That is, we set ref’ = (0, 1). Then the function f for which we are looking, the isomorphism of IVLS with IVLS’, is defined by formula (1) below.

(i) £G,p) = LABEL'(, p) = int’((O, 1), G, p)).

On the left of formula (i) the number-pair (i, p) is considered as an interval, a member of IVLS, while in the middle and on the right of the formula, the same number-pair is considered as a time span, a span being LABELed in GIS’ by its GIS’-interval from the referential object ref’ = (0, 1). The number-pair (i, p), as a pair of numbers, can be interpreted either way. Now we can write f(i, p)"'f(j, q)

= LABEL’(i, p)”'LABEL’(j,q), by formula (i). This = int’((i,p),Qj,q)) by 3.1.2. This = int’((0, p), G — 1,q)), since GIS’ enjoys Property (A) of Theorem 4.1.4 by supposition. And this = int’((0, 1), ((j — 1)/p,q/p)) since GIS’ enjoys Property (B) of Theorem 4.1.4 by supposition. And that

= f((j —1)/p.q/p) by formula (i) above. And that = f((i,p)~*(j,q)) by Lemma 4.1.6.2. Putting together the whole string of equalities we have just noted, substituting m for (i, p) and n for (j,q), we see that we have proved f(m)~' f(n) = f(m™' n)

for every m and every nin IVLS. By Lemma 4.1.6.1, we conclude that f is a

homomorphism. Since the LABEL’ function is 1-to-1 from TMSPS onto IVLS’ (3.1.2), the function f is 1-to-1 from IVLS onto IVLS’. Thus f is an isomorphism of IVLS with IVLS’.

It remains to prove that f(int(s,t)) = int’(s,t). Set s = (a,x) and t= (b, y). Then int(s, t) = ((b — a)/x, y/x) and

F(int(s, t)) = int’((0, 1), ((b — a)/x, y/x)) (formula (i) = int’((0,x),(b — a,y)) (since GIS’ enjoys Property (B)) = int’((a,x),(b, y)) (since GIS’ enjoys Property (A))

= int’(s,t). q.e.d. To recapitulate: Theorem 4.1.5 shows that GIS 4.1.3 is essentially the only possible GIS involving the family TMSPS whose function int is com-

pletely independent of the referential time-unit and referential zero time- 79

4.1.7 Generalized Interval Systems (3)

point. To put it more intuitively, GIS 4.1.3 is the only such GIS, essentially, that will return one and the same element of IVLS as the interval between the numerical time spans associated with two musical events in a piece, regardless of when you play the piece and what tempo you take. GIS 4.1.3 thereby has a privileged theoretical status, as well as a special plausibility for modeling events in the Carter passage and other pertinent music. Since GIS 4.1.3 is non-commutative, it will provide a useful example for illustrating and reviewing the work of sections 3.4 and 3.5 earlier, work that formulated the abstract theory of transpositions, interval-preserving operations, and inversions.

4.1.7 Notes: Within GIS 4.1.3, the following formulas and facts are true: (A): Given an interval (1, p) and a time span (a, x), the transposition of the given time span by the given interval is TG, py (as X) = (a + Ix, px).

(B): If we fix (0,1) as ref, a referential time span, then the number-pair (a,x), as a member of IVLS, is the LABEL for the time span (a, x): LABEL(a, x) = int((0, 1), (a, x)) = (a, x). (C): The (i, p)-transpose of the time span (a, x) is the number-pair given by the composition in IVLS of the two intervals (a, x) and (i, p). Ty, »)(a, x) = (a, x)(i, p).

(D): Using the number-pair (a, x) in the same way, as both a time span and an interval, we can show that the interval-preserving operation Py y) transforms the time span (a, x) into the time span Pon,w) (a x) — (h, u) (a, x) — (h + ua, ux).

(E): The only central member of IVLS is the identity interval (0, 1).

(F): No transposition preserves intervals, and no interval-preserving operation is a transposition, the identity operation Tio 1, = Pio, 1, excepted.

(G): The operation of (c,z)/(d, w) inversion, apphed to the time span (a, x), yields the time span

[o)(a, x) = (d + (¢ — a)w/x, zw/x) = (d, w) (a, x) *(c, 2).

(H): Given time spans s, t, s’, and t’, then I. = I! as an operation on TMSPS if and only if s’ = s and t' = t. (1): There are no interval-reversing operations on TMSPS. Proofs and commentary: (A): Via 3.4.1, the transposition of (a, x) by (i, p) is that time span (b, y) which lies the interval (1, p) from the time span (a, x), 1.e. which satisfies the equation int((a, x), (b, y)) = (i, p). Thus (b, y) satisfies the

equation ((b — a)/x, y/x) = (i, p); whence (b — a)/x =i and y/x = p. So b=

80 a+ix and y = px. The transposed time span (b,y) = (a + ix, px) can be

Generalized Interval Systems (3) 4.1.7 described as follows: b lies 1 x-spans later than a; y lasts p times as long as x. If we turn back to figure 4.4 (p. 74), we will see that the time span t, of the figure is the (4, 2)-transpose of s, : t, begins at b,, 4x,-spans after a,; t, lasts a duration of y,, 2 times the duration x, of s,. Likewise, t, on the figure is the

(4,2)-transpose of s,. We noted while studying the figure earlier that s, precedes s,, while t, follows t,. We may say that transposition operations, in this GIS, do not only fail to preserve intervals, they even fail to preserve chronology.

(B) of the Notes is a straightforward computation: LABEL(a, x) = int((0, 1), (a,x) = ((a — 0)/1, 1x) = (a,x). (C) of the Notes applies (B) to the formula of 3.4.3, and (D) applies (B) to the formula of 3.4.4. The interval-preserving operation Pq y, first blows up or

shrinks the sample time span (a,x) by a factor of u, transforming (a, x) to (ua, ux), and then moves the latter time span backward or forward in time by h or (—h) numerical units, transforming (ua, ux) to (h + ua, ux) = P(a, x). Re-

member that these interval-preserving operations are not formal “‘transpositions” in our non-commutative system! (E) of the Notes is proved as follows. Suppose the interval (1, p) is central

in IVLS, that is (i, p)(j,q) = Gq), p) for all (j,q). Expanding the binary composition on each side of that equation, we infer (i + pj, pq) = (j + qi, qp) for all j and all positive q. Then i + pj =] + qi for all such j and q, whence (p — 1)j = (q — 1)ifor all such j and q. Takej = 1 andgq = 1 as one suchjand q; then (p — 1)l = (1 — 1)1 or p — | = 0; we infer that p must be equal to I. Now we can go back to our general equation, (p — 1)j = (q — 1)1; substituting p = 1, weinfer that (q — 1)i = 0 forall positive q. But then 11s obviously zero. Sop = | andi = 0; our given central interval (i, p) must be the identity interval (0, 1).

(F) of the Notes then follows from Theorem 3.4.8. (G) of the Notes can be computed from 3.5.2 together with (B) of the Notes. Or it can be computed directly from the defining formula of 3.5.1, using the known group structure of IVLS here.

(H) follows from 3.5.3, where we proved that It. = I! if and only if t’ = I§(s’) and the interval int(s’,s) is central. Via (E) above, this will happen ifand onlyift’ = I,(s’) ands’ = s, Since I{(s) = t, this will happen if and only if

s’=sandt' =t.

(I) of the Notes simply restates 3.6.4 in the present context. We may use figure 4.4 yet once again to picture the effect of an inversion.

On that figure, we noted that int(s,,t,) = int(s,,t,). Hence, via Definition 3.5.1, tz is the t,/s, inversion of s,; that is, I{?(s,) = to.

structures. 8] This concludes our study of a non-commutative GIS which is also a rhythmic GIS of musical interest. We shall now study some timbral GIS

4.2.1 Generalized Interval Systems (3) 4.2.1 EXAMPLE: Given positive numbers s(1), s(3), and s(5), let the numbertriple s = (s(1), s(3), s(5)) denote the class of all harmonic steady-state sounds (i.e. periodic wave-forms) whose first, third, and fifth partials have respective power s(1), s(3), and s(5). Let S be the family of all such number-triples s, as s(1), s(3), and s(5) range over all positive values. Given positive numbers i(1), (3), and i(5), let us imagine at hand one or more ‘‘devices”’ (e.g. computer procedures) which have this property: Whenever a harmonic sound is led as input into such a device, the device outputs a

harmonic sound whose first, third, and fifth partials have respectively i(1) times, 1(3) times, and i1(5) times the power of the corresponding input partials. Given i(1), 1(3), and 1(5), let the number-triple i = (i(1), i(3), i(S)) denote the

class of devices that transform harmonic sounds according to these proportions for the first, third, and fifth partials. Let [VLS be the family of all such number-triples 1, as i(1), 1(3), and i(5) range over all positive values. IVLS is a group under the combination tj = (1(1)j(1), 1(3)j(3), 1(5)}(5)).

Given harmonic class s = (s(1),s8(3),8(5)) and harmonic class t= (t(1), t(3), t(5)), take int(s, t) to be that member 1 of IVLS for which i(1) = t(1)/s(1), i) = t(3)/s(3), and 1(S) = t¢(5)/s(5). Then (S, IVLS, int) is a GIS. That is, Conditions (A) and (B) of Definition 2.3.1 obtain. The GIS is commutative. When int(s, t) = 1, any sound in class t will have 1(1), 1(3), and 1(5) times the power of any sound in class s, at its first, third, and fifth partials respectively. Another way of regarding the statement “int(s, t) = 1” is to think: Any sound of class s, when led as input to any device of class i, will cause a sound of class t to be output.

The fundamental frequencies of the sounds are irrelevant here; we are concerned only with certain aspects of their timbral profiles. If a given sound is led through a device of class i, and if the resulting output is then led through a device of class j, the final output will be a sound of the same class as that which would have resulted, had the original sound been

led through a device of class ij. Or, more simply, a device of class i concatenated with a device of class j forms a device of class 1).

We can make many variations on the specific GIS just discussed. For example, instead of considering partials #1, #3, and #5, we could instead

consider partials #1, #2, and #4. Or we could consider partials #1through- #5, or # 1-through- #8, or # 1-through- # 8-except-for- #7, and so on.

We can use GIS structures of this sort to build more complex GIS structures of interest. For instance, let GIS, be the GIS of the sort just discussed which considers partials # 1-through- #8 of harmonic sounds. We shall callan elements = (s(1),...,8(8)) of GIS, a “pertinent spectrum.”’ Now let us take as GIS, a familiar GIS involving the space S, of “time points.” We imagine a referential zero time-point and a referential time-unit fixed, so that 82 wecan label the elements of S, by rea] numbers a. IVLS, is the additive group

Generalized Interval Systems (3) 4.2.1

of real numbers and, given time points a and b, int(a, b) is the number b — a of time units by which b is later than a. (b — a later = a — b earlier.) Let us

explore the direct product GIS, = GIS, &) GIS,. The elements of S, = S, X S, are pairs (s, a), where s = (s(1), ... , s(8)) is a pertinent spectrum and a is a time point. The pair (s, a) models a class of sounds having pertinent

spectral profile s at time a. A finite set of such pairs, say the set DVSP = ((S,, a,), (Sj, a), .-- + (Sys @y)), models a class of sounds that have spectrum s,

at time a,, spectrum s, at time a,,..., and spectrum s, at time a,. We consider DVSP to be an unordered set of S,-elements, since the chronological order of the time points a, imposes a natural ordering on the member pairs of DVSP, no matter in what order we list those pairs. For convenience, we shall assume the members of DVSP to be listed above in chronological order, that is with a, (4 SL Gd Cp 2S FIGURE 5.10

On figure 5.10 the four triangles are lined up beneath the tetrahedron, and the edges of each triangle are stacked up below that triangle. In the four resulting stacks, each edge of the original Z-tetrahedron appears twice. Thatis because each edge of the tetrahedron belongs to two of the triangles. (Edge AB, for instance, belongs both to triangle Y2 and to triangle Y7’.) As a result, when we count how many sticks at the bottom of the figure are in Forte-class 2—3, we must divide that count by two, to arrive at the number of sticks in that Forte-class we found on figure 9 earlier. For example, figure 5.9 counted two

110 ~—s edges-of-Z lying in Forte-class 2-3, namely AC and BD. EMB(class

Generalized Set Theory (1) 5.3.5.2 2-3, Z) = 2. Figure 5.10 counts as edges-of-faces-of-Z twice as many sticks of class 2—3, namely AC-as-edge-of-Y2, BD-as-edge-of-Y2’, AC-as-edge-of-Y7, and AC-as-edge-of-Y7’. What Theorem 5.3.5.2 does in this connection is to ADJUST the count of sticks at the bottom of figure 5.10, dividing it by two to conform with the

stick-count at the bottom of figure 5.9. The theorem knows that two is the

proper number to divide by here, because 1/2 is the present value of ADJUST = 1/COMB(N — M, N — L) = 1/COMB(, 2) = 1/2. If Z were a heptachordal object in six-dimensional space (N = 7) and the Y hyperfaces were pentachordal objects in four-dimensional space (M = 5) and we were again interested in counting edges (L = 2), then we would have to ADJUST our count of sticks analogously by 1/COMB(N — M, N — L) = 1/COMB(, 5) = 1/10. The probabilistic method we used to prove Theorem 5.3.5.2 will help us understand figures 5.9 and 5.10 in a somewhat different light. Inspecting figure 5.10, we see that if we peel a triangular face at random off the tetrahedron, the probability is 1/2 that the face will be in Forte-class 3—2 and 1/2 that the face will be in Forte-class 3-7. If we pull an edge at random off a triangle of Forte-class 3—2, our expectation is 1/3 that the edge will be in Forte-class 2-3. And if we pull an edge at random off a triangle of Forte-class 3—7, our expectation is 1/3 that the edge will be in Forte-class 2—3. Hence,

according to the theory of probability, our total expectation for pulling

an edge of Forte-class 2-3 off the tetrahedron by a random yank is ((1/3)(1/2) + (1/3)G/2)) = (1/6 + 1/6) = 1/3. And this agrees (as our work says it must) with the probability of that event which we infer from figure 5.9: There we see that of the six tetrahedral edges, two are of Forte-class 2—3; so we infer that our expectation of yanking an edge of that class in a random pull is 2/6, which is 1/3. To pursue farther what Theorem 5.3.5.2 has to do with figures 5.9 and

5.10, and with higher-dimensional analogs of those figures, would lead us deeply into a branch of mathematics called algebraic topology. That pursuit would be very much worth undertaking, but it would be out of place here. In discussing how our generalized embedding number applies to the example of figures 5.9 and 5.10, I have supposed that the reader is already familiar with Forte’s use of the interval vector in atonal theory. Now let us see how our generalized theory applies in a very different context, one with which

the reader is almost certainly unfamiliar. To that end, we shall study some examples in connection with the non-commutative GIS of time spans which we developed in chapter 4. The first thing we must do is fix the group CANON for our purposes. We shall take CANON to be the group of all interval-preserving operations here. We shall not allow transpositions, much less inversions, as canonical oper-

ations for this study. Our reason will become clear. Il]

5.3.5.2 Generalized Set Theory (1)

As we observed in 4.1.7(D), the generic interval-preserving operation

P=Po.y transforms the sample time span (a,x) into the time span (bh + ua, ux). The commentary on 4.1.7(D) elaborated upon this: “The interval-preserving operation Py, ,) first blows up or shrinks the sample time span (a, x) by a factor of u, transforming (a, x) to (ua, ux), and then moves the latter time span backward or forward in time by h or (—h) numerical units, transforming (ua, ux) to (h + ua, ux) = P(a,x).”’ So if X is a set of time spans

(S,,S5,-.-5,), Where s, = (a,,x,), then P(X) is the set (s|,s5,...sy), where s, = P(s,) = (h + ua,, ux,). We can imagine the set X here as modeling temporal aspects of a musical “passage” containing N events; then P(X) models analogous aspects of the passage played u times as slowly (1/u times as

fast), starting the tempo change from time-point zero, all this played h numerical time-units later (—h earlier). For example, let us take “the quarter note’’ as a numerical unit and ‘‘the beginning of the piece” as a numerical time-point zero. Imagine a motive consisting of an eighth, a dotted eighth, a sixteenth, and a quarter, played consecutively starting 10 quarters after the beginning of the piece. We could model some temporal aspects of this motive by the set X = ((10,4), (104, 3), (114.4), (114, 1)). Remember that X is formally an unordered set; we have listed its members “in order of appearance” only for convenience here. Let us take h = 1370 and u = 4. Then the transformed set Pa, ,)(X) first augments the entire rhythmic setting by a factor of 4, from time-point 0 on; then P(X) plays the augmented motive beginning h = 1370 quarters later, that is beginning 1370 quarters after time point 40, that is beginning at time point 1410. (The

augmented motive obtained as an intermediate stage began not at time-point 10, but at time-point u- 10 = 4-10 = 40.) So the transformed motive modeled by P(X) consists of a half note, a dotted half, a quarter, and a whole note, played consecutively starting 1410 quarters after the beginning of the piece. Expressing this in numbers, P(X) = ((1410, 2), (1412, 3), (1415, 1), (1416, 4)). One can check that each member of P(X) is mathematically related to the corresponding member of X via the transformation P(a, x) = (h + ua, ux), here = (1370 + 4a, 4x). For instance, the third-listed members of the sets X

and P(X) are related by the formula P(i1i,f)=(h+u-1l4i,u-)= (1370 + 4-114,4-4) = (1370 + 45, 1) = (1415, 1). There is no need to restrict our attention to sets modeling consecutive events, as X and P(X) did in the preceding example. We could for instance consider a passage in which a violin plays four consecutive quarters, while a viola plays three triplet halves, while a cello rests for an eighth and then plays

two consecutive quarters followed by a dotted quarter. We could model temporal aspects of this passage by a time-span set Y. Supposing that the

onset of the passage comes 16 quarters into the piece, we can write Y = ((16, 1), (17, 1), (18, 1), (19, 1), (16, 3), (17, 9), (184, 3), (162, 1), (173, 0), 112 ~~ (184, 14)). As P varies over the interval-preserving operations, P(Y) models

Generalized Set Theory (1) 5.3.5.2

the ensemble passage, played at (all) different tempos and at (all) different times. The elements of the unordered set Y above are listed, not “in order of appearance,” but “‘by parts,”’ as they were described in the text. Suppose the numerical time-span set Y, models the above passage for string trio at the precise time the music was first imagined clearly by the composer. Suppose the different numerical set Y, models the passage at the precise time it was played during the first performance. Suppose the still different numerical set Y, models the passage at the precise time my trio played it yesterday, taking a considerably faster tempo. Given one fixed referential time-point zero and one fixed referential time unit, the numbers denoting the members of the three sets Y,, Y,, and Y, will be very different. Our formalism, though, enables us to say that the three sets are all (approximately) canonically equivalent. That is one powerful methodological reason for choosing CANON here to be the group of interval-preserving operations. Another good reason for the

choice is provided by the way in which this group relates dyad structure to interval structure in the GIS at hand. We shall now explore that topic. By a “dyad” we understand a set containing two distinct members s and t. By an attack-ordered dyad (AOD) we shall mean a dyad containing say s and t, ordered in the following way: If s begins before t (as a time span), the order is (s, t); if t begins before s, the order is (t, s); 1f both time spans begin at the same time, the shorter of the two spans is listed first. Since s and t are distinct time spans, these criteria are sufficient to order the dyad. Given an AOD D = (s,t), let G, p) = int(s, t). Then t begins i s-durations after s begins, and t lasts p times as long as s. Because of the ordering criteria on D, the number i must be non-negative, and if i = 0 then the number p must be

greater than 1. Let us call an interval (i, p) of this form a forwards-oriented interval. We have seen that 1f D = (s, t) is an AOD, then int(s, t) is forwardsoriented. The converse is also easily seen: If s and t are time spans such that int(s, t) is forwards-oriented, then D = (s,t) 1s an AOD. We can define (j,q) to be a “backwards-oriented interval” in an analogous way: ] must be non-positive, and if} = 0 the number q must be less than 1.

Now in the group IVLS the inverse of the interval (i, p) is (—i/p, 1/p). It follows that the inverse of a forwards-oriented interval is backwards-oriented, and vice-versa. One sees quickly that the members of IVLS can be partitioned into three categories: the forwards-oriented intervals, the backwards-oriented intervals, and the identity interval (0, 1). Here now is the crucial manner in which our stipulated canonical group

comes into play: Given AODs D, = (s,,t,) and D, = (s;,t,), then D, and D, are canonically equivalent if and only if int(s,,t,) = int(s,,t,). It would take too long to include here a formal proof of that theorem; such a proof is appended to the end of the chapter as section 5.6. The theorem is by no means obvious or trivial. Once we have proved it, we can note that the 2-element set =: 113

5.4.1 Generalized Set Theory (1) classes correspond I-to-1 with the forwards-oriented intervals. If D = (s, t) is an

AOD, then the set class /D/ corresponds to the forwards-oriented interval (i, p) = int(s, t): Every member D’ = (s’, t’) of /D/, when attack-ordered, has int(s’, t') = Gi, p); furthermore, if s’ and t’ are any time spans such that int(s’, t') = G, p), then the AOD D’ = (s’, t’) is a member of the set class /D/. The forwards-oriented intervals thus play exactly the same role here that Forte’s “interval classes” play in his atonal theory: They can be used to label the distinct set-classes of dyads. They can be so used, that is, if we take the intervaJ-preserving operations as CANONical in constructing those setclasses.

As a result of this structure, we can develop a very strong forma! analog

for Forte’s interval vector in this particular system (NB). Let X be a set containing more than two members; let D be a dyad; then EMB(D, X), the number of forms of D embedded within X, is equal to the number of ways the forwards-oriented interval (i, p) can be spanned between members of X, where

(i, p) is the interval spanning the attack-ordered members of D. In other words, EMB(D, X) = IFUNC(X, X)(i, p). In this sense we can speak of EMB(/D/, X), when /D/ varies over the dyad-classes, as an “interval vector;” (i, p) will vary concomitantly over the forwards-oriented intervals. Let us study some actual interval vectors in this system by way of example.

b) c) d) FIGURE 5.1]!

5.4.1 Examp.e: Figure 5.11 shows the mensural skeletons for motives (b), (c), and (d) from the Chopin sonata studied earlier (in section 4.3). The rhythmic motives are modeled by sets of time spans, and their interval vectors are tabulated on figure 5.12. Forming and reading these interval vectors becomes easy with practice. The forwards-oriented interval (1, 1) labels the set-class of AODs D = (s, t) such that t begins right after s (1 s-length after s begins) and extends the same duration as s(1 times the length of s). Within set (b) we count three instances of

such AODs. The AODs are formed by the first-and-second notes of the motive, its third-and-fourth notes, and its fourth-and-fifth notes. Thus the number 3 is entered on the table of figure 5.12, in the row of the table headed by the interval (1, 1) and in the column of the table headed “vector of (b).”’ Set (c) includes only two AODs in the set-class (1, 1): the first two notes of motive

(c), and the last two notes of the motive. Remember: A pair of successive 114 quarter notes in any tempo at any point in the piece (or any other piece any

Generalized Set Theory (1) 5.4.1

(1, 4) l (1, %) 1 qd, 1) 3 2 2

interval vector vector vector

of (b) of (c) of (d)

(2, 1) 2 l l (3, 1) 2 1 1

(4, 1) 2 (5, 1) 1

(1, 2) 1 1 (2, 2) 1 (3, 2) 2 1 (4, 2) 1 1 (5, 2) I (6, 2) 1

FIGURE 5.12

time) is canonically equivalent to such a pair of successive eighths or successive halves or successive quintuplet sixteenths. All the AODs just indicated belong to the same set-class, the set-class determined by the interval (1, 1) between the first and second members of each AOD. The two AODs of class (1, 1) embedded in set (c) are counted on figure 5.12 by the number 2, entered to the right of the interval (1, 1) and in the column headed “vector of (c).” Set (d) also embeds two AODs of class (1, 1), namely the pair of half notes and

the pair of whole notes at the end of the motive. Let us now consider the set-class corresponding to the forwards-oriented interval (3, 2). An AOD D = (s, t) belongs to this class if t begins 3 s-spans | later than s begins, and lasts twice as long as s. The second quarter of (c) and the first half-note of (c) form such an AOD. So do the last quarter of (c) and the

second half-note of (c). These two dyads are tabulated by the entry 2 in the (3, 2)-row and the (c)-vector-column of figure 5.12. The entry of 1 in the (1%, \%)-row arises from the AOD formed by the first and third notes of motive (d). In section 4.3 we noted a progressive “expansion” from motive (b), through motives (c) and (d). The progressive broadening of note values, from eighths

to quarters to halves to whole notes, is obviously crucial. Our model does not address this aspect of the progression. But it does address and analyze well another aspect of interest, something we might call the “progres- = 115

5.4.2 Generalized Set Theory (1) sive diversification” of the motives in their internal rhythmic structures. As one sees from the first column of figure 5.12, motive (b) concentrates on only a few intervals, most of which appear more than once. Motive (c) projects only two intervals that appear more than once (in the second column of the figure); no interval appears thrice (in that column). Motive (c) projects more intervals, and more diverse intervals, than (b). Motive (d) projects only one interval that appears more than once; that interval appears only twice. Motive (d) thus continues the process of diversification. The insensitivity of our interval vector to changes in tempo, a defect in some ways, 1s useful here: It enables us to compare motives (b), (c), and (d), each in its own intrinsic context. We touched on this idea earlier in connection with IFUNC(Y, Y). Motive (a), the motive of the opening Grave, does not appear on figure 5.11 or figure 5.12. If one ignores the anticipation of Fb = E natural in the music, then motive (a) is canonically equivalent to motive (d). 5.4.2 In connection with figure 3.3 (page 41), we earlier studied an “‘unrolling interval vector” for a set in a different GIS, a set pertinent to Webern’s Piano Variations. The present GIS, like the earlier one, has an intrinsic chronology,

so we can “unroll” its interval vectors too. The abstract method of doing so will involve a number of technical finesses. To begin the abstract study let us consider the imaginary string trio we discussed a short time ago, and let us imagine another of its passages, which we can symbolize as in figure 5.13. [— 16th J after piece begins

Violin: J : ; , -——- 3 Viola: oO A Cello: 1 J d. FIGURE $4.13

Wecan model certain temporal aspects of this passage, as we did with the last one, by a set Y of time spans. The violin projects four time spans, (16, 1),

(17,1), (18,1), and (19,1); let us call these spans vnl, vn2, vn3, and vn4 respectively. The viola projects the two time spans (16, 8) and (18%, 4); let us call these spans val and va2. The cello projects the two time spans (164, 2) and (184, 14); let us call these spans vcl and ve2. We can list the members of Y “‘in

parts” as vnl, vn2, vn3, vn4, val, va2, vcl, vc2. Of course that is not their 116 ~=0 “order of appearance” in the music. But what is?

Generalized Set Theory (1) 5.4.2

We might try attack-ordering to list the members of Y “in order of appearance.” Then we would list them as vn1, val, vcl, vn2, vn3, vc2, va2, vn4. For many purposes attack-ordering is natural, and we have seen how cogent it 1s in connection with dyads, intervals, and the canonical group. But

we shall not want to use attack-ordering in connection with the way we perceive the members of Y “appearing.” To see why not, imagine that we stop the music of figure 5.13 just after the attack of vn2, that is, just after time-point

17, and suppose that we ask just which time spans we have perceived up through that time. Obviously we have perceived vn!. But we have not yet perceived any other spans. True, we have heard both the other instruments attack other spans. But we do not yet know how Jong those spans are going to be, so we cannot claim to have perceived them as spans, using them e.g. to form intervals in an unrolling interval vector. For all we know as we listen at time-point 17, the viola may be intending to hold onto its note for 24 quarters. Now when we unroll the interval vector for Y in connection with figure 5.13, we are going to want precisely to “stop the music’”’ of the figure at various stages, asking at each stage what intervals we have heard so far. As we have just seen, when we stop the music at time-point 17, we can be sure of having perceived only one span, namely vn1; hence we cannot say we have perceived any proper intervals at all so far. The attack-ordering for Y 1s deceptive in this connection. That ordering, beginning vn1, val, vcl, vn2,..., makes it seem as

if the spans val and vcl have “already occurred” by the time vn2 occurs, attacking at time-point 17; hence it seems (wrongly) as if we ought to count the

forwards-oriented intervals int(vnl, val) and int(vn1, vcl) and int(val, vel) as “having already occurred” by the time we “‘get to” vn2. But this inference is wrong. As we have seen, no intervals have yet “‘occurred”’ by time-point 17, so

far as our perceptions of spans and their interrelations are concerned. To reflect the true order of our span-perceptions, we shall want to use a different system of ordering, not attack-ordering but release-ordering. Given distinct spans s and t, s precedes t in the release-ordering if s ends before t ends, or if they end simultaneously and s is longer. If s and t correspond to musical

events event] and event2, then s precedes t in the release-ordering when we perceive the time span during which event! has happened before we perceive the time span during which event2 has happened, or if we perceive both spans simultaneously and recall that event! began first. Release-ordering for the set Y thus enables us to articulate the music of figure 5.13 into stages that correspond to our evolving perceptions of time spans “having happened” as we listen. The members of Y in the releaseordering are vnl, vn2, vcl, val, vn3, vc2, va2, vn4. Furthermore, when we articulate the music into such stages, we shall want to demarcate the stages by the time points at which various spans are released, not at which they are attacked. Thus the finest possible articulation of the set Y = (vnl, vn2, vel, val, vn3, vc2, va2, vn4) into stages for our purposes can be realized as follows. =‘ 1/7

5.4.2 Generalized Set Theory (1) Stage 1: We have heard Y, = (vnl, vn2) at time-point 18, the release of vn2. Stage 2: We have heard Y, = (vnl, vn2, vcl) at time-point 183, the releasepoint of vcl. Stage 3: We have heard Y, = (vn1, vn2, vel, val) at time-point 184, the release-point of val. Stage 4: We have heard Y, = all of Y at timepoint 20, the simultaneous release for vc2, va2, and vn4. By calculating how the interval vectors for Y,, Y,, Y;, and Y, develop, each expanding the counts of the last among the various intervals counted, we shall be able to model how our sense of intervallic structure evolves as we listen to the musical passage. We shall be able to use our formal model analytically,

just as we used analogous machinery earlier in connection with the Webern passage and the expanding interval-counts of figure 3.3. Our work above can now be generalized. Given any set Y of time spans, first list Y as (S,,82,--., Sy) in the release-ordering. Next identify N or fewer “stages” associated with certain subsets of Y as follows. Stage 1 is articulated at the release of's,; it is associated with a certain subset Y, of Y. Y, = (s;,8,) unless s, releases simultaneously with s,;in thatcase Y, = (s,,8,,83) unless s, also releases simultaneously at that time; in that case ... (etc. etc.). After Y, = (S1,82,---, Sy) has been found, Stage 2 is articulated by the release point of S44 ,- Stage 2 is associated with a certain subset Y, of Y. Y, = Y, + (Sys) unless S442 releases simultaneously with s,,, (etc. etc.). And so on. Eventually one attains the release point of s, and exhausts the set Y. We can regard the stages as developing in a simple serial rhythm as stage /, stage 2, stage 3, and so forth. Or we can regard them as developing in a “perceptual rhythm,” the rhythm of the various release-points at which the stages articulate. (This is interesting but it oversimplifies the psychology of what is going on.) As the stages develop rhythmically, the evolving interval vectors of Y,, Y>, etc. can be studied. Care must be taken here because the release-ordering of Y does not necessarily coincide with the attack-ordering. It is possible for s,, to precede s,, in the release-ordering, but to follow s, in the attack-ordering. (The different listings of Y in connection with figure 5.13 illustrate the possibility.) Should

this happen, when we get to the stage that notices (the release of) s, in the release-ordering, we shall want to tabulate the forwards-oriented interval int(S,,S,) in our updated interval vector, not the backwards-oriented interval int(s,,,S,): The reader who likes to fool with computer programming and who has a home computer with a color monitor will enjoy writing an “unrolling interval vector” program. The program will take a set Y of time spans, arrange it in release-ordering, determine the articulation-points of the various stages, and find the corresponding subsets Y,, Y>,..., Y. The program will then compute

the interval vector for Y, and display it on the screen as follows. For each forwards-oriented interval (1, p) that is counted, a colored dot appears at the point (i, log p) on a half-plane grid. (iis always non-negative; log p is positive, 118 + ~—=zero, or negative.) If the interval appears only once in the set, the dot is violet;

Generalized Set Theory (1) 3.4.3

the more times the interval appears, the more the color of the dot moves toward the red end of the spectrum. (The background of the screen is either white or black.) After the program has computed the interval vector for Y,, it will update the count of various intervals so as to obtain the interval vector for Y,, changing the color of some dots on the screen as pertinent. Then it will update the count of various intervals to obtain the interval vector for Y,, and so on. The updating can be done quickly following the method of figure 3.3. (Remember that you may have to adjoin more than one releasing time-span at any new stage. Also remember to adjust for any new dyads that may be release-ordered but not attack-ordered.) The rhythmic updating of the screen can follow either the serial rhythm of the stages or their “perceptual rhythm”’ as discussed above, either in real time or suitably scaled for visual effect.

oo

5.4.3 EXAMPLE: The technique of unrolling can be applied to EMB-related functions beyond the interval vector.

Stages: ] 2 3 4

2 ls |J |

3P3ee 3ddd3ye3Odd 3

CPP | OOF | Ler) Loe Me CPF om OL

FIGURE 5.14

The “‘set”’ Y of figure 5.14, for example, is articulated into four stages. (We could articulate it farther, but we shall not do so here. Since neither the time unit nor the point zero is specified, Y is not strictly a numerical “‘set’’ within TMSPS, but I am assuming the reader will not mind a certain looseness

in discourse at this point.) Figure 5.14 also displays “sets” X,, X,, X, and X%,, all of which can be found embedded within Y. X‘, isa canonical form of X,

and X‘, is a canonical form of X,. Figure 5.15 shows how the embedding numbers of the set-classes /X,/ and /X,/ within Y develop, as Y develops over the four stages. The values rise at Stage 4 because the dotted half releases there and the appearances of X’, and X3, augmented (canonical) forms of X, and X,, can now be counted as fully “embedded” in Y. Figure 5.15 shows us how the set-class /X,/ comes on late

Stage 4. 119

and strong, pulling ahead of /X,/ at Stage 3 and then decisively ahead at

]423 3]2 J532 5.4.4 Generalized Set Theory (1)

for Y at EMB (/X,;/, Y) and EMB (/X2/, Y)

stage equals equals

FIGURE 5.15

No doubt the reader has recognized Y as interpreting the opening of Brahms’s G-Minor Rhapsody. The idea that “/X,/ comes on late and strong” is reinforced by the end of the closing group in the music, where the closing theme is liquidated rhythmically down to a succession of X,-forms alternating on the tonic and dominant of D minor. The rhythmic interpretation of figure 5.14 does not exclude other possible rhythmic readings of this music. E.g. one could read triplet eighths where there are triplet rests on figure 5.14; then /X,/ and /X,/ would come out ina tie on figure 5.15. But such other interpretations just as clearly do not exclude the reading of figure 5.14. The reader who consults the score will find ways enough in which the relation of quarter note to accompaniment changes after Stage 2 so as to support the reading of the figure.

5.4.4 Note: Through section 5.4 so far, we have focused upon the interval

vector and more generally the EMB function, in connection with our non-commutative GIS of time spans. The technique of “unrolling in stages”

which we applied to this study could also be applied in connection with IFUNC(X, Y), as we unroll either X or Y or both in stages.

5.5 Notes: Let us return now to the most general abstract setting, that of a family S and a group CANON of operations on S. Following the suggestions

of my writings elsewhere, we can explore numbers of interest beyond EMB(X, Y).’ We may define COV(X, Y), for example, the covering number of X in Y, as the number of forms of Y that include X. This is not necessarily the same number as EMB(X, Y), the number of forms of X that are embedded

in Y. E.g. in atonal theory take X =(C,E) and Y = (C,E,G#); then EMB(X, Y) =3 but COV(X,Y)=1. If S is finite then COV(X, Y) = EMB(Y,X), where Y and X are the complements of Y and X. 7. “Some New Constructions Involving Abstract Pesets, and Probabilistic Applications,” Perspectives of New Music vol. 18, nos. 1-2 (Fall-Winter 1979 and Spring-Summer 1980),

£20 433-44.

Generalized Set Theory (1) 5.6

We may also consider SNDW(X, Y, Z), the sandwich number of Y between X and Z; this is the number of forms of Y that both include X and are

included in Z. If we write @ for the empty set, then SNDW(Q, Y, Z) = EMBC(Y, Z); if S is finite then SNDW(X, Y,S) = COV(X, Y). If Y’ is a form

of Y then SNDW(X, Y,Z) = SNDW(X, Y’,Z); hence we can write SNDW(X, /Y/, Z) without ambiguity. But we cannot use /X/ or /Z/ for sandwich arguments in this way; SNDW(X, Y, Z) depends very much on the specific forms of X and Z being used as arguments. For example let Z be the C-major scale; let /Y/ be Forte-class 3—4, which we could write as /(B,C, E)/. Let X, = (C, E). Then, allowing both transpositions and inversions as canonical, SNDW(X,,/Y/,Z) = 2: There are 2 forms of (B,C, E) that can be sandwiched between (C, E) and the scale, namely (B, C, E) and (C, E, F). Now let X, = (F,A). X, and X, belong to the same set-class, but SNDW(X,,/Y/, Z) = 1, not 2: Only 1 form of (B,C, E) can be sandwiched between (F, A) and the scale, namely (E, F, A). Another interesting number 1s ADJOIN(X, Y, Z). This 1s the number of forms Y’ of Y satisfying both (A) and (B) following. (A): Y’ 1s disjoint from X. (B): There is some form of Z that includes both X and Y’. To illustrate what this number is inspecting, let X = (C, E), Y = (D, G), Z = the C-major scale. (C, E) + (D, G) lies within some major scale; so does (C, E) + (F, Bp); so does (C, E) + (F#, B); so does (C, E) + (A, D). Exactly the 4 fourths (forms of Y) metioned in the preceding sentence have both the desired properties (A) and (B); other fourths (forms of Y) either contain C or contain E or do not add up

together with (C,E) to lie within any major scale. So ADJOIN((C, E), (D, G), C-major scale) = 4. Inspecting properties (A) and (B), one sees that

we can write ADJOIN(X,/Y/,/Z/); it follows that we can even write ADJOIN(/X/, /Y/, /Z/). 5.6 APPENDIX (optional): We prove here the crucial theorem stated earlier,

on the relation of dyads and intervals in our non-commutative GIS for TMSPS, using the group of irtterval-preserving operations as CANONical. Here ts that theorem stated again: Given attack-ordered dyads D, = (s,,t,)

and D, = (s,,t,), then D, and D, are canonically equivalent if and only if int(s,,t,) = int(s,,t,). The proof follows. Set 5) = (a,,X1), ty — (b:,Y1), S2 = (a2,X2), ty = (b,, Y2). Suppose first

that D, and D, are canonically equivalent; we shall show that int(s,,t,) = int(s,,t,). Say that P = Py, ,, is the canonical operation mapping D, onto D,. P maps the members of the first dyad somehow onto the members of the second; conceivably the operation might transform s, into t, and s, into t,. But in fact that cannot happen in this situation: P transforms s, intos, and t, into t,. To see that, we use the fact that both dyads are attack-ordered. Since

D, 1s attack-ordered, either a, < b, or(a, = b, andx, < y,). Ifa, < b, then ua, < ub, andh + ua,

FIGURE 6.3

The F-with-a-question-mark, notated above the upper beam at Z,, indicates a “‘missing’”’ F that breaks an otherwise consistent pattern of wedging and inversional balance. In this respect the missing F is exactly like the missing F on figure 6.2 earlier. It is like that F in breaking an E-wedge; it is like that F, too, in leaving the Eb of chord Z, bereft of its I-partner, just as the Ep of chord Y on figure 6.2 was bereft of its I-partner. The note-against-note relations of

figure 6.3(b) show that INJ(Z,, Z,) (1) = 2 for n = 1 and 2; INJ(Z;3,Z;)(D) actually = 3. (Within the set Z, of pitch classes, D and F# invert into each other, while Bb inverts into itself.) Hence the abrupt inversional imbalance at Z, 1s all the more strongly felt, since INJ(Z,, Z,)(1) = only 1. The metaphor of “imbalance” caused by “‘something missing” interacts well with the text: The singer is in a state of emotional discombobulation caused by the lover’s absence. Continuing along figure 6.3(b), we hear how, when the progression Z, etc. recurs as the progression Z; etc., the singer’s final sign-off supplies the

required F natural at the right moment, within the arpeggiated set Z,. The required Z;, which continues the patterns of wedging and inversion consistently, sets the word begehre = require. (The singer claims nor to require the consolation of any friend.) Once the begehrte Z, has appeared, the wedgecan 127

6.2.3 Generalized Set Theory (2)

and does converge all the way to the E of Z,, across the pickup chord Z), in the music, The level of progressive wedge-projection and internal I-projec-

tion is thus restored: INJ(Z,,Z;)(w*) = 2, INJ(Z,,Z,)(w®) = 3; INJ(Ze, Zs) (1) = 2, INJ(Z,, Z,) 1) = 3. The convergence of the wedge to its focal point E within Z, coincides with

and supports a big structural downbeat. The sonority of Z, has earlier been associated with the word Seufzer, during the text, ‘meine Worte sich in Seufzer dehnen (my words trail off in sighs (or groans)).” Figure 6.3.(b) portrays the sighing and trailing-off ideas very well. It also shows how the beamed sighing-progressions are framed by the elements (C, Ab) and (E) of the Seufzer-chord, which (for this reason and for others too) takes a big downbeat when it appears as Z,. Figure 6.3(b) shows how the wedging commences yet once more after Ze, continuing through X to Y, which is the end of the piece. Within X and Y, the pitches D5 and Eb5S of the music are brought down an octave, to be displayed as D4 and Eb4 noteheads on the figure; this shows clearly how the pitch classes

D and Eb contribute to the final wedge. In particular, it brings out strongly how the final progression, Z,—-Z,—X-—Y, recapitulates the initial wedgestructure of the opening progression Z,—-Z,—Z,-Z, on figure 6.3(b). The

‘blue note” Fb of the final Y, shown on figure 6.3(a), takes on added significance in its “‘tonic” function, as it prolongs the downbeat E from the Seufzer-Z,. (Dehnung).’

Figure 6.3(c) sketches in a format similar to (b) the influence of a subordinate wedge and inversion over this passage. That is wedging-to-F#

and the inversion-operation J = In = Ic. The symbols “Ap—G-F#//,” at the beginning at bottom of the figure, show how the inner voice of the chords attains the goal and center F# of the wedge, getting there “from above” over

chords Z, through Z,. Then from Z; right on, all the way to X, the outer voices almost succeed in converging to F#(Gp), except that F is missing in the lower voice. Again we run into the thematic and structural “missing F!”’ The

missing F now has a new structural function: It is missing as a semitone neighbor to Gb in a wedge converging to Gb, the bass of the Angst-chord X. The earlier missing F was missing (inter alia) as a semitone neighbor to E in wedges converging to E, the bass of the Hoffen-chord Y. The T , p-relation between the bass note of X and the bass note of Y is thus expanded into a larger T, -relation, a relation involving the respective wedges and inversions about those notes. The larger relation can be expressed by the 1. Idiscuss these ideas and others of the same sort elsewhere, exploring more systematically rhythm, meter, text setting, registers, doublings, and other features of the music. The reader who would like to go deeper into the piece itself will be interested by that article, which 1s less preoccupied than we must be here by theoretical constructions of various sorts. The article is “A Way Into Schoenberg’s Opus 15, Number 7,” Jn Theory Only vol. 6, no. 1 (November 1981),

128 3-24.

Generalized Set Theory (2) 6.2.3

transformational equations w= = T,)w'*Tj¢; I= T,.JTj¢. Essentially, these equations “modulate by T,,.’ the wedging and inversional transformations centering on F#, to obtain the analogous transformations centering on E, We shall explore the idea of such “modulation” more generally and more formally later on. The present remarks are meant to prepare that later exploration. The reader can hear in connection with the example at hand that the relation T,)(Gb) = E has to do both with the bass line in the chord-succession X-Y, and also with the relation of w* and J, in figure 6.3(c), to w® and I, in figure 6.3(b). The reader will then be able to summon that musical experience to mind during the abstract discussion of transformational “‘modulation” later on (in connection with Formula 6.7.2(c)). The formalities of INJ, applied to the transformations w® and I on S, have engaged figure 6.3(b) only partially. The visual layout of that figure conveys a good deal more than our formal machinery has so far described. For

instance, the figure shows by its note-against-note layout that we do not simply have an Ab in Z, wedging to a G in Z, and, independently, a C in Z, wedging to a Cf in Z,; rather the “I-partnership” of (Ab, C) within Z, wedges as a whole to the “I-partnership” of (G, C#) within Z,. In a similar sense, the missing F of Z, 1s not just a missing note; it is a missing I-partner for Eb, without which the wedge cannot converge; as a missing partner it symbolizes the absent lover. And in a similar sense, the last line of text, “(that) I require the consolation of no friend,” 1s symbolized exquisitely by the pitch class E as goal of the wedge and center of inversion: The pitch class has and needs no partner; it gets along by itself, perfectly self-centered in its Seufzer. The visual layout of the figure brings out such ideas; our transformational machinery has not as yet adequately engaged them. But it can be formally developed so as to do so. The operation I partitions the family S into distinct “transitivity sets” (Bb), (A,B), (Ab, C), (G, C#), (Gb, D), (F, Ep), and (EB). I transforms the members of each transitivity set

among themselves: I(Bhb) = Bp; I(B) = A and I(A) = B; and so on. Such transitivity sets enable us to engage the notion of “I-partnerships” in our formal machinery. Many of the chords under consideration embed an entire

transitivity set; some chords even embed two (e.g. Z, = X = Angst and Z, = Seufzer). Frequently a transitivity set embedded in one chord is transformed as a whole by w® into a transitivity set embedded in the next chord. Such ideas can be developed very abstractly in connection with the generalized INJ function. Here, it is formally important that the transformations I and w®

commute. It is not remarkable that the visual aspects of figure 6.3 can be described by formal aspects of the INJ machinery when suitably extended. After all, one could hardly conceive the layout of the figure without some prior intuitions of w* and I as transformations; the formal extensions of the

explicit. 129 machinery, in that connection, amount only to making the relevant intuitions

6.2.3 Generalized Set Theory (2) Our investigations so far have focused on harmony and voice leading, as we explored the structural functions of various transformations in this song, using INJ. Now let us use INJ to explore some melodic functions of various transformations therein.

2Mat Lun Sr vi $.W smite

“ 4ee#2ee 3 4nl5Sa 6cl.F 8 9 10 $,14,3.”

FIGURE 6.4

Figure 6.4 will help us in this endeavor. The figure transcribes the pitches

from the opening of the voice part, where they set the first line of text. Schoenberg’s spelling projects throughout the phrase a strong visual inversional symmetry about the third line of the staff, where B>4 would appear as a center of I. Ordinal numbers 1, 2, ... , 10 appear under the first, second, ... tenth notes of the figure. The events of the melody are modeled here in a space

S whose members are pairs (n, p), n being an ordinal number and p a pitch class. As a serial structure, the “melody” is modeled by an unordered set of ten such pairs; the elements of this set are the pairs (2, Gb), (1, D), (10, Eb, (3, Eb), and so on. “(2, Gb)” can be interpreted as saying “The second note is Gb.”

Arrows on the figure indicate transformational relations that will interest us here, Each arrow is labeled by a pair of symbols comprising a number (1, 2, 3, 5, or 6) and a letter @ or w). The number indicates how many ordinals later the transformed pitch class appears. The letter indicates a pitch-class transformation, w standing for w®, Thus the arrow labeled “6, w” which issues from the third note of the series, Eb, indicates that the note is transformed into a note appearing 6 order positions later, via the wedge transformation. The arrow from A to Ch, labeled “2, I,” indicates that the A is transformed into a note appearing 2 order positions later, via the 7 transformation. More formally, the transformation 6, w maps the element (3, E>) of the melody into the element G + 6, w(E})) = (9, F)); the transformation 2, I maps the element

(6, A) into the element (6 + 2, I(A)) = (8, Cb). The transformations (6, w) and (@, I) are well defined by this formal method on the space S$ of pairs (n, p). (6, w) maps the pair (n, p) into the pair (n + 6, w(p)); (2, D maps the pair (n, p) into the pair (n + 2, I(p)). The trans130 formations are not operations. w itself is not an operation on the twelve pitch

| Generalized Set Theory (2) — 6.2.3 classes; even beyond that, the ordinal aspect of the mappings prevents the transformations from mapping S onto itself. E.g. there is no (n, p) in S (withn a positive integer) such that (2, I)(n, p) = (1, Ab).

The dotted arrows and question marks on figure 6.4 arise from the possibilities of considering the Fbs as substitutes for F naturals. If the flats were naturals, the dotted arrows would be solid and the question marks would disappear. We shall denote by X", the set comprising the m" through n"™ events of the series (i.e. melody-set). Using the subsets X",, we can apply familiar “‘unrolling”’ techniques to the situation, now using our injection numbers. For instance INJ(X7, X7)(f) = 2, where f is either (2, w) or (1, I), assuming we allow the dotted lines. The equation states: The set comprising (1, D), (2, Gp), (3, Eb), and (4, F(p)) contains “‘2”” members that transform into the set under the transformation (2, w), and ‘‘2”” members that transform into the set under the transformation (1, I), supposing that Fb is read “‘asif” F natural.

This equation engages a significant “unrolling” when compared to the equation INJ(X?, X?)(f) = (only)1 for the same transformations f. We can compare these internal transformations of X} with the internal transformations of X§, the next 4-element subset of the melody. X2 comprises (5, C), (6, A), (7, Ab), and (8, Ch). INJCX8, X8) (2, I) = 2: With respect to the new tetrad X8, the transformation (2, I) plays the same internal role that (1, 1)

did in connection with the first tetrad X7. And, as the figure shows, (3, w) plays the same role with respect to X8 that (2, w) played with respect to X¢. That is so even though INJ(X2, X8) (3, w) is only 1, not 2. Our arrow diagrams

capture a certain picture of X7 on the figure, as it appears bound together internally in a certain way by (J, I) and (2, w) arrows. The same kinds of arrow

shapes capture a similar picture of X? on the figure, as it appears bound together in a similar way by (2,1) and (3, w) arrows. Our model enables us to observe an interesting augmentation of ordinal

distances, from the arrow transforms binding X{ to the arrow transforms binding X§. That is, within Xf I-relations occur 1 note apart and w-relations

occur 2 notes apart; within X$ these ordinal distances are expanded: Irelations occur 2 notes apart and a w-relation occurs 3 notes apart. This serial augmentation is particularly interesting because X8 takes only half the time to sing as did X7, in the clock time of the music. Our discussion of X{, X§, I, and wis enhanced by the observation that no I-arrows and vo w-arrows lead events of the first tetrad to events of the second, on figure 6.4. In our terminology, INJ(X*¢, X8)(f) = 0 when fis either (n, I) or

(n,w), for any n. This observation specifically enhances our sense that the melody articulates into Xf + X§ + X$°, when heard in the context of w and I relations. The italicized phrase is meant to recall our earlier discussion in connection with the various contexts of a melodic phrase within the Webern violin piece. Our sense of Xf + X8 + X2° in this context is further enhanced 131

6.2.3 Generalized Set Theory (2) by the (6, w) arrow on figure 6.4, extending from inside Xt over X8 to X20. INJ(X+, X§°) (6, w) = 1, after INJ(X{, X2)(n, w) had been zero for all n. The ordinal distance over which w functions continues to grow: The first w-arrow(s) had ordinal span 2 within X{; the next w-arrow had ordinal span 3 within X§; now a w-arrow has ordinal span 6, between X* and X3°. In this context the F flat of (9, Fp) in the melody is “correct’’; it is in fact the goal of the wedge. One notes the care with which the high Fb is distinguished by the composer from the low Fp. When we hear the penultimate Fb as “correct” and ignore the last dotted arrow on figure 6.4, we get a sense of “ordinal expansion’”’ over the phrase as regards not just w spans but also I spans. First I projects at ordinal distance 1 and w at distance 2; next I projects at distance 2 and w at distance 3; finally I and w both project at distance 5, and w projects at distance 6.

a) 6, Ty

$9 by a et 1 2 3,7 3 4s46te7 8 ‘10

9) | 2, Ts |

! 2 3 4 1,5 Tg 6 12,8Ts9 10 FIGURE 6.5

The injection function, like IFUNC earlier, enables us to discover and explore relationships our ears might not otherwise pick up quickly. Figure 6.5(a), for instance, elaborates on the observation that there are 4 transformations f of form (n,T,) such that INJ(X¢, X$)(f) is non-zero, allowing Fb to represent F natural. So there are four T, arrows on the figure, projecting all of X+ progessively into all of X$ at a variety of ordinal distances. This contrasts sharply with the absence of any I or w arrows on figure 6.4 that led from anywhere within X+ to anywhere within X°8. So far as the two tetrads in the

vocal melody are concerned, we may put it that I and w are “internal” transformations, while T, is “progressive.” The structure of figure 6.5(a) 1s

hard to pick up by ear alone because its predominant ordinal rhythm of ‘*3 Jater’’ conflicts both with the motivic rhythm of the music, and with the 132 —_-2-later rhythm established in figure 6.4 within X¢.

Generalized Set Theory (2) 6.2.3

The structure of figure 6.5(b) is also easy to pick up by inspecting INJ(X2, X32), hard to pick up at first by ear alone. It is reasonable to pay some attention to the hexad X{ here because that set spans the transfer of the low Fb to the high Fb in the melody, the first repeated pitch-class of the series. Inspecting the internal structure of the hexad with our machinery, one notes

that there are relatively many transformations f of form (n, T,)-for-some-n such that INJ(X2, X2)(f) is high or positive. The arrows on the figure show how this works out analytically. For transformations f involving other T,, not as many analogous arrows would appear. The strong ordinal rhythm of the arrows on figure 6.5(b) is supported by the contour of the pitches—C, Ab, and both the Fps are all turning points—and also by the rhythm and meter of the music. In this context, the melodic rise over the octave Fp takes place through

an ornamented arpeggiation of an augmented triad Fb, C, Ab, Fp. One recognizes the triadic set of pitch classes from earlier discussion. It is Z,, the Seufzer triad into which the E-wedge will converge at the big downbeat near the end of the piece. And so on. One could combine the pitch classes or the pitches of the melody into formal pairs not only with the ordinal numbers 1 through 10, but also with the durations of the written notes, or with the time points at which the notes are attacked, or with the time spans of the written notes in our noncommutative GIS, and so on. One would get interesting results in each case. I have used ordinal numbers here because they furnish some new kinds of ideas about non-mensural rhythm, and because they guarantee that none of our transformations on the space of elements (n, p) can be operations. Even if OP is an operation on pitch classes, like I, the transformation (5, OP) cannot map our pair-space onto itself: There is no (n, p) such that (n + 5, OP(p)) = (2, q). So using the ordinal-number model gives INJ another opportunity to show how smoothly it handles transformations that are not operations.

So far as the “new kinds of ideas about non-mensural rhythm” are concerned, we can take note that our pair-space is one useful way to model serial melody. Later we shall explore other interesting models for representing series of pitches or pitch classes (or anything else). As I mentioned earlier, the reader who is interested can find a more ample analysis of “Angst und Hoffen”’ for its own sake elsewhere in my writings. The interested reader might also wish to consult my analytic remarks elsewhere on

“Die Kreuze,” Number 14 from Pierrot Lunaire.* Ideas of wedging and inversion are also engaged there. The pertinent wedge transformations are wedging-to-(C/C#) and wedging-to-(F#/G). w“'“* maps F#-to-F-to-E-to-Ep-

to-D-to-C#-to-C# and G-to-Ab-to-A-to-Bh-to-B-to-C-to-C. wF#© transforms the pitch classes in analogous wise with respect to the focal goal-dyad 2. “Inversional Balance as an Organizing Force in Schoenberg’s Music and Thought,” Perspectives of New Music vol. 6, no. 2 (Spring-Summer 1968), 1-21. The discussion of “Die

Kreuze”’ is on pages 4-8. 133

6.2.4 Generalized Set Theory (2) Fit/G. Figure 6.6 sketches a sense of how these transformations pertain to the opening of “Die Kreuze.” w C/CE:

,_ -o— ba

—————

_. O wFH/G: $—w——b a

{o—s— bo

FIGURE 6.6

No pitch class p satisfies w/“#(p) = F#. Likewise no p satisfies w'*/"(p) = Dp. This feature of the transformations is actually projected by the music, as the figure shows: The first F# and the last Db sit out their respective wedgegames. IG = IS* naturally functions prominently in connection with the two wedges. IB also figures in the music; it structures the second chord of figure 6.6 as an “internal” transformation.

Let us stand back for a moment and think about the analytic uses to which we have put INJ so far. Nowhere in the discussion of the Schoenberg pieces have we used the word “interval” or even invoked the concept, except so far as it is implicit when we label certain operations as T,9, Ts, and so on. Nowhere, therefore, have we needed to use the fact that the family of pitch

classes is a GIS. Nor did we need to suppose that our melodic space of elements (n, p) was a GIS, which in fact it was not. We have nowhere needed to

suppose that the transformations we were inspecting were 1-to-! or onto; many in fact were not. From all this we get some idea of how generally the INJ construct can be applied in how great a variety of situations. We shall increase

our sense of that variety now by studying another application of INJ toa situation not directly involving a GIS for the space S of elements. 6.2.4 EXAMPLE: Our space of elements for this study will be the family PROT

of protocol pairs. A protocol pair is an ordered pair (p,q) of distinct (NB) chromatic pitch classes.*7 There are thus 132 = 12 times 11 protocol pairs. A twelve-tone row can be regarded as a certain set within PROT: The pair (p, q) 3. More generally, we could consider protocol pairs of distinct objects from any finite family, and the mechanics of our discussion coming up would obtain, so far as the theory was

134 concerned.

Generalized Set Theory (2) 6.2.4

is in the set if and only if p precedes q in the row. Note that while the row imposes a certain ordering on the twelve pitch classes, the set under consideration is an unordered subset of PROT, 1.e. an unordered collection of pitchclass pairs. Besides rows, we can consider other subsets of PROT that are consistent with our intuitions of “ordering pitch classes.” To be consistent in this way, a set X must satisfy two conditions. First, we cannot intuit both p-preceding-q and q-preceding-p. Second, if we intuit p-preceding-q and g-preceding-r, then we intuit p-preceding-r. These two conditions translate into the two formal properties following, POI and PO2, which X must satisfy as a collection of pairs. (PO1): There is no (p,q) in PROT such that X contains both (p, q) and (q, p).

(PO2): If (p,q) and (q,r) are members of X, then so ts (p,r). Mathematically, a subset X of PROT that satisfies (PO1) and (PO2) is called a (strict) partial ordering of the pttch classes. The specia! partial order-

ings that correspond to rows are the “linear’’ or the “stmple” orderings L; these subsets of PROT satisfy in addition the condition (SIMP) below. (SIMP): For any (p,q) in PROT, either (p, q) or (q, p) belongs to L.

The set-theoretic condition matches our intuition that either p will pre- , cede q in the row, or q will precede p. Representing twelve-tone rows as linear orderings ts attractive in many ways. For one thing it makes all rows conceptually equal. That is, it does not assign explicit or implicit priority to one row (e.g. the chromatic-scale row), from which other rows are explicitly or implicitly derived. The model assumes no a priori ordering of the pitch classes; any row orders them as well as any other row. This is very much 1n the spirit of the classical twelve-tone method. Other attractive features of the model will become apparent presently. In connection with the melody from Schoenberg’s “Angst und Hoffen’”’ a little while ago, we brought attention to the way in which series of pitches, pitch classes, and the like could be represented by pairs (n, p) consisting of ordinal numbers n and objects p. Now we have a different way of representing such series, provided they are non-repeating (NB). Our new representation allows us to apply set theory to a linear ordering L, together with its various transforms and other partial orderings X of interest as subsets of PROT. The old model represents the row of Schoenberg’s Fourth Quartet by a family of pairs (1, D), (4, Bb), (3, A), (2, C#), ... and so on: The first note of the row is D, the fourth note is Bp, the third note is A, the second note is C#, and so on. The new model represents the same row by the family of pairs (A, Bb), (D, Bb), (Ce, A), (D, C#), ... and so on: A precedes Bb, D precedes Bb, C# precedes A,

D precedes C¥#, and so on. 135

6.2.4 Generalized Set Theory (2) No matter which formal model we use, it will still be convenient to use the notation D-C#—A-—Bb-... for quick perusal. Partial orderings that are not rows can model many structures of interest in twelve-tone theory. The partial orderings X, and X, on figure 6.7 exemplify only two such types of structure from among many.

Xi : E-A-Bb L,: A-B)-E-D-Eb-C#-G-F-F¥ -G#-B-C

X,: B-D-Eb G-Bb-F E-Cg-F%

C-A-Ab

L,: Bb-F¥-B-Ab-G-A-Eb-Db-D-F-C-E FIGURE 6.7

X, models the small linear motive E-A~—Bp. As a subset of PROT, X, contains the three protocol pairs (E, A), (E, Bp), and (A, Bp). The reader may check formally that this 3-element set satisfies conditions (PO1) and (PO2). X, models our being sure of the three cited precedence relations, and unsure of or indifferent to any other precedence relations. L, is what I take to be “the row” of Schoenberg’s Moses und Aron; as a subset of PROT, L, contains the pairs

(A, Bh), (A,E),..., (A,C); (Bb, E), (Bb, D),..., (Bb,C), (E,D),..., (E,C);...; (B,C). X, models an aggregate governing the soprano, alto, tenor, and bass voices of the four-part texture at the beginning of Variation 3 in Babbitt’s Semi-Simple Variations.* X, contains the twelve pairs (B, D), (B, Ep), (D, Ep);

(G, Bb), (G, F), (Bb, F); and so on. L, models what I take as the row of the piece, that is, the succession of pitch classes formed by the first twelve notes in the soprano voice.

We shall be examining various numbers INJ(L,, X,)(f) and INJ(L,, X,)(f) in connection with a few analytic observations on the two pieces. The transformations f on PROT which will attract our interest are of types (1)-(4) following. (1) Transpositions of pairs, T;(p, q) = (T,(p), T,(q)); (2) inversions of pairs, I{p,q) = (I(p), I(q)); (3) retrogression of pairs, R(p,q) = (q, p); (4) combinations of types (1) through (3). These transformations are all well defined on PROT; they do not map any protocol pair into some pair (q,q) whose members are not distinct. The transformations are in fact oper4, Christopher Wintle provides a very useful analytic study of the piece in “Milton Babbitt’s Semi-Simple Variations,” Perspectives of New Music vol. 14, no. 2 and vol. 15, no. | 136 (Spring—Summer/Fall—Winter 1976), 111-54.

Generalized Set Theory (2) 6.2.4

ations on PROT. They form a group isomorphic to the twelve-tone group of operations on rows. Furthermore, when we identify rows with subsets L of PROT, the sense of any row-operation coincides with the sense of the corresponding set-operation. E.g. if the set L corresponds to a certain row, then the set T,RI(L), as determined by the operations T,, R, and I on PROT, corresponds to the row formed by inverting, retrograding, and T,-ing the given row.

Let J = JE, the inversion operation that maps the pitch class A to the pitch class KH. Then the small linear motive X, can be extracted from the row J(L,). We can see this by writing out the inverted row and italicizing the entries

FE, A, and B} of the motive X, as they come along in X,-order: J(L,) = E-E}—A—B-B}b-C-F#-.... Another way of expressing the phenomenon is to point out that all three of the member pairs of X, are member pairs of the inverted row: E precedes A in X, and also in J(L,); A precedes B> both in X, and in J(L,); E precedes B} both in X, and in J(L,). That is, the three mem-

bers of X,, (E, A), (A, Bb), and (E, Bb), are also all members of J(L,. We reformulate this observation once more: There are three members of L, whose J-transforms are members of X,. And finally, we can reformulate the observa-

tion into our present terminology most concisely: INJ(L,, X,)J) = 3. Are there other inversion operations I such that INJ(L,, X,)() = 3? As it turns out here, there are not: J is the only one of the twelve inversion operations with that property. Working backwards through the semantic equivalencies of our observations in the paragraph above, we can interpret our most recent observation as telling us that J(L,) is the only inverted form of L, that serially embeds the small linear motive X,. We might say that X, has a high “signature value” for J(L,) among the twelve inverted forms of L,: If we sense that an inverted form is at hand and we intuit the three protocol pairs of

X, clearly, that is enough information, abstractly, to identify J(L,) as the specific form at hand. This property of X, was noted by Michael Cherlin in connection with events near the opening of act 1, scene 2 in Moses und Aron.> The scene portrays the brothers meeting in the desert; it begins with a lot of Aron music, light-textured, grazioso, piano and pianissimo, scored for solo flute accompanied by violins, harp and pianissimo horns. Then, just before Aron starts to sing, there 1s one measure of Moses music, scored for loud trombone and string bass. The trombone plays the short linear motive X,, extracted from the row-form J(L,). The “signature motive” X, is here attached to Moses as he steps forth on stage. Aron immediately thereafter begins to address Moses, singing the prime row-form combinatorial to J(L,) and then J(L,) itself. X, sounds particularly powerful here because it rearranges the opening trichord

Yale University, 1983). 137 5. The Formal and Dramatic Organization of Schoenberg’s Moses und Aron (Ph.D. diss.,

6.2.4 Generalized Set Theory (2) of L,, a trichord which had a strong tonic character as a harmony during the preceding (opening) scene of the drama. Babbitt has published a discussion of just such small linear signature motives in his own music.® He gives the row of his composition Reflections as L = C-B—D-A-Db-Bb-—E-F-—G—Ep-—Gp—Ab. Then he points out that the

ordered trichord X = B~D-A is “uniquely characteristic [of the row] to within the transpositional sub-array, and of [one inverted form] to within the inversional sub-array.” In our terminology, INJ(L, X)(T;) is less than 3 unless 1= 0; also INJ(L, X) (IJ) is less than 3 unless I is the one specific inversion J which Babbitt singles out. He then goes on to discuss the ordered tetrachord Y = B-D-A-—Dp. He notes that Y “is unique ... for the total array.” In our terminology, INJ(L, Y)(f) is less than 6 for the forty-seven twelve-tone oper-

ations f other than f = Ty; L itself is the only twelve-tone form of L that contains all of the six protocol pairs of Y. Y ts a signature for L among its forty-eight forms; X is a signature for L among its twelve transposed forms; X is also a signature for J(L) among the twelve inverted forms of the row.

Now let us turn out attention back to X, and L, on figure 6.7. The ageregate X,, considered as a subset of PROT, contains 12 member pairs. Hence INJ(L,, X,)(f) can be at most 12, iff is an operation. (In that case f will map N distinct members of L, /-to-/ into N distinct members of X,, so that N must be 12 or Jess.) The forty-eight specific transformations f that interest us here are in fact operations. As it turns out, none of our forty-eight operations f actually embed X, in some form of L,, satisfying INJ(L,, X,)(f) = 12. However there are operations f that do 45 of the job, satisfying INJ(L,, X.)(f) =

11. These operations are f = T,,f = RT,,f = J, and f = RT,J, where J is the inversion Top Since the row is its own retrograde at the tritone, these four operations generate only two forms of L,, namely T,(L,) = RT,(L,) and J(L,) = RT,J(L,). Figure 6.8 shows how well the ordering of the aggregate X, fits into each of these row-forms.

(a) X2 (b) X2-in-T, (L2) (c) X,-in-J (L2)

B-D-Eb |B D Eb BD Eb

G-Bb-F G Bb F G (F Bb)

E-Cf-F¢ E (FE C#) E Ct Fe C-A-Ab C A Ab C A Ab FIGURE 6.8 6. “Responses: A First Approximation,” Perspectives of New Music vol. 14, no. 2 and 138 vol. 15, no. | (Spring—Summer/Fall~Winter 1976), 3-23. The discussion is on page 10.

Generalized Set Theory (2) 6.2.4

The figure makes it visually clear how X, fits ‘4 within’’ either row. Figure 6.8(b) shows how only the pair (C#, F#) of X, is not within the row T,(L.,); the row contains instead the protocol pair (F#, C#). Figure 6.8(c) shows how only the pair (By, F) of X, is not within the row J(L,); the row contains instead the protocol pair (F, Bb). If only the tenor voice of X, went E-—F#-—C# instead of E-C#—F¥#, then the embedding of figure 6.8(b) would be perfect. Or, if only the alto voice of X, went G-F-—Bb instead of G—Bb-F,

then the embedding of figure 6.8(c) would be perfect. Or, yet again, if only the tenth and eleventh notes of all the rows involved were exchanged, then both embeddings would be perfect. This urge to ““make small adjustments”’ with one set or another will be further discussed later. It is a typical feature of situations in which an INJ function almost attains a theoretical maximum possible value. X,, as mentioned before, is the aggregate governing soprano, alto, tenor, and bass voices at the opening of Variation 3 in the Semi-Simple Variations. Various (12—tone) forms of X, govern SATB relations throughout Variation 3. SATB aggregates of similar format govern other variations, but none of

those fit more than ‘42 within’ any form of the row L,. To put it in our terminology, if X 1s an aggregate governing SATB anywhere in the piece outside Variation 3, the INJ(L,, X)(f) is at most 10, for each operation f we are considering. Thus we can say that the SATB-aggregates of Variation 3 are maximally compatible with forms of L,, compared to such aggregates from other variations. Statements of this sort are very useful to express structural differences among sections of a piece that sounds at first extremely homo-

geneous in texture throughout. That is particularly so when the statements can be backed up by precise measurements like 44, 75, and the like. INJ helps us pinpoint and explore precisely other structural differences

among sections of the composition. For example, let V and V’ be SATBaggregates from any one variation; then INJ(V, V’)(T,) = either 0 or 4. That is, V and V’ will either have no common pairs or exactly 4 common pairs. Let

Vi, V2,---, Vs be SATB-agegregates from the first, second, ..., fifth variations; then with two exceptions INJ(V,, V,,)(To) is less than or equal to 2. That ts, with two exceptions, aggregates V,, and V, from different variations will have only 2 or fewer common pairs. Since two distinct aggregates in this format could theoretically share as many as 11 common pairs, we can say that the level of “ordering cross-talk’”’ between variations is very low, half as low as the level of cross-talk within each variation (INJ(V, V’)(T,) = 4). Indeed that

latter level (4 pairs out of a possible 11) is itself none too high. The two exceptions are these: INJ(V,, V2)(T,)) = 4and INJ(V,, V2)(T,)) = 5. We can say that Vartations 5 and 4 thus “‘talk with’ Variation 2, so far as SATBaggregate ordering goes, at a level equalling or even surpassing the level of cross-talk within each individual variation.

INJ(V4, V2)(To) = 5 is a maximum compared with other values of 139

6.3 Generalized Set Theory (2)

INJ(V,,,V,)(T,). This observation suggests we devote special attention to aggregate-relations between V, and V.,. And when we do so, we shall notice a feature of the composition we might not quickly have come to notice otherwise. Whichever V, and V, we select to represent Variations 4 and 2 respectively, the two SATB-aggregates will share exactly one 3-note linear segment

from among the four segments D#-B-E, Ab—C-—G, C#-FH-D, and Bp—F-A. If V,, and V, come from variations other than 4 and 2, the two SATB-aggregates will not share any 3-note linear segments. Let us call the family of four segments listed above the “pivot aggregate.” Figure 6.9 shows how the pivot aggregate controls the tenor and bass voices of Variation 2, and the soprano and alto voices of Variation 4. The bar lines on the figure mark off SATB-aggregates within each variation. (a) Variation 2 S: A:

T: D#-B-E | C$-F#-D | Ab-C-G | Bb-F -A B: Ab-C-G | Bb-F -A | D#-B-E | C#-F#-D (b) Variation 4

S: Ab-C-G D%-B -E | Bb-F-A C#-F%-D A: Bb-F-A C%-F#-D | Ab-C-G D$-B -E T: B:

FIGURE 6.9

INJ numbers bring quickly and effortlessly to our attention the fact that the relationship of figure 6.9 between Vartation 4 and Variation 2 is a unique relationship between variations in the piece; it is not a ubiquitous feature of a large-scale design. As noted before, this sort of observation is very useful in bringing to our attention special discriminations within a composition that sounds at first extremely homogeneous.

6.3 In this section we shall explore further the notion of “if-only adjustments”’ in connection with INJ. The abstract notion can be formulated as 140 ~— follows. Suppose INJ(X, Y)(f) is near its theoretical possible maximum in a

_ Generalized Set Theory (2) 6.4

certain situation. The number might be close to the cardinality of X for instance, so that almost all of X is mapped into Y by f. Or f might be 1-to-1 and

the injection number might be very close to the cardinality of Y, so that the transformed set f(X) comes close to embedding Y. In such cases, a small adjustment in X or in Y might enable us to remove the ‘‘almost’”’ component of

the situation, bringing the injection number up to its theoretical maximum value. The parts of X or Y that do not quite fit may come under pressure to conform, giving rise in the music to urges for generating new material. We encountered an if-only situation in discussing the Semi-Simple Vari-

ations. Figure 6.8 earlier showed how the SATB-aggregate X, was almost embedded in the rows T,(L,) and J(L,); more precisely ‘43 embedded.” While discussing the figure, we mused about making if-only adjustments in either the aggregate or the row, to make the embedding work completely. So far as I can tell, the speculative adjustments do not correspond with musical pressures in Babbitt’s piece.

But in the Schoenberg song we examined earlier, similar speculative adjustments do correspond with strong musical pressures. In examining the progression of the 3-note chord X = Angst to the 3-note chord Y = Hoffen, we observed that INJ(X, Y)(w*) = 2 out of a maximum possible 3, and that INJ(X, Y)(D) also = 2 out of a maximum possible 3, ‘I’? here meaning IE = Ip). “If only” the Fb of chord Y were adjusted to F natural, we noted, both the injection values of 2 above would rise to the maximum 3. In this connection the if-only speculation led to fruitful analytic ideas about the “missing F natural,” the “missing I-partner,”’ the missing lover, begehren, Fb as functional substitute for F natural, and so on. The interested reader will find an extensive treatment of if-only adjustment and its analytic implications elsewhere in my writings, in connection with the opening two chords of Schoenberg’s piano piece op. 19, no. 6.7 I call the chords “rh” and “lh” for right hand and left hand. 6.4 The cited article goes on to discuss “‘progressive”’ and “‘internal’’ transformations in connection with the succession rh—lh. “Progressive” transfor-

mations make rh into something much like lh; “internal” transformations make rh into something much like itself, or lh into something much like itself, or both. The reader will recall that we used this nomenclature earlier, in connection with various transformations pertaining to harmony and melody in the song “Angst und Hoffen.”’

We shall now extend the nomenclature and put it into a completely 7. “Transformational Techniques in Atonal and Other Music Theones,” Perspectives of New Music, vol. 21, nos. 1-2 (Fall~Winter 1982/Spring-Summer 1983), 312-71, especially 336— 42. lam indebted to Michael Bushnell for having observed the rewards that this sort of approach brings in analyzing the music. He worked with transposition operations only; I enlarge his aural

field to include inversions as well. 14]

6.4 Generalized Set Theory (2)

general theoretical setting, invoking only a family S of elements, certain transformations f on S, and the INJ function. Given sets X and Y, suppose we are inspecting the values of INJ(X, Y)(f), INJCX, X)(f), and INICY, Y)cf) as f varies over a certain family INSPECT of transformations.

For certain transformations f within the family INSPECT, the value of INJ(X, Y)(f) will be maximal, or at least relatively high subject to the constraints of the situation. We shall call these transformations progressive. They

map a lot of X into Y, For certain transformations f within the family INSPECT, the value of INJ(X, X)(f) or INICY, Y)(f) will be high. We shall call these transformations

X-internal or Y-internal accordingly. A transformation which is both Xinternal and Y-internal can be called “internal (for the progression X-—Y).” An X-internal transformation maps a lot of X into X.

Intuitively, an X-internal transformation tends to extend/elaborate/ develop/prolong X in the music, while a progressive transformation tends to urge X onwards, to become something else (like Y). Progressive and internal transformations will tend to combine mathematically in certain interrelated ways, by their very natures. If I transform X to be much like itself, and then transform the result to be much like itself, it is likely that the composition of the two gestures will make X much like itself. That is, the composition of two X-internal transformations will tend to be X-internal. Similarly, the inverse of an X-internal operation will tend to be

X-internal. Similarly, an X-internal transformation followed by an X-Yprogressive transformation will tend to be an X-—Y-progressive transformation; and an X—Y-progressive transformation followed by a Y-internal transformation will tend to be X—Y-progressive.

As a result, when we inspect the families of progressive and internal transformations pertinent to a given X-—Y situation, we shall find those families tending to interrelate algebraically according to the considerations just surveyed.

We can introduce other useful nomenclature. An f such that INIJ(X, X)(f) is minimal or at least relatively small, given the constraints of X and INSPECT, can be called X-external. Such an f maps X largely outside itself. We can also define a dispersive transformation to be one that maps X largely outside Y, makeing the value of INJ(X, Y)(f) minimal or relatively small. These definitions avoid mentioning the complements of the sets X and Y in S, which may not be “sets” according to our definition if S is infinite.

External and dispersive transformations tend to enter into typical algebraic relations with themselves, with each other, and with internal and progressive transformations. An X-internal transformation followed by an Xexternal one will tend to be X-external; a progressive transformation followed by a Y-external one will tend to be dispersive; and so on.

142 A good example of dispersive transformations is furnished by measure 8

Generalized Set Theory (2) 6.4

a) b) c)

of Schoenberg’s op. 19, no. 6. This is the cryptic, very dense measure that precedes the final return of the chords rh and lh in measure 9. Figure 6.10(a) reproduces measure 8.

aan mi Sehr Jangsam (¢ )

-—3— T2(th) Ts(th) T7(th) To(th) To(th) Ih

P+. Ce 2 eee po iJ 2 | | | NS

a L333 A SC————— S| Laer Sf

FIGURE 6.10 , genau im Takt (no common tones)

Forte has noted that the music embeds many forms of the rh chord.® For our purposes, we can observe that the music embeds four transposed forms of

rh. Those are the four shown in figure 6.10(b): T,(rh), T,(rh), T+(rh), and To(rh). Now if we let the interval 1 range from 0 through 11, we shall find that there are six values of i for which INJ(rh, lh) (T;) = 0; these values arei = 2, 4,

5, 7, 9, and 0. That is, T,, T,, T;, T,, To, and Ty are the six dispersive transposition operations, given the progression rh-to-lh. To put it another

way, T,(rh), T,(rh), T;(rh), T;(rh), T,(rh), and T,(rh) = rh are the transposed forms of rh that have no common tones with lh. As figure 6.10(b) shows us, four of these six forms are embedded within the music of figure 6.10(a).

(The rest of that music does contain common tones with lh.) And, as figure 6.10(c) shows us, a fifth dispersive form, T,(rh) = rh, ensues immediately thereafter, joining the parade of dispersive forms and thereby linking measure 8 to the downbeat of the final reprise at measure 9.

The relevance of external transformations to traditional theory is illustrated by the “‘semi-combinatorial hexachord.” If X is a 6-note set of pitch classes and I is an inversion operation that transforms X into its complement X, then I is an X-external transformation: INJ(X, X)(I) = 0. This is a good place to think about exploring how the injection function relates with set-conplementation when S is finite. We shall soon carry out that exploration, in section 6.6. Before that, it will be helpful to prove a theorem and a corollary about INJ(X, Y)(f) when f is an operation OP.

8. The Structure of Atonal Music, example 102 (p. 99). 143

6.5.1 Generalized Set Theory (2)

6.5.1 THEOREM: If f is an operation OP, then INJ(X, Y)(OP) is the cardinality of OP(X) cq Y, that is the number of common members shared by the sets OP(X) and Y. We took note of this theorem informally in the commentary following

Definition 6.2.1 earlier. The fact is important enough to warrant formal verification.

Proof: Let M = INJ(X, Y)(OP); let N = card(OP(X) qn Y). Let x,,x,,..., Xx, be the distinct members of X that map into Y via OP. Since OP is 1-to-1, the

elements OP(x,), OP(x,), ..., OP(X,y) are distinct members of OP(X)n Y. Thus OP(X) Y contains at least M distinct members; N is greater than or equal to M. Now let y;, y2, .--, Yn be the distinct members of OP(X) 7 Y. Then

OP~'(y,), OP “(y2), ..., OP (yy) are N distinct members of X, each of which maps into Y under OP. So there are at least N distinct members of X that map into Y under OP; M ts greater than or equal to N. q.e.d. The injection function applied to operations thus generalizes Regener’s Common-Note Function, which was developed for the special case in which X and Y are sets of pitch classes and OP runs through the twelve transposition operations.”

6.5.2 Coro vary: If fis an operation OP, then INJ(Y, X)(OP) = INJCX, Y)(OP*).

Proof: An element z is a member of the set OP! (X) OY if an only if OP(z) is in X and z is in Y; this is the case if and only if OP(z) is in X and

OP(z) is in OP(Y), that is if and only if OP(z) is a member of the set Xn OP(Y). So the transformation OP maps the set (OP~*(X) q Y) 1-to-1 onto the set (X MOP(Y)). Therefore the two sets have the same cardinality: card(OP(Y) m X) = card(OP7!(X) mY). Applying Theorem 6.5.1 to both sides of this equation, we infer the formula of the Corollary. Now we are ready to explore set-complementation in connection with INJ. It will simplify matters greatly to restrict our attention to transformations f that are operations OP in this context. We shall then be able to use the theorem and the corollary we have just proved. We must restrict the abstract setting by supposing S to be finite; then the complement X of a set X will bea formal “set” by our criterion, 1.e. a finite subfamily of S. 6.6.1 THEOREM: Suppose S is finite. Given sets X and Y with complements X

and Y; given any operation OP; then formulas (A) through (E) below obtain. 9. Eric Regener, “On Allen Forte’s Theory of Chords,” Perspectives of New Music vol. 13,

144 no. 1 (Fall-Winter 1974), 191-212. The Common-Note Function is defined on page 202.

Generalized Set Theory (2) 6.6.2

(A): INJ(X, Y)(OP) = cardX — INJ(X, Y)(OP). (B): INJ(X, Y)(OP) = cardY — INJ(X, Y)(OP). (C): INJ(X, Y)(OP) = cardY — cardX + INJ(X, Y)(OP). (D): If cardY = cardX, then INJ(X, Y)(OP) = INJ(X, Y)(OP). (E): (Generalized Babbitt Hexachord Theorem) If cardX = 4cardS, then INJ(X, X)(OP) = INJ(X, X)(OP). Proofs: (A): The operation OP maps each member of X either into Y or into Y. So

cardX = INJ(X, Y)(OP) (the number of X-members mapped into Y) plus INJ(X, Y)(OP) (the number of X-members mapped into Y). The formula follows. This argument works just as well for any transformation f. (B): Here it is essential that OP be an operation, so that we can apply

Corollary 6.5.2. We write INJ(X, Y)(OP) = INJ(Y, X)(OP“?), via 6.5.2. This, via formula (A) just proved, = cardY — INJ(Y, X)(OP™). And, applying 6.5.2 again, we infer that this number is indeed card Y — INJ(X, Y)(OP), as claimed.

(C): INJ(X, Y)(OP) = cardY = INJ(X, Y)(OP), via (B); this is cardY — (cardX — INJ(X, Y)(OP)), via (A); this is cardY — cardX + INJ(X, Y)(OP), as desired. (D) is an obvious corollary of (C). And (E) is an obvious corollary of (D),

setting Y= X. q.e.d. The methods of proof I have used are essentially Regener’s. I have called formula (E) the Generalized Babbitt Hexachord Theorem because Babbitt’s theorem, somewhat disguised, is a special case of this formula. For readers who may feel the disguise is perfect, I shall show the connection. Let X be a sixnote pitch-class set; let OP be a transposition operation T,. Theorem 6.6.1 (E) above tells us that INJ(X, X)(T,) = INJ(X, X)(T,). Theorem 6.5.1 then tells

us that the cardinality of (T,(X)- X) is the same as the cardinality of (T,X) mn X). Now the cardinality of (T,;(X) m X) is the number of members of X that lie the interval i from some member of X. In other words, the number is IFUNC(X, X) (i). Similarly, the cardinality of (T,(X) 7 X) is IFUNC(X, X) (i).

We have shown, then, that IFUNC(X, X)(i) = IFUNC(X, X) (i) for every 1: the complementary hexachords contain the saine number of i-dyads for each 1. This 1s Babbitt’s theorem. One sees that 6.6.1(E) is a very broad generalization. 6.6.2 EXAMPLE: Let us see how the Generalized Hexachord Theorem applies in another specific context. The reader will recall the space PROT of protocol

pairs which we constructed earlier (6.2.4) in examining sometopicsfromserial 145

6.6.2 Generalized Set Theory (2) theory. We noted that the various twelve-tone rows can be regarded as those subsets L of PROT that are linear orderings on the pitch classes. In this model, the retrograde of the row L corresponds to the set-theoretic complement L of the set Lin PROT. Fora pair (p, q) is in the complement of the set if and only if p does not precede q in the row, which is the case if and only if p precedes q in the retrograde of the row, which 1s the case if and only if (p,q) lies in that portion of PROT corresponding to the retrograde row. The family PROT has 132 = 12 times 11 members. And any row, as a subset of PROT, contains exactly 66 protocol pairs. To see that, suppose the

pitch classes of the row come in the serial order p,, p,,..., Py. Then the subset L of PROT contains 11 pairs of form (p,,p,), and 10 pairs of form (p.,p,),--., and | pair of form (p,,,p,). The cardinality of the set L is thus 11+ 10 +--+ + I, which is 66.

We are thus in a setting to which 6.6.1(E) pertains. card(S) = card(PROT) = 132; card(X) = card(L) = 66 = 4card(S). In this setting, a row and its retrograde (complement) play a set-theoretic role formally analogous to that of a hexachord and its complement in traditional atonal theory. Here, 6.6.1(&) tells us the following. Let OP be any operation on PROT. Given any row L with retrograde L, let N be the number of pairs (p,q) in L such that the pair OP(p, q) is also a precedence relation in L; let N’ be the number of pairs (p’, q’) in L such that OP(p’, q’) is also a precedence relation in

L. Then N = N’. In this connection, OP(L) need not itself be a row. Indeed L itself can be replaced by any set of cardinality 66 within PROT and the theorem remains

ture, “L” now being simply the set-theoretic complement of L. But the application is of particular interest when we interpret the complementary sets L-and-L as row-and-retrograde.

The twelve-tone operations T; and I are induced on PROT by corresponding operations on individual pitch classes: T,(p,q) = (T;(p),T,(q)); I(p,q) = (1(p), I(q)). But in general there need not be any operation op on pitch classes such that OP(p, q) = (op(p), op(q)). There is no such op, for instance, in the case of the retrograde operation R on PROT: R(p, q) = (q, p). Nor is there such an op for any of the operations to which R contributes, e.g. RT, and RI. One can also construct fancier operations on PROT not induced by pitch-class operations op. For example: If p and q are in the same whole-

tone scale, OP(p, q) = (p,q); if p and q are in opposite whole-tone scales, OP(p, q) = (T3(q), T9(p)). When L is a row and OP(L) ts also a row, then INJ(L, L)(OP) measures

the size of the largest partial ordering on the pitch classes which can be embedded in both OP(L) and L, i.e. whose protocol pairs are compatible with

both those rows. That is because INJ(L,L)(OP) is the cardinality of 146 OP(L)L, as we know from 6.5.1.

Generalized Set Theory (2) 6.7.2

Now we shall show formally how INJ completely generalizes IFUNC when there is a GIS at hand. 6.7.1 THeorem: Let (S, IVLS, int) be a GIS. Then for each interval i and for all sets X and Y, IFUNC(X, Y) (i) = INJ(X, Y)(T;,).

Proof: Let IFUNC(X, Y)(i) = M; let INJ(X, Y)(T;) = N. We shall see that N must be at least as big as M, and that M must be at least as big as N.

Since IFUNC(X, Y)(i) = M, there are M distinct pairs (x,,y,), (X5,¥>),---, (Xu Yu) Such that x,, lies in X, y,, lies in Y, and int(x,,, y,,) = 1. For each such pair, y,, = T;(x,,). For m and n distinct, x,, and x,, are distinct members of X. (Otherwise we would have y,, = T,(Xn) = Ti(Xa) = Yx» Whence the pairs (xX, Yn) and (x,, y,) would not be distinct, contrary to supposition.)

Thus X has at least M distinct members whose i-transposes lie in Y. That 1s, INJ(X, Y)(T;) is at least as big as M. Or: N is at least as big as M. Now let z,, Z2,..., Zn be the N distinct members of X whose i-transposes

lie within Y. (There are N such, since INJ(X, Y)(T;) = N.) For each such Z,>Z, 18 in X, u = T,(z,) is in Y, and int(z,, u) = 1. Therefore the pair (z,, u) was counted as some (X,,, Ym) above. So every one of the N elements z,,..., Zn is one of the M elements x,,..., Xy. Hence M is at least as big as N._ q.e.d. The logic of Theorem 6.7.1 can be visualized through the following aid. Imagine X and Y as two finite configurations of points in the Euclidean plane. Suppose 11s the vector (directed distance) “‘to the right and up 30 degrees for a distance of 5 inches.” We can ask: “From points of X to points of Y, how many distinct arrows can J draw that go to the right and up 30 degrees for a distance of 5 inches?’”’ The answer to this question is IFUNC(X, Y)(). We can also ask: “If I move the whole X-configuration to the nght and up 30 degrees for a distance of 5 inches, how many points of the displaced configuration will

then coincide with points of Y? The answer to that question is INJ(X, Y)(T;,). One intuits easily that the two questions are logically equivalent. Next we shall explore what happens to INJ when the sets X and/or Y are transformed by some operation A.

6.7.2 THEOREM: Given a family S of objects, given sets X and Y, given a transformation f on S and an operation A on 8S, then formulas (A), (B), and (C) below obtain. (A): INJ(A(X), Y)(f} = INJ(X, Y) (fA). (B): INJ(X, ACY))(f) = INJ(X, Y)(A“'f).

(C): INJ(A(X), ACY)) (f) = INICK, Y)(A7!fA). 147

6.7.2 Generalized Set Theory (2)

Proofs: (A): INI(A(X), Y)(f) is the number of t within A(X) such that f(t) is a member of Y. Set t = A(s); then the family of such t is in 1-to-1 correspondence, via A, with the family of sin X such that fA(s) is a member of Y. And the number of such s is exactly INJ(X, Y) (fA). (B): INJ(X, ACY)) (f) is the number of's in X such that f(s) is a member of A(Y). Now f(s) is a member of A(Y) ifand only if A~’ f(s) isa member of Y. So

the number at issue is the number of s in X such that A7'f(s) belongs to Y. And that is exactly INJ(X, Y)(A7'f). (C): The formula follows at once from (A) and (B). _ q.e.d. Formula 6.7.2(C) is of particular abstract interest. We can imagine that

the shift from X-and-Y to A(X)-and-A(Y) reflects a “modulation” of the system by the operation A. For instance, if we are in a GIS and A is a transposition or an inversion, we are transposing or inverting (the sets of) the system accordingly. It is natural to ask: “If we modulate the system by A, what

effect does that have on the INJ function?” At first one might suppose that INJ would remain unaffected by the modulation: INJ(ACX), A(Y)) = INJ(X, Y). But, as formula 6.7.2(C) shows, that is not in fact the case. Rather the INJ function ts itself “‘modulated”’ according to the formula. We noted a specific instance of this general phenomenon earlier, during

our analysis of “Angst und Hoffen.” The wedge-structure of figure 6.3(b) converged on E; it was studied in connection with the transformations w* = wedging-to-E and I = inversion about E. The wedge-structure of figure 6.3(c) converged on F#; it was studied in connection with the transformations wF* — wedging-to-F# and J = inversion about F#. To get from the situation of (b) to the situation of (c), one ‘“‘modulates the system by T,.” The following

equations obtain: Ft = T,(E); w'* = T,w®T Z!; J=T,IT>!. The first equation relates the focal points of the two wedges, also the bass notes of the Angst and Hoffen chords. The second equation leads, via 6.7.2(C), to the relationship INJ(T,(X), T,(Y))(w**) = INJ(X, Y)(Tz!w*?T.) = INJ(X, Y)(w*). The third equation leads via the same formula to the relation-

ship INJ(T,(X), T.CY)) (J) = INICK, Y)(Tz'JT,) = INI(CK, Y)(D. = Thus, when we modulate the system from E-centricity to F#-centricity via the operation T,, then the wedge w** plays the role, with respect to the modulated sets T,(X) and T,(Y), that the wedge w® originally played with respect to the sets X and Y. Similarly, the inversion J plays the role, with respect to the modulated sets, that the inversion I originally played with respect to the unmodulated sets. The reader will recall, perhaps, our earlier remarks on this subject

by way of preparation (pp. 128-129).

Here is another, more abstract, example of system-modulation. Let X be an atonal hexachord that inverts into its complement via the inversion I=I1§$. Then INJ(X, X)(1) = 0. Suppose some music projecting X-and-

148 —its-complement “modulates” to a new section projecting T,(X)-and-its-

Generalized Set Theory (2) 6.7.3

complement. Here I plays the role of “f” and T, plays the role of “A” in Formula 6.7.2(C). We cannot suppose that INJ(T,(X), T,(X))Z) = 0: the new hexachord T,(X) will not invert into its complement via the inversion I,

B/C inversion. Rather, T.(X), the modulated hexachord, inverts into its complement by the inversion J = T,IT~{. Formula 6.7.2(C) tells us this: INJ(T,QO), T,QO)0) = INI(X, X(T-3IJT.) = INI(X, XD) = 0.

Using formulas 3.5.6 (A) and (B), we can compute J = T,I$T, = I¢T, = If. Thus T,(X) inverts into its complement about E and F (or about Bb and B). The system having modulated by T,, the transformation J = T,IT~ now plays the role that the transformation I originally played. Formula 6.7.2(C) tells us this sort of thing in great generality: When a system modulates by an operation A, the transformation f’ = AfA7! plays the structural role in the modulated system that f played in the original system, in the sense that INJ(modulated X, modulated Y(f') = INJCX, Y)(f).

6.7.3 Theorems 6.7.1 and 6.7.2 enable us to generalize the abstract questions about IFUNC we asked earlier, toward the end of the section 5.1. We asked, for instance, under what circumstances in a GIS we would have IFUNC(X,, X,) = IFUNC(X,, X,). Via 6.7.1, we can rephrase the question, asking under what circumstances in a GIS we shall have INJ(X,, X,)(T,) = INJ(X,, X,)(T,) for every transposition T,. And that question can easily be generalized: Given any family S of objects and any family INSPECT

of transformation on S, under what circumstances shall we have INJ(X,, X)G) = INICK,, X,)) for every member f of INSPECT? We do not have to demand that S be in a GIS, or that INSPECT be a group; indeed, the question makes sense even if INSPECT is not a closed family (semigroup) of transformations.

Likewise, we earlier asked under what conditions in a GIS we would have IFUNC(X,, X,) = IFUNC(Y,, Y,). We can generalize that question analogouly: Given any family INSPECT of transformations f on a family

S of objects, under what conditions shall we have INJ(X,, Xf) = INJ(Y,, Y,)(f) for every member f of INSPECT? The ideas of 6.7.2 enable us to expand that question even farther. Suppose we have a family S of objects, a family INSPECT of transformations f on S, and a group MDLT of “modulating operations” A on S. Under what conditions, given sets X,, X, Y,, and Y., will there exist some modulation A such that when we modulate Y, and Y, by A, obtaining Y’, and Y’,, we

shall have INJ(X,, X,)(f) = INJ(Y’,, Y’,)(f) for every f in INSPECT? Via 6.7.2(C) this amounts to demanding that INJ(X,, X,)(f) and INI(Y,, Y¥,(A7'fA) be equal numbers, for each member f of INSPECT. In the special case where we are in a GIS, where INSPECT is the family of trans-

positions, and where MDLT is the group of transpositions, the question asks under what conditions, given the four sets, there will exist some interval 149

6.7.4 Generalized Set Theory (2)

j such that for every interval 1, IFUNC(X,, X,)(i) = IFUNC(T,(Y,), T,(Y¥2))@) = IFUNC(Y,, Y2)(jij™*). (5.1.6 gives us the last equality. It is consistent with 6.7.2(C) because T,,,-. = Tj*T,T,;: The group of transpositions iS anti-isomorphic to the group of intervals.) Besides using 6.7.1 and 6.7.2 to generalize earlier questions, we can also use them to help out with computations we earlier found difficult to execute. The formula following is a good example.

6.7.4 THEOREM: In any GIS let I = I’. Fix any referential element and set j = LABEL()). Then for any sets X and Y, and for each interval 1, IFUNC((X%), I(Y))G) = INJ(X, Y)(P,) where P, is the interval-preserving operation labeled by k = ji7+j™?.

Proof (optional): IFUNC(I(X), ICY))@) = INJWCX), ICY))(T;), via 6.7.1. This, via 6.7.2, = INJ(X, Y)(I-'T, I). Set I-/T,I = OP. It now suffices to prove that OP = P,, where k = ji7’j7?. I7* = 12(3.5.9). So OP = I¥T,I¥. We can compute T,I% by 3.5.6(A); the result is IX, where x = T,(u). So OP = IVIX. And we can compute the

composition of the two inversions by 3.5.8. It is P,,T,, where m= LABEL(u)LABEL(x)7? and n = LABEL(v) 'LABEL(v). Here n =e, so we have computed that OP =P,,, where m = LABEL(u)LABEL(x)"!. Since x = T,(u), LABEL(x) = LABEL(u):1(3.4.3). Replacing LABEL(u) by j, we then have m = j(ji)™’ = ji+j7? = k. Thus OP = P, as desired.

6.8 To demonstrate further the generalizing powers of INJ, we shall now use it to generalize Forte’s K and Kh relations. For our generalization we suppose only a family S of objects and a group of operations on S which we shall call “‘canonical” for reasons known only to us. We shall denote the group by CANON. The complement of the set X will be denoted X. If S is infinite, X will not be finite, hence not a “‘set’’ in our terminology. Nevertheless we shall speak of its “cardinality,” understanding the value infinity for the expression cardX. If S is finite, cardX will mean the finite cardinality of the set X. We shall restrict our attention to sets X and Y such that cardX < cardX and cardY < card Y. We can do this because Forte’s K and Kh relations are

not affected by the restriction. If cardX should be less than cardX, we can simply exchange the roles of the sets X and X as they do or do not enter into K or Kh relations with other sets. Having made that restriction, we may also suppose that cardX < cardY. Otherwise, we can simply exchange the roles of the sets X and Y in the arguments coming up. So our restrictions, in sum, are these: cardX < cardX, cardX < cardY