straus - uniformity, balance, and smoothness in atonal voice leading

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Uniformity, Balance, and Smoothness in Atonal Voice Leading Author(s): JOSEPH N. STRAUS Source: Music Theory Spectrum, Vol. 25, No. 2 (Fall 2003), pp. 305-352Published by: {oupl} on behalf of the Society for Music TheoryStable URL: http://www.jstor.org/stable/10.1525/mts.2003.25.2.305Accessed: 21-10-2015 01:48 UTC

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305

Uniformity, Balance, and Smoothness in Atonal Voice Leading

joseph n. straus

This paper offers a broadly applicable model for atonal voice leading, a model of pitch-class coun-terpoint to connect any two harmonies. Voice leadings are evaluated by three criteria: (1) unifor-mity: the extent to which the voices move by the same interval distance and thus approach tradi-tional transposition; (2) balance: the extent to which the voices move by the same index numberand thus approach traditional inversion; and (3) smoothness: the extent to which the voices travelthe shortest possible distance. The most uniform, most balanced, or smoothest way of movingfrom one set to another in pitch-class space, or from one set class to another in a proposed voice-leading space, provides a standpoint from which to assess any specific compositional realization inpitch space.

i. introduction

Theories of atonal music have traditionally been better atdescribing harmonies—at devising schemes of classificationand comparison—than at showing how one harmony movesto another.1 In recent years, however, a number of studieshave attempted to shift the theoretical focus from harmony

to voice leading.2 Approaches to atonal voice leading havegenerally had one of three distinct theoretical orientations.First, theorists have sought to adapt or extend models oftonal voice leading, including particularly the Schenkerianmodel, to atonal music.3 A second approach has describedthe linear organization of atonal music in terms of lines ofpitches linked by shared register, timbre, metrical placement,or other contextual means.4 A third approach takes recentwork by David Lewin as its point of departure and adopts

Earlier versions of this paper were presented at The Graduate Center ofthe City University of New York, New York University, the Summer2000 meeting of the Princeton Theory Group, and the 2001 AnnualConference of the Society for Music Theory. I benefited from com-ments from the audience on each of those occasions. I also receivedguidance from several friends and colleagues who read earlier drafts:Cynthia Folio, Ed Gollin, Henry Klumpenhouwer, Philip Lambert,and Shaugn O’Donnell. In creating the computer programs that under-pin the methodology presented in this article, Christopher Ariza sug-gested numerous important refinements in my approach. Programs areavailable at Ariza 2002

1 As Maisel 1999 has observed: “For tonal music, we have two theoriesthat stand in metatheoretical relation to each other: that is, harmonyand counterpoint. While we have in set theory a working theory of har-mony for post-tonal music, we hardly have the barest beginnings of atheory of counterpoint” (177–8).

2 For a valuable survey of recent approaches, see Morris 1998.3 Straus 1997a calls this approach “prolongational” and provides an ex-

tensive bibliography. More recent contributions to this literature in-clude Vaisaala 1999. From a non-Schenkerian point of view, Lerdahl1989, 1999, 2001 derive a model of atonal prolongation from Lerdahl& Jackendoff 1983. A related perceptual perspective is adopted inHuron 2001. The prolongational approach is critiqued in Straus 1987.See further discussion in Larson 1997 and Straus 1997b.

4 Straus 1997a calls this approach “associational” and provides an exten-sive bibliography. For exemplary instances of this approach, see Forte1988 and Morris 1987, which presents a theory of compositional de-sign in which “lynes” of pitch-classes are projected musically in a variety of ways.

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what Lewin calls a “transformational attitude.”5 This ap-proach has flowered in a number of directions.6

ii. transposition and inversion

The present study adopts a transformational approach,and identifies transposition and inversion as sources of atonalvoice leading.7 As shown in Example 1, when Set X is trans-posed (a) or inverted (b) onto Set Y, each note in X mapsonto a corresponding note in Y. These mappings take placein pitch-class space, and are thus independent of the registralorder of the notes in X and Y. As such, they comprise a po-tential “pitch-class counterpoint” for atonal music.8 Indeed,

I propose to understand such mappings as transformationalvoices within a transformational voice leading that arises fromharmonic successions motivated by transposition or inversion.9

The kind of pitch-class counterpoint I am describing hasalways been implicit in atonal set theory. Any time we saythat two chords are related by transposition or inversion, weare also saying that each note in the first chord maps onto acorresponding note in the second, thus creating a network oflinear connections between the chords. The approach Iadopt here can thus be thought of as set theory in which asense of linearity and movement has been brought to thefore.

The transformational voice leading I describe here distin-guishes pitch-class voices from actual musical lines, realizedin register. This basic distinction is analogous to the distinc-tion in tonal theory between voice and part.10 Schenkerian

306 music theory spectrum 25 (2003)

5 Straus 1997a calls this third approach “transformational” and providesan extensive bibliography. The phrase “transformational attitude” andmuch of the theoretical apparatus associated with it are found in Lewin1987. Lambert 2000, 45–6, refers to “the basic reorientation thatLewin’s approach requires—the shift of focus away from equivalencerelations among individual pitch-class collections and toward transfor-mational processes that relate them.”

6 The two principal current offshoots that bear on atonal voice leadinginvolve Klumpenhouwer networks and Neo-Riemannian Theory (seespecific references later in this article).

7 The present study takes as its point of departure Straus 1997a, as well asKlumpenhouwer 1991, O’Donnell 1997, and especially Lewin 1998. Ina more general sense, I am indebted to Morris 1995b and Morris 1998.

8 The phrase “pitch-class counterpoint” is taken from Benjamin 1981, 4:“As I understand it, a harmonic progression is not a succession of verti-cal complexes so much as it is a counterpoint of lines. What separatessuch a counterpoint from what we ordinarily call counterpoint is thatthe lines in a harmonic progression are PC lines. Whereas the struc-tures of traditional counterpoint—such as Fuxian species counterpoint—are defined in pitch-specific terms, the structures of harmony arehere, by definition, conceived of in PC terms, that is, without referenceto the octave placement or irreducible elements—scale degrees—in anindividual harmonic progression. Therefore, whereas traditional coun-terpoint distinguishes between the bass voice and upper voices, or be-tween perfect fourths and perfect fifths, harmony as I conceive it makesno such distinctions and regards even the matter of melodic contour, sobasic to counterpoint, as foreign.”

9 My sense of atonal voice leading as the pitch-class counterpoint in-duced by certain transformations is derived from Klumpenhouwer 1991and O’Donnell 1997, and explored at length in Straus 1997a. See alsoLewin 1998, Gollin 1998, and O’Donnell 1998. In some of thesesources, the transformations in question are not the traditional transpo-sitions and inversions.

10 As Wen 1999, 277, observes: “I should like to establish a distinction be-tween ‘voice’ and ‘part.’ Although these terms are often used inter-changeably (and indeed frequently overlap), it can be helpful to employthem in two different ways. In this divided usage, the term ‘part’ is de-fined by a work’s performing forces or textural elements, and is usuallybound by the constraints of a particular tessitura (e.g., SATB). A ‘voice,’ on the other hand, is not dependent upon the limitations of themedium expressing the musical idea. A part can usually be identified by simply viewing the score; a voice is a linear succession that mighttraverse two or more parts. Such an ‘ideal’ conceptual entity can be anobject of musical experience because its continuity results from basicproperties of the tonal system: the melodic fluency produced by step-wise succession, the need to prepare and—especially—resolve disso-nances, the affinity subsisting between members of the same chord, andthe connection of high and low registers effected by octave equiva-lence.” Similarly, Karpinski 2000, 125–6, states: “The distinction be-tween part and voice is an important one. A part is a feature of the mu-sical surface, a line performed as a continuity. In ensemble music, a part

679-125.MTS.Strauss_pp305-352 8/28/03 2:24 PM 06



theory, for example, is replete with melodic techniques thatdepend on this distinction, including substitution, unfolding,reaching over, and motion into and out of an inner voice.More elementary tonal phenomena, like the transfer of reso-lution of a dissonance, also suggest the conceptual indepen-dence of voice and part.

In tonal theory, generally speaking, as in the transforma-tional voice-leading theory described here, voices are opera-tional and systematic whereas parts or lines are contextual. Intonal music, of course, the relevant transformations are notthe Tn and In that are so characteristic of atonal music, butrather idiomatic tonal harmonic progressions that are mod-eled in recent literature as DOM, SUBD, MED, REL,PAR, LT, and so on.11 These idiomatic transformations in-duce a leading of voices that may or may not correspond to

the motion within any particular registral or instrumentalpart.

I do not want to insist too much on the analogy betweentonal and atonal voice leading, however, as there are also sig-nificant differences between tonal voices as generally under-stood and the transformational voices I describe here. In thisstudy, a voice is simply the mapping of pitch-classes inducedby transposition or inversion. As I will use the terms here,voices are transformational (they result from pitch-classmappings in pitch-class space) while lines are contextual(they result from associations of register or timbre in pitchspace).12 An underlying counterpoint of pitch classes, in-duced by transposition and inversion, is what I will calltransformational voice leading.

Example 2 shows the transformational voice leading induced by transposition in two atonal passages. In (a), thefirst two chords are related by transposition at T8, and, infact, each note in the first chord simply moves down four

uniformity, balance, and smoothness in atonal voice leading 307

is just that—the notes played or sung from a single printed part (e.g.,“Bassoon I”; “Altos”). In keyboard music, parts are those lines separatedfrom one another by the musical texture (often pointed up by com-posers or editors through the use of markings such as stem direction,slurs, and dotted lines). A voice, however, is a theoretical construct.Upper voices are the abstract lines formed by the smoothest possiblevoice leading from one prevailing harmony to another.”

11 See Lewin 1987, 175–8, for definition of these transformations, whichalso figure prominently in the Neo-Riemannian literature.

12 Gollin 1998 makes the same distinction between “transformationalvoice leading” (i.e., the linear connections induced by transformationalmapping) and “registral voice leading” (i.e., what I am referring to as“registral lines”). Similarly, O’Donnell 1998 speaks of “transformationalvoices,” by which he also refers to the mappings induced by some transformation.

(a) (b)(n is an ordered pc interval) (n is an index number)x1 n y1 x1 n y1x2 n y2 x2 n y2x3 n y3 x3 n y3

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example 1. Voice leading from set X to set Y via transposition (Tn ) or inversion (In ).




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example 2. Voice-leading induced by transposition in (a) Webern, Movements for String Quartet, op. 5, no. 5,mm. 2–6 and (b) Webern, Movements for String Quartet, op. 5, no. 2, mm. 1–3.

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semitones onto its counterpart in the second chord. Thelines beneath the score show the note-to-note mapping in-duced by the transposition. In a similar fashion, the third,fourth, and fifth chords are also related by transposition, andagain each instrumental line simply follows the transposi-tional path: T7, followed by T2, composing to T9.

13 For eachof the transpositions in Example 2(a), the transpositionalmappings are realized by the registral, instrumental lines—meaning that the voices and lines discussed above here coin-cide. It is also worth noting that the quasi-ostinato melodyin the cello includes frequent reiterations of a closely relatedtranspositional sequence: T4 (the complement of T8) fol-lowed by T9.

In Example 2(b), the second and third chords are relatedby transposition at T4.

14 But there is a voice crossing: thehighest note in the second chord, A, can be heard as moving,via T4, onto the lowest note in the third chord, C�. Mean-while, the other two notes in the second chord (E � and G)ascend four semitones to their destinations in the thirdchord (G and B). This transpositional gesture, and particu-larly the voice-leading from G to B, repeat the initialmelodic leap in the viola.

The registral lines are also of interest: the highest de-scends 2 semitones (A–G); the middle descends 4 semitones(E �–B); and the lowest descends 6 semitones (G–C�). Thesethree intervals (2, 4, and 6), heard descending in the progres-sion of chords, are the intervals heard ascending in the viola’sinitial three-note figure (G–B–C�).15 Furthermore, becausethe chords in mm. 2–3 are related by transposition to the ini-tial motive and to each other (the final chord has the same

pitch-class content as the initial motive), the intervals be-tween the chords duplicate the intervals within the chords.In this progression, the transformational voices (A–C �,E �–G, and G–B) and the registral lines (A–G, E �–B, andG–C �) do not coincide, but each provides a valuable per-spective on the linear organization of the passage.

In both passages in Example 2, transposition maps eachnote in one set onto some corresponding note in another. Theresulting pitch-class counterpoint comprises a transforma-tional voice leading for each progression. Musical lines can beformed in many different ways—through association basedon register or instrumentation, for example—but transfor-mational voices result from an operation of some kind. Inthis case, it is transposition that leads the voices from chordto chord.

Inversion works the same way: it maps notes in one har-mony onto corresponding notes in the next and thus createstransformational voices. In Example 3(a), the first and sec-ond tetrachords are related at I9.

16 The motion in the bassline, F–E, defines an axis of inversion around which theother voices flip. Each note in the first chord is mapped ontoits inversional partner in the second: F–E, C� –G�, D–G, andF �–E �.17 In the lower two parts, these inversional mappings

16 The first two tetrachords in Example 3(a) can also be understood as re-lated by transposition at T2—this kind of transpositional/inversionalambiguity always obtains when the harmony in question is inversionallysymmetrical. The voice leading induced by the transpositional hearingof this progression involves a complete registral inversion of the tri-chords in mm. 2 and 5 (F–G, C � –E�, and D–E) and supports themelodic motion F � –G � that comes between the trichords. My segmen-tation of the piano part into tetrachords is not the only, or the most ob-vious, possible parsing. Nonetheless, it is not difficult to hear the pas-sage in this way, and this hearing provides the benefit of set-classrecurrence.

17 Note that when two chords are related by inversion, the distance tra-versed by the voices is identified as an index number (sum), not an in-terval. In Example 3(a), each voice moves by index 9; in Example 3(b),each voice moves by index 7. The idea of moving by an index numbermay seem strange initially—we normally think of voices as moving by

13 No connection is shown between the second and third chords, or between the fifth and sixth chords, because they are not related bytransposition. This is an important theoretical problem to which wewill return.

14 The first chord is left out of the discussion for now because it is not re-lated to the others by transposition.

15 The three-note motive G–B–C� is identified by Lewin 1982–83 as a“Hauptmotiv” for this passage.




take place within a registral line; in the upper two parts, thevoices cross from one registral line to another.

The F–E in the bass is simultaneously a transformationalvoice (in which F moves onto E via I9) and a registral line, inwhich F descends a semitone to E. In that motivic, interval-lic sense, the descending semitone is also featured promi-nently in the cello’s melodic line, in mm. 2–3 and 9–10. Thetransformational voice leading and the registral lines thusboth contribute to the linear organization of the progression.18

In Example 3(b), the soprano and bass lines are alsotransformational voices, moving by I7, but the tenor and altocross. In addition, there is a voice exchange: E and D � in the


intervals, not by sums. In talking about inversion, the index numbersoffer simply an arithmetic convenience. The actual motion should beunderstood as a flip around an axis. In Example 3(a), to say that thevoices move by index 9 is to say that the voices flip around a shared axis,which, in this case, is defined by the pitch classes E and F. Similarly, inExample 3(b), the voices share an index number of 7, which is to saythey flip around a common axis defined, in this case, by D � and E.

18 As in Examples 2(a) and (b), one chord has been left out of the discus-sion temporarily because it is not related to the others by transpositionor inversion.

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example 3. Voice-leading induced by inversion in (a) Webern,Three Pieces for Violoncello and Piano, op. 11, no. 3 and (b)Ravel, Le Tombeau de Couperin, “Forlane,” m. 1.




first chord exchange with D � and E in the second.19 Ofcourse, it is also true that in this progression the accompany-ing chord, an augmented triad, simply moves down 3 semi-tones, in contrary motion with the melody. Indeed that isprobably one’s first impression of the linear connection be-tween these chords. At the same time, however, the inver-sional relationship, and the linear connections it induces, arealso audibly present. As with transposition, the voices cre-ated by inversion interact in interesting ways with the regis-tral and instrumental lines. Sometimes the lines and voicessupport each other; at others, there is tension between them,and the voices cross.

iii. uniformity and balance

Transposition and inversion, however, are inherently lim-ited in their ability to forge connections among chords. Inthe atonal literature, it is quite common to find apparentprogressions of harmonies not related by transposition or in-version and thus not members of the same set class. It is de-sirable, then, to have a theoretical model that accommodatesprogressions among diverse harmonies and describes thevoice leading that results.

One possible solution would be simply to abandon trans-position and inversion and look for other sources of voice-leading continuity.20 But I am reluctant to do so. Transpo-

sition and inversion lie at the core of most of our theories ofatonal harmony, and it is appealing to keep them at the coreof a theory of atonal voice leading. Their musical bindingpower has been amply demonstrated. But clearly the con-cepts of transposition and inversion must be expanded andgeneralized to accommodate voice leading between non-equivalent harmonies.

To lay the groundwork for such a generalization, it is useful to think about voice leading in a basic way. Imaginepitch-class sets X and Y of cardinality n, their elementsarranged in registral order from highest to lowest, and avoice leading that connects each note in X to some note in Y.If X and Y are singleton sets, as in Example 4(a), only onevoice leading is possible: x1—y1. That is, x1 (a pitch class inset X) moves onto y1 (a pitch class in set Y). If X and Y aredyads, as at (b), two voice leadings are possible (x1—y1, x2—y2 and x1—y2, x2—y1). If X and Y are trichords, (c), six voiceleadings are possible; if X and Y are tetrachords, (d), 24 voiceleadings are possible; and so on. In other words, there are n!(n factorial) voice leadings, where n is the cardinality of thesets involved.21

Example 5 shows cases in which X and Y are of differentcardinality. Here, a single note in the smaller set may splitonto two or more notes in the larger one. Conversely, two ormore notes in the larger set may fuse into one note in thesmaller set.22 Although the sets in Example 5 differ in sizeby only one, they may in principle differ by any amount. Anynote in the smaller set may split onto any two or more notesin the larger set; any two or more notes in the larger set mayfuse onto any note in the smaller set. As many splittings andfusings will occur as are necessary to map each note in one


21 For practical reasons, Example 4 considers only sets of cardinality 1through 4.

22 I adopt the terms “split” and “fuse” from Callender 1998. But whereasCallender’s splittings and fusings involve exclusively motion by semi-tone, I impose no such restriction. Also, unlike Callender, I permit anote to split onto more than two destinations and more than two notesto fuse into one.

19 In tonal music, a voice exchange normally suggests the prolongation ofa harmony that contains the exchanged notes, but in atonal music, voiceexchanges normally suggest pitch-class inversion around the axis cre-ated by the exchanged notes. See Straus 1997a for discussion.

20 This is the approach taken in Roeder 1994, which conceives of voiceleading as “the intervals between registrally corresponding members ofsimultaneities, which are collections of simultaneously attacked pitches”(41). See also Roeder 1984 and 1989. Indeed, most recent approachesto atonal voice leading have seen only a limited role for transpositionand inversion. This is doubtless due to the inherent limitation of theseoperations: they permit connections only between sets that are mem-bers of the same set class.




(1)x1 y1

(a) X and Y are singletons (one voice-leading)

(1) (2)x1 y1 x1 y1x2 y2 x2 y2

(b) X and Y are dyads (two voice-leadings)

(1) (2) (3) (4) (5) (6)x1 y1 x1 y1 x1 y1 x1 y1 x1 y1 x1 y1x2 y2 x2 y2 x2 y2 x2 y2 x2 y2 x2 y2x3 y3 x3 y3 x3 y3 x3 y3 x3 y3 x3 y3

(c) X and Y are trichords (six voice-leadings)

(1) (2) (3) (4) (5) (6)x1 y1 x1 y1 x1 y1 x1 y1 x1 y1 x1 y1x2 y2 x2 y2 x2 y2 x2 y2 x2 y2 x2 y2x3 y3 x3 y3 x3 y3 x3 y3 x3 y3 x3 y3x4 y4 x4 y4 x4 y4 x4 y4 x4 y4 x4 y4




(d) X and Y are tetrachords (twenty-four voice-leadings)

example 4. Voice leading between set X and set Y (same cardinality).




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y3 y3 y3 y3 y3 y3

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example 5. Voice leading between set X and set Y (different cardinality).




set onto some note in the other. For practical reasons,Example 5 is confined to sets of cardinalities 1 through 4.

Strictly speaking, the voice leadings of Example 5, involv-ing sets of different cardinality, are no longer transforma-tions, as the mappings involved are not one-to-one.23 None-theless, they can still be thought of as “transformational” in a looser sense to suggest some kind of dynamic action (asopposed to a static, “relational” mode of conception). In theapproach adopted here, something happens to Set X thatmaps its pitch classes to the pitch classes of Set Y, with allthe notes of X and Y covered in the process—no note is leftout.24 In the process, notes split or fuse to accommodate dif-ferences in cardinality between X and Y.

When X is a singleton and Y is a dyad, there will be onlyone voice leading, as the single note in X splits onto the twonotes in Y. When X is a dyad and Y is a trichord, there willbe six voice leadings, as either of the two notes of X maysplit onto any of three pairs of notes within Y. When X is atrichord and Y is a tetrachord, there will be thirty-six voiceleadings, as any of the three notes in X may split onto any ofsix pairs of notes within Y, and the remaining notes of Xhave two ways of moving onto the remaining two notes ofY.25 The chart in Example 6 summarizes the number ofvoice leadings between any two sets X and Y of cardinalityone through six.

Amid this profusion of possibilities, this study will evalu-ate voice leading by two criteria: uniformity and balance. Uni-formity refers to the extent to which the voices move by thesame intervallic distance. The more uniform the voice lead-ing, the more closely it approximates traditional transposi-

tion. Balance refers to the extent to which the voices fliparound the same axis of inversion, that is, the extent towhich they move by the same index number. The more bal-anced the voice leading, the more closely it approximates tra-ditional inversion. Uniformity and balance thus representgeneralizations of transposition and inversion for the purposeof describing voice leading.26

iv. uniformity

Example 7 addresses the voice leading between twopitch-class sets of the same size: {F, F �, B} and {G, B �, D},two trichords that are related by neither transposition norinversion.27 As with any two trichords, there are six distinct


23 See Gollin 2000, 347–69, for definition and discussion of functions, op-erations, and transformations.

24 My approach thus differs from that taken in Lewin 1998, which doesnot insist that every note in X have a destination in Y.

25 The formula for calculating the number of voice-leadings (the numberof “onto functions”) from an n-element set to a k-element set, and itsproof, may be found at: www.research.att. com/cgi-bin/access.cgi/as/njas/sequences/eisA. cgi?Anum=019538.

26 To deal with the large numbers of voice leadings and to sort them accu-rately as to their relative uniformity or balance obviously requires exten-sive computational assistance. The initial computer work on this projectwas done by Dave Smey. More recently, Christopher Ariza, in thecross-platform Python language, has developed sophisticated, flexible,and fast software for all of the calculations described in this article. Theprogram is free, runs on every platform, and is licensed under the GPLopen-source license. Downloads are available at his website, Ariza 2002.

27 Note that these are pitch-class sets moving in pitch-class space. Thesets are arranged on the staff purely for visual convenience. No voiceshould be understood as conceptually higher or lower than another.

1 2 3 4 5 61: 1 1 1 1 1 1

2: 2 6 14 30 623: 6 36 150 540

4: 24 240 15605: 120 1800

6: 720

example 6. The number of voice leadings between any two setsX and Y of cardinalities 1–6.




ways of leading the notes of the first set onto the notes of thesecond. Within each voice leading, each voice traverses anordered pitch-class interval.

Example 7 arranges the six voice leadings in order accord-ing to their relative uniformity, measured by the extent towhich the voices move by the same or nearly-the-same in-terval. The first voice leading closely approximates a transpo-sition at T3. The bass actually does move by T3. The soprano,however, moves by T2 (a semitone “too low”) and the altomoves by T4 (a semitone “too high”). This first voice leadingis defined as offset by 2 semitones from a transposition at T3—indicated by the (2) at the bottom of the analytic diagram.

This paper relies on offset as the principal means formeasuring degree of uniformity. But there are two othermeasures that seem equally plausible, or nearly so. The firstjudges voice leadings as relatively uniform depending on thenumber of voices that move by the same interval, a conditiontermed consistency. If all of the voices move consistently by

the same ordered pitch-class interval, then the voice leadingis entirely uniform (and the two sets are related by transposi-tion at that interval). If all but one of the voices move by thesame ordered pitch-class interval, then the voice leading canbe understood as nearly uniform because of the inconsistentbehavior of the one voice.28 In general, the greater the num-ber of voices that participate in an actual transposition, themore uniform the voice leading. In Example 7, the secondvoice leading is an instance of “near-transposition”: two ofthe three voices move by the same interval. It is nonethelessconsidered less uniform than the first because it has a higheroffset, (3).

Another way of measuring voice-leading uniformity in-volves the extent to which the voice-leading intervals diverge.


28 When all but one of the voices move by the same interval, the two setsare related by what Straus 1997a calls “near-transposition” and Lewin1998 calls “pseudo-transposition.”

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F 2 G F 5 G F 9 G F 5 G F 9 G F 2 GF � 4 B � F �8 B � F �1 B � F �1 B � F � 4 B � F � 8 B �B 3 D B 8 D B11 D B 3 D B 8 D B11 D

*T3 *T8 *T11 *T3 *T8 *T11(2) (3) (4) (4) (5) (6)

Most uniform Least uniformMost transposition-like Least transposition-likeMost parallel motion Least parallel motion

example 7. Voice-leading uniformity in the progression from {F, F �, B} to {G, B �, D}.




If we imagine such intervals in some voice leading displayedon a twelve-place vector, with entries available for each ofthe twelve ordered pitch-class intervals, then that voice lead-ing whose entries are clustered together within the smallestspan on the vector would be judged most uniform. Thesmaller the range of difference among the voice leading in-tervals, the more uniform the voice leading. Conversely, themore widely dispersed the voice-leading intervals among the possibilities, the less uniform the voice leading.

All three measures of uniformity I have proposed—offset,consistency, and span—capture at least part of our intuitivesense of what transposition is, namely, the uniform move-ment of all the voices. But they do not always produce thesame results. Offset provides certain systematic advantages,particularly in its relationship to the notion of displacementdeveloped later in this paper. But I will also invoke consis-tency and span as the analytical situation warrants.29

The offset is the total number of semitones by which the voices would have to be adjusted to bring them all into conformity with each other.30 For the first voice leading inExample 7, I call T3 the convergence point, which is the trans-positional level from which there is the least possible offset.A voice that moves by the transpositional interval that de-fines the convergence point is indicated with a solid line;voices that diverge are indicated with a dotted line. The as-terisk attached to the T3 label indicates that the transposi-tion is not exact, with the offset number in parentheses mea-suring the extent of the deviation. Of the six voice leadings,the first is the one that is most nearly transposition-like according to the offset. The remaining voice leadings have

progressively higher offset values. The last of the six is offsetby 6 semitones from a transposition at T11—this is thus theleast transposition-like of these six voice leadings.

Example 8 assesses the uniformity of the voice leadingbetween two sets of different size: {D, G} and {A, C, C � }. Inthe first, fifth, and sixth voice leadings, the G in the first setsplits as it moves onto two notes in the second set. In theother three voice leadings, the D in the first set splits ontotwo notes in the second set. If the order of the two sets werereversed, two notes in the larger set would be understood tofuse onto either G or D in the smaller.

The first voice leading in Example 8 is the most uniformbecause its three voice leading intervals scarcely differ at all—the voice leading is offset from T6 by only two semitones.The disparity among the voice leading intervals increases asone moves through the example from left to right, with thesixth voice leading the least uniform: its three voice leadingintervals are maximally dispersed among the twelve orderedpitch class intervals, and a large adjustment of 8 semitoneswould be required to repair the deviation.31

In the six progressions of Example 8, if the order of thetwo sets were reversed (that is, if {A, C, C �} preceded {D,G}), each transposition number would be replaced by itscomplement mod 12. But the offset numbers and the rank-ing of relative uniformity would be unchanged. When themapping from X to Y has a certain offset, and thus a certaindegree of uniformity, the retrogression of that mapping willhave the same offset and thus the same degree of uniformity.Y can move to X simply by reversing the pathways from X to


29 Ariza 2002 permits exploration of the different results produced bythese three different approaches to voice-leading uniformity.

30 Different voice leadings may produce the same offset. In Example 7,the third and fourth voice leadings have the same offset (4), but pro-duce it in different ways. I will not try to establish any mechanism forbreaking ties of this kind. Rather, I will simply point them out as theyoccur.

31 In the sixth voice-leading of Example 8, there is thus ambiguity as tothe convergence point: *T2, *T6, and *T10 will all produce the same off-set (8). As with the ambiguities previously discussed (transposition orinversion [see note 16] and voice leadings sharing the same offset num-ber [see note 30]), I will not create any mechanism for resolving thisambiguity. It is simply a fact of life, to be identified and explored as itoccurs.




Y. All of the voice leadings discussed in this study are re-versible in this sense.32

There have been a number of recent theoretical attemptsto measure degrees of divergence from true transposition.Most of these have been concerned with situations in whichall but one of the voices move by the same interval.33 Of par-

ticular importance to the present inquiry, Lewin develops anotion of “maximally uniform voice-leading”: “Given pitch-class sets X and Y, a voice-leading V from X into Y will be


sets may be related by unary voice-leading transformation if all but oneof their pitch classes are held in common (that is, are related by T0 orwhatever In maps them onto themselves). Straus 1997a designates“near-transposition” and “near-inversion” to generalize this idea to otherintervals of transposition and indexes of inversion: “Two harmonies arerelated by near-transposition or near-inversion if all but one of theirnotes are related by actual transposition or actual inversion.” The possi-bility of relating sets by the simultaneous application of two or moreoperations (e.g., some notes move by Tn, some by Tn+x) is explored indetail in O’Donnell 1997 and 1998. See also Roig-Francoli 2001,which develops a notion of “pitch-class-set extension” that includes thepossibility of intervallic expansion or contraction, and Vishio 2002,which develops a concept of “skew” to measure deviation from transpo-sition and inversion.

32 Strictly speaking, my voice leadings are not retrogradable because theirmappings are not necessarily one-to-one. As a result, the functions I amdescribing can have no well-defined inverse. Nonetheless, it is perfectlyreasonable, and in conformance with musical intuitions, to welcome thepossibility that if, for example, the voice leading from X to Y has a cer-tain quality (such as a certain degree of uniformity), that the reversal ofthat voice leading in the progression from Y to X will have precisely thesame quality.

33 Forte 1988 has described what he calls a “unary transform”: “A unaryvoice-leading transformation results in the mutation of one pitch-classset into another by a change of a single element.” For Forte, then, two

C C C C C CD 7 D10 D10 D7 D 11 D

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AG 6 G 6 G2 G5 G 2 G 6

C� C� C� C� C� C�*T6 *T7 *T11 *T7 *T2 *T2,6, or 10(2) (4) (4) (6) (6) (8)

Most uniform Least uniformMost transposition-like Least transposition-likeMost parallel motion Least parallel motion

example 8. Voice-leading uniformity in the progression from {D, G} to {A, C, C �}.

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called maximally uniform if it differs as little as possible froma straight transposition.”34 He also presents the related concept of an “offset number”: “Given a maximally uniformvoice leading V from pcset X to pcset Y, the offset number ofV is the positive (absolute) real number of semitones (eitherup or down or both) by which Y differs from Tn(X), in a pertinent pitch manifestation of V.”35

More broadly, my notion of voice-leading uniformity,with its attendant generalization of the concept of transposi-tion, has its roots in Quinn 1996. Quinn imagines actualtransposition as a “crisp” extreme at one end of a continuumof transpositional effects. For Quinn (and in the presentstudy), any two sets are related by transposition in some measurable degree. If that degree is sufficiently low, it may notbe musically advantageous to invoke the relationship. Butthere will be instances where two sets are not related by traditional, crisp transposition but are still related by trans-position to some significant degree. I will say that the rela-tionship between such sets is highly transposition-like. Theconnections created by such fuzzy transpositions may serveto link harmonies that would be judged as incomparable bytraditional, crisp atonal set theory. And once two harmoniesare understood as linked by fuzzy transposition, we can tracethe voice leading that results.

v. balance

Just as transposition can be generalized into a concept ofvoice-leading uniformity, inversion can be generalized into aconcept of voice-leading balance, that is, the extent to whichthe voices can be understood to flip symmetrically aroundsome common axis. Uniformity refers to the transposition-like quality of the voice leading; balance refers to its inversion-

like quality. Example 9 returns to the progression of Example7 but reorders the six voice-leadings in order of their relativebalance. Each transformational voice now describes an indexof inversion.

The first voice leading is the most balanced, the most inversion-like, because two of its voices describe index 1(they flip around a shared axis) and the axis of the remainingvoice is offset by only two semitones. The sixth voice leadingis the least balanced, the least inversion-like, because itsthree voices describe axes of inversion that are widely dis-persed among the possibilities and would require an adjust-ment of 6 semitones to repair the deviation.

Example 10 returns to the progression from Example 8,and reorders the six voice leadings in order of their relativebalance. The offsets range from a minimum of (2) to a maxi-mum of (8). Offset numbers work for voice-leading balancejust as they do for voice-leading uniformity: they measurethe degree of deviation from normal, crisp inversion ortransposition.

Throughout this discussion of uniformity and balance, Iam imagining that each transformational voice may be heardas though it were seeking to do, to the greatest extent possi-ble, just what the other voices are doing at the same time. Indiscussing voice-leading uniformity, I invoke a potential urgetoward parallel motion—in moving from harmony to har-mony, each voice may be heard to imitate the behavior of theother voices and move by roughly the same distance. The re-sult is a kind of pitch-class organum, where each voice de-rives its sense of direction from the parallel (or nearly paral-lel) activity of the others in pitch-class space.36 In discussingvoice-leading balance, I imagine that each voice may beheard to move so as to create a sense of symmetry and bal-ance, as each note in the first chord appears to seek an inver-sional partner in the second. If the first chord is asymmetri-cal, the voices may be heard to create a sense of symmetry by


34 Lewin 1988, 31.35 Lewin 1988, 33. As noted earlier, the approach taken in this paper dif-

fers from Lewin 1998 in my insistence that every note in X have a des-tination in Y, that is, that every note in both sets be covered by, or ac-counted for by, the voice leading. Neither X nor Y can be incomplete.

36 The phrase “pitch-class organum” is Michael Friedmann’s (privatecommunication).




moving to a chord related by inversion to the first. If the firstchord is symmetrical, the voices may be heard to maintainthe sense of symmetry. Depending upon the context, trans-formational voices may be heard as though seeking to movearound a shared axis of inversion.

The idea that a pitch class may be understood as desiringto emulate the behavior of the pitch classes around it is ex-pressed with particular vividness by Lewin, who speaks ofthe “urge” and the “lust” that a note in one set might feel tobecome the transpositional (or inversional) partner of a cor-responding note in another set: “I find it suggestive to thinkof these generative lusts as musical tensions and/or poten-tialities which later events of the piece will resolve and/or realize to greater or lesser extents.”37

Traditional atonal set theory is based on the transposi-tional and inversional equivalence of pitch class sets. Two setsare either equivalent or non-equivalent—a stark either/or—and the relationships are crisp. But the methodology I havepresented here makes transposition and inversion fuzzy, notcrisp. Now, it appears, any pair of sets is related by transpositionin some degree; any pair of sets is related by inversion in somedegree. If the degree of equivalence is sufficiently low, it may


to desire partnerships with other notes. If all music-theoretical dis-course involves what Guck 1994 refers to as an “analytical fiction,” thenthe fiction of the tones having a will strikes me as a relatively attractiveone, enlivening the discourse with an appealing sense of motion andpurposeful activity. The danger of this metaphor, or any such metaphor,is that one will be seduced into believing it too literally, and thus be ledinto various kinds of conceptual distortions—Harrison 2001 discussesthis issue in interesting detail. In the present paper, however, my ascrip-tion of will to tones is always anchored in substantial, technically trans-parent descriptions. Behind every metaphorical urge and lust lies a fullytheorized musical relationship.

37 Lewin 1982–83, 341. To imagine that notes have desires, urges, lusts, orvolitions of any kind—even to imagine that notes are capable of moving—is to indulge in a familiar music-theoretical metaphor. Our discourseoften anthropomorphizes notes, endowing them with a limited degreeof human agency—we permit them to move from place to place, even

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F 3 G F 7 G F 0 G F 0 G F 3 G F 7 GF �1 B � F � 4 B � F � 8 B � F � 4 B � F � 8 B � F � 1 B �B 1 D B 6 D B 9 D B 1 D B 6 D B 9 D

*I1 *I6 *I9 *I1 *I6 *I9(2) (3) (4) (4) (5) (6)

Most balanced Least balancedMost inversion-like Least inversion-likeMost symmetrical Least symmetrical

example 9. Voice-leading balance in the progression from {F, F �, B} to {G, B �, D}.




not be musically advantageous to invoke the relationship.But there will be many instances where two sets are not re-lated by traditional, crisp transposition or inversion, but arerelated by transposition or inversion to some significant de-gree. We might say that the relationship between such sets ishighly transposition-like or highly inversion-like. The connec-tions created by such fuzzy transpositions and inversionsmay link harmonies that would be judged as incomparableby traditional, crisp atonal set theory. Thus fuzzified, trans-position and inversion can serve as an impetus for voice lead-ing, creating a counterpoint of pitch classes bound togetherby their shared, or nearly-shared, intervals of transposition orindices of inversion.

vi. smoothness

There is one additional factor to consider. It can be use-ful, as I have suggested, to imagine the pitch classes in set X

driven onto the pitch-classes in set Y by transposition or in-version (crisp or fuzzy). But it can also be useful to thinkabout the total distance the pitch classes in set X have totravel to get to their destinations in Y. Example 11 illustrateswith respect to the sets from Examples 7 through 10, withthe voice leadings now arranged in the order of their relativesmoothness.38

Voice-leading smoothness is measured by the total dis-placement, the sum of the intervals traversed by each notefrom its origin in X to its destination in Y. The intervals in


38 What I call voice-leading smoothness is also referred to in the literatureas parsimoniousness, efficiency, nearness, closeness, and proximity. Noticethat I use the term here to refer to proximity in pitch-class space, not inregistral, pitch space. The voice leadings that I am describing in pitch-class space as relatively smooth could thus be realized in relatively un-smooth ways in pitch space, and vice versa. The same is true, of course,of the voice leadings I describe as relatively uniform or relatively balanced.

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D23 D 11 D 3 D2 D2 11 D11

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C� C� C� C� C� C�*I3 *I8 *I4 *I4 *I11 *I3,7, or 11(2) (4) (4) (6) (6) (8)

Most balanced Least balancedMost inversion-like Least inversion-likeMost symmetrical Least symmetrical

example 10. Voice-leading balance in the progression from {D, G} to {A, C, C �}.





C C C C C CD2

1 D 1 D2 D5 D25

D5

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Smoothest Least smoothMost efficient Least efficientClosest, nearest Farthest, most distant

example 11. Voice-leading smoothness in the progression from (a) {F, F �, B} to {G, B �, D} and (b) {D, G} to {A, C, C �}.

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F 3 G F 2 G F 2 G F 5 G F 3 G F 5 GF �1 B � F �4 B � F � 4 B � F �1 B � F �4 B � F �4 B �B 1 D B 1 D B 3 D B 3 D B 4 D B 4 D

5 7 9 9 11 13

Smoothest Least smoothMost efficient Least efficientClosest, nearest Farthest, most distant

displacement:

displacement:

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question are unordered pitch-class intervals (interval classes).39

Given two voice leadings, the one that has the lower totaldisplacement is the smoother.40 Unlike transposition and in-version, smoothness is a quality, not an operation. It cannot,therefore, propel the voices, but it can usefully characterizethe voice leading that results from transposition or inversion.41

Voice-leading smoothness has long been of interest totheorists.42 In discussing it, I invoke an implicit desire oftones to conserve their energy, to move by the shortest possi-ble distances. Notes exert a gravitational pull on each other,and the attraction increases as the distance between themshrinks. Notes move only through an expenditure of effort,and it requires greater effort to move a greater distance. Inmoving from Chord X to Chord Y, there are many routesavailable, some smooth, short, and direct; others that are dis-tant and more difficult. There is at least potential expressivemeaning in this contrast, as composers plot their voice lead-ing courses from harmony to harmony, either following apath of relatively little resistance by moving smoothly, orpushing their harmonies along a harder path, forcing themto move relatively great distances.


39 Cohn 1998b has proposed a related definition of what he calls “totalvoice leading distance” or “directed voice leading sum,” but these arebased on the ordered pitch-class interval within each voice. This has realadvantages for systematization, as Cohn shows, but is not an effectivemeasure of the actual exertion or intervallic “work” involved. For exam-ple, if three voices move respectively by 3, 4, and 6 semitones, I wouldcall that a considerable total exertion of 13 semitones, near the theoreti-cal maximum for trichords, which is 18. Cohn’s “sum,” however, whichinvolves arithmetic modulo 12, would consider the total voice leadingdistance to be only 1, and thus equivalent to the voice leading betweentwo trichords where two of the notes are retained as common tones andthe third moves by semitone. Furthermore, in Cohn’s approach, orderedpitch-class intervals can effectively cancel each other, so if one voicemoves up a semitone and another moves down a semitone, Cohn con-siders the voice leading displacement to be 0; I think it should be 2. Ifollow Lewin 1998 and Alegant 2001 in measuring voice leading prox-imity according to the sum of the interval classes involved. In the mea-sure employed here, total displacement values are not to be takenmod12. The total displacement values for all trichords, for example,range from 0 (when set X maps onto set Y at T0 and all three notes ofthe first chord are retained as common tones) to 18 (when set X mapsonto set Y at T6 and all three voices move by tritone).

40 It is possible for two or more voice leadings to share the same total dis-placement, as in the third and fourth voice leadings in Example 11(a).It would be possible to devise an algorithm for breaking ties (for exam-ple, by seeing which voice leading has the greatest number of small in-tervals or the smallest number of large intervals), but I will not do thathere. Instead, I will simply note such ties as they occur.

41 There is a close relationship between smoothness (measured by dis-placement) and uniformity or balance (measured by offset), but it is onethat becomes apparent only at the level of the set class. Both offset anddisplacement measure deviation from an ideal state, either transposition/inversion or pitch-class identity (that is, either T0 or a self-mapping in-version). The convergence of smoothness with uniformity and balancewill be discussed later in this paper.

42 An interest in voice-leading smoothness apparently goes back as far as Marchetto of Padua’s Lucidarium of 1317–18; see Cohen 2001. Theclassic formulation is “the law of the shortest way” described in Schoen-berg 1987: “The first of these directions [for connecting chords] re-quires that in the voice leading, at first, only that be done which is ab-solutely necessary for connecting the chords. This means each voice willmove only when it must; each voice will take the smallest possible stepor leap, and then, moreover, just that smallest step which will allow theother voices also to take small steps. Thus, the voices will follow (as Ionce heard Bruckner say) the ‘law of the shortest way ’ ” (39). An interestin smooth voice leading is a central aspect of the recent outpouring ofNeo-Riemannian theory—see Cohn 1996, 1997, 1998a, 1998b, and theextensive bibliography provided in the Journal of Music Theory 42(1998): 335–41. The Neo-Riemannian enterprise is particularly rele-vant to the present project in that it detaches voice leading fromacoustical or tonal/functional issues. Lewin 1998 is an even more im-mediate source in its discussion of “maximally close voice leading,”which depends on a concept of “total shift”—a measure of the total dis-tance traveled by all of the voices added together: “the total number ofsemitones traversed by the three voices, as they move from their notesin one chord to their notes in the next chord” (24). Lewin engages fa-miliar metaphors of musical exertion in referring to “melodic hus-bandry” (30, 31), and the possibility of moving between sets “with aslittle overall strain as possible” (38).




The methodology set forth here is designed to establishvoice-leading connections between any two pitch-class sets.The sets may in principle be of any size and they need not bethe same size. Transposition (generalized as uniformity) andinversion (generalized as balance) impel the voice leading—they lead the voices. I imagine that each note in one chordseeks a path in pitch-class space onto some note in anotherchord, and that these paths are distinguished by their unifor-mity, their balance, and their smoothness. These paths com-prise a potential counterpoint of pitch classes. An actual musical passage may realize a particular counterpoint in a va-riety of ways in pitch space. A kind of dialectical relationshipbetween pitch-class counterpoint and compositional realiza-tion characterizes the analyses that follow.

vii. analyses

Example 12 revisits a passage from Example 2(a).43

Above a quasi-ostinato in the cello, the upper three partsplay a progression of six chords representing three differentset classes. As we observed previously, the first and secondchords are related by crisp transposition, as are the third,fourth, and fifth chords. The analytical problem is to connectthe second chord to the third and the fifth chord to thesixth. In the progression from the second to the third, threeof the voices deviate from the prevailing T2, but they do so aslittle as possible, by only a semitone each (Example 12[a]).In moving from the fifth chord to the sixth, the deviationfrom an actual transposition is even smaller—only one voiceis off, and it’s off by only one semitone.

The individual transpositional moves combine to create asingle, larger transpositional move shown in Example 12(b):from the first chord to the last is a fuzzy-transposition at*T5, with an extremely small offset of only two semitones.Notice that the offsets are as likely to cancel each other out

as they are to accumulate. In the second-lowest voice, for ex-ample, the motion from F � to A is a semitone “too high” forthe prevailing *T2 while the motion from F � to D � is a semi-tone “too low” for the prevailing *T10. As a result, that lineconforms in its entirety to the larger *T5, even though itcontains two smaller deviant moves. The progression as awhole, embracing members of three different set classes, canthus be heard as a single, unified gesture. Furthermore, theprogression-spanning *T5 can be heard as a large-scale state-ment of two smaller T5’s at the opening of the cello melody:the motion from F � to B, and from {G, G �, B} to {C, C �, E}.

The Stravinsky passages in Examples 13 and 14 involve asimilar confluence of fuzzy-transpositional voices with regis-tral lines.44 In Example 13, from the first chord to the thirdis an actual transposition at T9, diagrammed at (b), with eachof the three registral lines moving down 3 semitones. But, asExample 13(a) shows, they move down at different rates: thesoprano and alto move first by semitone and then by whole-tone, while the bass moves first by whole-tone and then bysemitone. This difference accounts for the small offsets inthe chord-to-chord succession.

The passage in Example 14 contains four different chordsrepresenting three different set classes. The second andfourth chords are related by transposition at T10, but the firstand third are related neither to the second and fourth nor toeach other. The voice leading that connects all four chords,however, is highly transposition-like.45

Consulting Example 14(a), we note that in moving fromthe first chord to the second, three of the four instrumentallines descend by semitone—only the viola does not partici-pate. This voice leading thus diverges from T11 with an offsetof only (3). This is also the maximally smooth voice leadingbetween these chords. The progression from the second


44 This passage has been extensively discussed in the literature, beginningwith Schenker 1926/1996. See relevant citations in Straus 1997a,which discusses it in similar terms. More recently, see Traut 2000.

45 Straus 1997a studies this passage in a related way.43 This passage is discussed in related ways in Lewin 1998 and Roig-

Francoli 2001.





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chord to the third is also maximally smooth, with a total dis-placement of only 4 (all four instrumental lines move up ordown by semitone). In terms of uniformity, the progressionmight be described either as *T1 or *T11, with an offset of(4) in either case.46 The progression from the third chord tothe fourth is not maximally smooth, but once again the most

uniform voice leading follows the instrumental lines. As withthe progression from the second chord to the third, two ofthe voices move by T11 while the other two are “off ” by twosemitones, resulting in a total offset of (4).47

If one takes the first violin as the leading part, then thesuccession of chords is *T11–*T1–*T9. If one takes the celloas the leading part, then the succession of chords is*T11–*T11–*T11. In either case, the three fuzzy-T’s combineto create a single fuzzy-*T9 that spans the progression from


47 Again, there is an ambiguity regarding the convergence and again theoffset of (4) can be produced by other voice leadings.

46 In other words, the convergence point is ambiguous—either T11 or T1produces the same offset. In addition, the offset of (4) can be producedby several other voice leadings, but these all involve intervals that di-verge more widely (that is, they occupy a larger span among the twelvepossible ordered pc intervals).

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the first chord to the fourth (Example 14[b]). Stravinsky’sregistration and instrumentation of these chords reflects theuniformity of their progression. Each instrumental line thusacts also as a transformational voice.

Example 15 revisits a passage previously discussed in connection with Example 2(b).48 The passage consists of alyrical melody in the viola accompanied by three differentchords in the second violin and cello. The second and thirdchords are related by transposition at T4, as we noted earlier.The analytical problem is how to handle the first chord,which has four notes, not three, and is not related in any ob-vious harmonic or intervallic way to the second and thirdchords.

But, as seen in Example 15(a), the first chord is related tothe second by fuzzy transposition at *T1. The tritone in thefirst chord, A�–D, moves up by T1 to the tritone in the sec-ond chord, A–E�. The other two voices are slightly off. Thenote A moves onto A at T0 (a semitone “too low”), and Fmoves onto G at T2 (a semitone “too high”). The result is afuzzy-transposition with an extremely small offset of (2).

The fuzzy-*T1 that connects the first chord to the secondcombines with the crisp-T4 between the second chord andthe third to create a fuzzy-*T5 that spans the entire three-chord progression, represented in Example 15(b). In thislarger fuzzy-*T5, as in the smaller fuzzy-*T1 it contains, twovoices move by five semitones, while a third is a semitone“too high” and the fourth is a semitone “too low.” So, fuzzy-*T1 plus crisp-T4 equals fuzzy-*T5, and the first chord isthus integrated into a coherent three-chord progression.

The transpositions of the tritone, shared among all threechords, can be heard to guide the larger progression. In thefuzzy-*T1 that connects the first two chords, the tritone A �–D moves by crisp-T1 onto the tritone A–E �, and the othertwo voices tag along as closely as they can. Similarly, in thefuzzy-*T5 that connects the first chord with the third, the

tritone A �–D moves by crisp-T5 to the tritone C �–G, andthe other two voices tag along with minimal deviations.

Furthermore, the idea of combining T1 with T4 to makeT5 is reflected in the viola melody also, once in the overallmelodic shape (G–A �–C) and twice in closely related inter-vallic cells within the melody (C–C �–A � and F �–B–G),which involve the complements of these transpositional in-tervals. But neither the harmonic progression itself nor itsrelationship to the melody can be understood without ac-cepting the progression from the first chord to the second asa fuzzy-*T1.

Offset measures the degree to which some set Y can beheard as a deformed X—that is, the amount of work thatwould be required to repair the deformation, to transform Xinto Y. It is thus a measure of the distance between two setsand, by extension, between the set-classes to which they be-long. In the passage in Example 15, the first chord is withina two-semitone adjustment of being a member of 3-8[026].If the A were an A � and the F an F �, the first chord wouldbe Tn-equivalent (at T1) to the second. Conversely, the sec-ond chord is within a two-semitone adjustment of being amember of 4-18[0147]. If the A in the second chord wereboth an A and a B �, and the G an F �, the second chordwould be Tn-equivalent to the first at T1. So, offset measuresboth the relationship between two sets and between the setclass to which they belong. The relationship between set andset class is one to which we shall return.

Voice-leading smoothness provides an additional interest-ing perspective on this progression. In moving from the firstchord to the second, there are three voice leadings that produce a minimum total displacement of (4). Webern hasarranged the chords to project one of these in register, withthe bass moving A�–G, the tenor F–E �, the alto A–A, andthe soprano D moving to an implied E�.49 In moving from


49 Actually, the voice-leading model shows the F and D in the first chordfusing onto E � in the second. But it is easier to grasp that relationship,in this particular context, if the pitch class E � is imagined as represented

48 This passage is discussed in related ways in Lewin 1982–83, Roeder1994, and Straus 1997a.




the second chord to the third, Webern could have arrangedthem to project the same minimal displacement of 4 (G–G,E �–C �, A–B). Instead, his registral lines project a voice

leading that is minimally smooth, with a maximum total displacement of (12).50 Chords 1 and 2 are harmonically dis-parate, and Webern connects them smoothly; chords 2 and 3


by two different pitches: E �3 and, an octave higher, an implied E �

4. Theother two voice leadings that produce the same total displacement of 4are D–E �, A–A, F–G, A �–A and D–E�, A–A, F–G, A �–G.

50 As noted earlier, the registral lines (G–C �, E �–B, A–G) state the inter-vals (6, 4, and 2) found in 3-8[026], the set class of the first three notesin the viola melody as well as the second and third accompanyingchords.

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are harmonically uniform (they are related by transposition),and Webern connects them non-smoothly. In this way, theregistral lines, moving with varying degrees of smoothness,interact in interesting ways with the voice-leading voices,motivated by transposition.

Example 16 revisits an entire short movement previouslydiscussed in connection with Example 3(a). The first twochords are related by I9 and, as observed earlier, the bass line(F–E) defines an axis of inversion around which the remain-ing three voices flip (see Example 16[a]).51 The third chordis not related to the second by crisp-inversion, but it is re-lated by a fuzzy-inversion at *I3, with an offset of (3). Thatoperation induces a complete registral inversion: bass and soprano exchange, as do tenor and alto. Those two inver-sional moves—crisp I9 followed by fuzzy-*I3—compose to afuzzy-*T6 that connects the first chord with the third, shownat (b).

That fuzzy-*T6 can also be understood in transpositionalterms, as a crisp-T2 followed by a fuzzy-*T4, which is shownat (c). In moving from the first chord to the second, eachnote moves onto a transpositional partner two pitch-classsemitones higher, a gesture that is reinforced by the melodicmotion F � –G � in mm. 3–4. In moving from the secondchord to the third, the transformational voices, induced byfuzzy-*T4, correspond to the registral lines. That is, eachnote in the second chord moves up by four pitch-class semi-tones (more or less) onto a note in the same registral positionin the third chord. In moving from the second chord to thethird, then, the inversional hearing (via *I3) induces a regis-tral inversion (all of the voices cross), while the transposi-tional hearing (via *T4) induces registral invariance (all ofthe voices move in parallel).

The *T4-induced voice leading from the second chord tothe third is the least smooth way of connecting these twochords. As shown at (d), it involves a displacement of (17).Webern’s registral lines, however, describe the smoothest wayof connecting the first chord to the second, with a displace-ment of only (8).52 The linear organization of the progres-sion cannot be attributed to transposition, inversion, orsmoothness alone. Rather, it resides in some interactive com-bination of them. All are present, and all shape the linearflow in some degree.

Example 17 revisits a passage from Example 3(b) andplaces it in a more complete context.53 This passage is obvi-ously in E minor in some sense, and it ends with a strong,modal V–I cadence. But it contains elements that resist thepull of E minor and that create their own, distinctively non-tonal musical logic. At (a), eight strong-beat harmonies areconnected by inversion and transposition. It is worthwhile tofollow a single voice through the progression. As the firstchord moves to the second by I7, E moves to D �. The move-ment of the highest registral line thus defines an inversionalaxis, and the sense of inversion is confirmed by the voice ex-change: E–D � and D �–E. The second chord moves to thethird by T9, and again that gesture is reflected in the move-ment of the highest registral line: D � to B �. At the sametime, of course, the progression from the second chord to the third could be understood as a fuzzy-*T1, with the aug-mented triad in the left hand moving up 1 semitone and themelody moving against it in contrary motion. This fuzzy-*T1portrays the accompanying chords as being in control of themotion with the melody deviating from them. The crisp-T9portrays the melody as in control, bringing about a crossing


52 There are four other voice leadings between the first and second chordsthat share the same displacement of (8). One of these is the T2-inducedvoice leading: all four voices move by 2 semitones.

53 This passage is discussed from a Neo-Riemannian perspective in Gollin2001. See also Hyde 1996, which describes the voice leading in tonalterms (using figured bass, with numerous registral transfers and dis-placements).

51 Lambert 2000 observes that all three pieces of Webern’s op. 11 are “sat-urated with [0145]s” and constructs relevant transformational net-works. Among the [0145]s identified by Lambert in the third piece,shown in Example 16, are the first four notes in the cello part, {B, C,E �, F �} and the four upper notes in the piano right hand in mm. 2–5,{D, E �, F �, G}.





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of the voices within the accompanying chords. Both inter-pretations are easily audible and useful in characterizing thelinear relationship of the harmonies.

As the progression continues, the soprano moves into aninner voice, first to G � (via *T8) and then to A (via *I5). Inthe motion from the fourth chord to the fifth, as from thefirst chord to the second, the inversion is confirmed by avoice-exchange: G �–A and A–G �. Example 17(b) shows thecomposite motion from the first chord to the fifth, in whichthe soprano E has moved 5 pitch-class semitones to the bassA, confirming the motion from the first chord onto the fifthby T5. In the remainder of the progression, three fuzzy-transpositions, largely coincident with the registral lines,compose to a larger fuzzy-*T7. The cumulative result ofthese motions, shown at (c), is a fuzzy-*T0 with an offset ofonly (1) that spans the whole progression. At this level, weare able to hear the final E-minor triad as a minimally dis-torted version of the initial 4-19[0148].54 The reverse is alsotrue: the initial 4-19[0148] can be heard as a minimally dis-torted E-minor triad, and one can imagine the progressionas an extended effort to rectify the distortion, to transformthe 4-19[0148] into an E-minor triad.

Example 18 involves a progression of four chords repre-senting four different set classes.55 It is hard to generalizeabout them as harmonies: the first three are octatonic, butfourth is not; the first and third are symmetrical, but secondand fourth are not; the first, second, and fourth embed tri-ads, but third does not. The analytical task is to describe thevoice leading that binds these disparate chords.

Ruggles has connected the first two chords and the lasttwo as smoothly as possible, as shown at (a). The smoothest

connection between the second and third chords involves aregistral inversion, as does the connection from the firstchord to the fourth that is shown at (b). Notice that dis-placement numbers, like offset numbers, do not necessarilyaccumulate; they may cancel each other out instead, as theydo here. The progressions from chord to chord involve dis-placements of 6, 10, and 7 semitones, but the progressionfrom the first chord to the last involves a displacement ofonly 1; the first and last chords share three out of four pitchclasses, and the one note that is different differs by only onesemitone.56

The transformational voice leading induced by transposi-tion and inversion, shown at (c), follows the same smoothpath. In the motion from the first chord to the second, threeof the registral lines move down two semitones, while the re-maining line sustains a note. Something very similar hap-pens in the motion from the third chord to the fourth, wherethree of the four lines move by an index of 5, while the re-maining line sustains a note. In moving from the secondchord to the third, *I3 induces a registral inversion, as doesthe *T0 that connects the first chord with the fourth, shownat (d).57

Notice that, as with displacement numbers, the offsetnumbers do not necessarily accumulate. In the chord-to-chord succession, the offsets are (2), (4), and (5). But theoverall offset as the first chord moves to the last is only (1).That is because offsets are as likely to cancel each other out


54 It is important to note again that, over the course of an extended pro-gression, the offset numbers are as likely to cancel each other out asthey are to accumulate. In the chord-to-chord successions, the offsetnumbers range from (0) to (3) and most of the successions involve somedegree of offset. But in the progression from the first chord to the last,the offset is only (1).

55 See Slottow 2001 for a related discussion of this passage.

56 For each of the chord-to-chord successions, there is another voice lead-ing with the same minimum displacement value, but the connectionshown from the first chord to the fourth is uniquely smooth.

57 For each of the chord-to-chord successions, there is at least one otherwith the same minimal offset value. The same is true for the connectionfrom the first chord to the fourth, which could be modeled as *I3 withan offset of only (1). The voice leading induced by *I3 follows the regis-tral lines (D � to D, F to B�, and D to C � by crisp-I3, with B �–F � deviat-ing by semitone) and involves a voice-exchange (D � to D in the sopranoand D to C � in the bass). Interestingly, I3 is also the transformation thatmaps the inversionally-symmetrical first chord onto itself.





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as they are to accumulate. In this example, and in all of theexamples in this paper, each voice leading involves the small-est possible offset. The moves from chord to chord are astransposition-like or inversion-like as they can be. And thelarger gestures, the middlegrounds, are also as transposition-like or inversion-like as they can be. Indeed, these examplesare chosen precisely because the chord-to-chord successions,with their minimal offsets, combine into larger middle-ground gestures that also involve minimal offsets.

Schoenberg’s Little Piano Piece, op. 19, no. 2, is punctu-ated by the three big chords shown in Example 19.58 Differ-ent though they are in size and internal structure, thesechords can be connected by transformational voice leadingresulting from fuzzy-transposition. As shown at (a), the firstchord is connected to the second by *T11, which sends thediminished triad at the bottom of the first chord to the topof the second, and also sends the top three notes of the firstchord fuzzily to the bottom three notes of the second (theG � in the middle of the first chord splits up to F and downto F �). The second chord is connected to the third by *T4,which exchanges the upper four and bottom two notes in thesecond chord with the top two and bottom four in the third.In both moves, there is an offset of (4), the same as in themotion from the first chord to the third by *T3, shown at(b).59 This voice leading is guided by the motion of theminor third (C–E �) at the bottom of the first chord, whichmoves at T11 to the minor third (B–D) at the top of the sec-ond chord, and then by T4 to the minor third (E �–F �) in themiddle of the third chord, where it acts as a center of inver-sional symmetry. There is a strong motivic aspect to thesetranspositions: they effectively compose out the intervals of


58 See Vaisaala 1999 for a prolongational interpretation of the voice lead-ing in this piece and among these chords.

59 The progression among these three chords can also be understood interms of inversion, with *I6 and *I5 composing to *T11, with an offset of(4) in each case. But the transpositional interpretation has a significantmotivic aspect and produces a voice leading that conforms more closelywith the registral arrangement of the chords.

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3-3[014], such a prominent feature of the motivic surface ofthe piece, as in its opening six-note melody, some intervalclasses of which are shown in Example 19(c).

viii. sets and set classes

The discussion up to this point has focused on the voiceleading between actual sets of pitch classes. It has probablybeen apparent, however, that the voice leading between setsis highly constrained by the structure of the set classes towhich these sets belong. The nature of the constraints is bestillustrated with an example. Example 20 shows the voiceleading between a set X {C, C�, E} and all transpositions andinversions of a set Y {C, E, F �}. For each of the twenty-fourpairs of sets, Example 20 shows the voice leading that pro-duces the smallest offset, whether by *Tn or *In. In everycase, the smallest offset is the same: (2). In other words, onecan get from X (or any of its transpositions or inversions) toY (or any of its transpositions or inversion) by a voice leadingthat approximates, to within two semitones, either transposi-tion or inversion. The voice-leading patterns shown inExample 20 and the offset of (2) constitute a kind of finger-print for the set classes that contain sets X and Y.

It is thus possible to speak of a best fit, or what I will callan optimal offset, between two set classes. In Example 20, Xbelongs to set-class 3-3[014] and Y belongs to set-class 3-8[026]. We can say that the best possible fit between thesetwo set classes involves an offset of (2). That is, any memberof 3-3[014] is related to any member of 3-8[026] by eithertransposition or inversion with an offset of (2). Optimal off-set is a convenient way of measuring the distance betweentwo set classes.

At this level, smoothness converges with uniformity andbalance. The optimal offset is also the minimum displace-ment value associated with the smoothest way of moving between two sets. Thus it is possible to talk about smoothnessto within fuzzy-Tn or fuzzy-In. In the case of the two sets inExample 20, the voice leading from X to I5(Y) (circled in

the Example) involves a total displacement of 2 semitones.That is, it deviates by only 2 semitones from the absolutesmoothness of a crisp-T0—the displacement value, whichmeasures smoothness, is thus the same as the offset value,which measures uniformity and balance. Both measure basi-cally the same thing, namely the amount of deviation fromthe perfect smoothness of crisp-T0.

It is possible to calculate the optimal offset, the best andsmoothest fit, between any two set classes. Example 21 pro-vides the optimal offset between any two trichords.60 Theinformation is provided in the form of a chart: the optimaloffset value for any two trichord-classes can be found at theintersection of the relevant row and column. (The customaryparentheses around the offset numbers have been removedfor the sake of legibility.) The first row of the chart, for ex-ample, shows 3-1[012] in relation to all twelve trichord-types. The optimal offset between 3-1[012] and itself is ofcourse (0), since it is possible to map any member of 3-1[012] onto any other member of the set class by crisp trans-position or inversion. It is not so easy, however, to get from3-1[012] to 3-8[026]. From any member of 3-1[012], mov-ing by transposition or inversion, it will require an offset ofat least (4) to map onto a member of 3-8[026]. It is even far-ther from 3-1[012] to 3-12[048], with an optimal offset of(6). In fact, an optimal offset of (6) is as far apart as two tri-chords can get.

The contents of the chart in Example 21 can also be pre-sented as the spatial map of Example 22.61 Offset values thusdefine what Morris (1998) calls a “voice leading space.”62


60 The same information for sets of any size, cardinality 2 through 6, andfor sets of different size, is provided in Ariza 2002.

61 See Callender 2002 for a similar trichordal map.62 See also Morris 1995b, which describes more general “compositional

spaces,” which he defines as “out-of-time networks of pcs that can un-derlie compositional or improvisational action.” The creation of suchcompositional spaces is a traditional activity of music theorists (see, forexample, the Riemannian Tonnetz, described in many of the sourceslisted in note 42 and the “pitch space” described in Lerdahl 2001).




Lines link trichord classes connected by an optimal offset of(1). 3-2[013], for example, is offset by (1) in relation to 3-1[012], 3-6[024], and 3-3[014]. Each move on the map thusrepresents an offset of (1), and the moves are cumulative.From 3-2[013], for example, it requires two steps on themap, representing an optimal offset of (2), to get to 3-4[015]and 3-7[027]. The two set classes most widely separated on

the map (3-1[012] and 3-12[048]) are separated by 6 steps,representing an optimal offset of (6). 3-1[012] and 3-12[048]are the most isolated set classes on the map—each is in di-rect communication with only one other set class. 3-8[026]and 3-7[025] lie right at the center of the map and are rela-tively promiscuous in their relations—they communicate directly with 6 and 5 other set classes respectively.


X T0(Y) X T1(Y) X T2(Y) X T3(Y) X T4(Y) X T5(Y)E 4 C E 5 C� E 6 D E 7 D� E 8 E E 9 FC � 5 E C � 6 F C� 7 F � C � 8 G C� 9 G� C� 10 AC 6 F � C 7 G C 8 G� C 9 A C 10 A� C 11 B

*I5 *I6 *I7 *I8 *I9 *I10(2) (2) (2) (2) (2) (2)

X T6(Y) X T7(Y) X T8(Y) X T9(Y) X T10(Y) X T11(Y)E 10 F � E 11 G E 0 G� E 1 A E 2 A� E 3 BC� 11 A � C� 0 B C� 1 C C� 2 C� C� 3 D C� 4 D�C 0 C C 1 C� C 2 D C 3 D� C 4 E C 5 F

*I11 *I0 *I1 *I2 *I3 *I4(2) (2) (2) (2) (2) (2)

X I0(Y) X I1(Y) X I2(Y) X I3(Y) X I4(Y) X I5(Y)E 8 C E 9 C� E 10 D E 11 D� E 0 E E 1 FC� 7 G� C� 8 A C� 9 A� C� 10 B C� 11 C C� 0 C�C 6 F � C 7 G C 8 G� C 9 A C 10 A� C 11 B

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X I6(Y) X I7(Y) X I8(Y) X I9(Y) X I10(Y) X I11(Y)E 2 F � E 3 G E 4 G� E 5 A E 6 A� E 7 BC� 1 D C� 2 D� C� 3 E C� 4 F C� 5 F� C� 6 GC 0 C C 1 C� C 2 D C 3 D� C 4 E C 5 F

*T1 *T2 *T3 *T4 *T5 *T6(2) (2) (2) (2) (2) (2)

example 20. Voice leading from X {C, C �, E} to all transpositions and inversions of Y {C, E, F �}.

679-125.MTS.Strauss_pp305-352 8/28/03 2:24 PM 6



Example 23 provides the same information for tetra-chords in chart format, and Example 24 presents the sameinformation as the map of a voice-leading space.63 Like Ex-amples 21 and 22, these show the smallest amount of adjust-ment (offset, displacement) that is required to get from oneset-class to another via transposition or inversion. As withthe trichords, the tetrachords on the periphery of the mapare relatively isolated from the others. 4-1[0123], 4-9[0167],and 4-28[0369], for example, have an optimal offset of (1)with only one other set class, while those at the center arerelatively promiscuous in their relationships (4-22[0247] and4-27[0258] are in direct communication with the largestnumber of other set classes). Example 25 shows trichords inrelation to tetrachords in chart format. Unfortunately it isdifficult to present the contents of this chart in the form of aspatial map, in the manner of Examples 22 and 24, since thevoice-leading connections among the set classes are too nu-merous and intertwined. The same is true, to an even greaterextent, for pentachords and hexachords, both within them-

selves and in relation to set classes of different size. For setclasses with more than six notes, the situation becomes sim-pler again: the maps for octachords, nonachords, and deca-chords will be identical to those for tetrachords, trichords,and dyads—each set class in the map is simply replaced byits complement. In the absence of a legible two-dimensional


63 Cohn 2003 elegantly displays the tetrachordal space in three dimen-sions as a tetrahedron. See Quinn 2003 for additional discussion.

3-1 3-2 3-3 3-4 3-5 3-6 3-7 3-8 3-9 3-10 3-11 3-123-1 0 1 2 3 4 2 3 4 5 4 5 63-2 1 0 1 2 3 1 2 3 4 3 4 53-3 2 1 0 1 2 1 1 2 3 2 3 43-4 3 2 1 0 1 2 1 1 2 2 2 33-5 4 3 2 1 0 3 2 1 1 2 2 33-6 2 1 1 2 3 0 1 2 3 2 3 43-7 3 2 1 1 2 1 0 1 2 1 2 33-8 4 3 2 1 1 2 1 0 1 1 1 23-9 5 4 3 2 1 3 2 1 0 2 1 23-10 4 3 2 2 2 2 1 1 2 0 1 23-11 5 4 3 2 2 3 2 1 1 1 0 13-12 6 5 4 3 3 4 3 2 2 2 1 0

example 21. Optimal offsets for trichords (as a chart).

example 22. Optimal offsets for trichords (as the map of a voiceleading space).

3-1012

3-2013

3-3014

3-4015

3-5016

3-6024

3-7025

3-8026

3-9027

3-10036

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3-12048




-1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20 -21 -22 -23 -24 -25 -26 -27 -28 -29

4-1 0 1 2 2 3 4 4 6 8 2 3 3 4 4 5 7 4 6 5 6 4 5 6 6 8 6 7 8 54-2 1 0 1 1 2 3 3 5 7 1 2 2 3 3 4 6 3 5 4 5 3 4 5 5 7 5 6 7 44-3 2 1 0 2 3 4 2 4 6 2 1 3 2 4 3 5 4 4 5 6 2 3 4 4 6 4 5 6 34-4 2 1 2 0 1 2 2 4 6 2 1 1 2 2 3 5 2 4 3 4 2 3 4 4 6 4 5 6 34-5 3 2 3 1 0 1 3 3 5 3 2 2 1 1 2 4 3 3 2 3 3 2 3 3 5 3 4 5 24-6 4 3 4 2 1 0 4 4 4 4 3 3 2 2 3 3 4 2 3 4 4 3 4 2 4 4 3 4 14-7 4 3 2 2 3 4 0 2 4 2 1 3 2 4 1 3 4 2 3 4 2 3 2 4 4 4 3 4 34-8 6 5 4 4 3 4 2 0 2 4 3 3 2 4 1 1 4 2 3 2 2 3 2 4 2 4 3 4 34-9 8 7 6 6 5 4 4 2 0 6 5 5 4 4 3 1 4 2 3 2 4 3 2 4 2 4 3 4 34-10 2 1 2 2 3 4 2 4 6 0 1 1 2 2 3 5 2 4 3 4 2 3 4 4 6 4 5 6 34-11 3 2 1 1 2 3 1 3 5 1 0 2 1 3 2 4 3 3 4 5 1 2 3 3 5 3 4 5 24-12 3 2 3 1 2 3 3 3 5 1 2 0 1 1 2 4 1 3 2 3 1 2 3 3 5 3 4 5 24-13 4 3 2 2 1 2 2 2 4 2 1 1 0 2 1 3 2 2 3 4 2 1 2 2 4 2 3 4 14-14 4 3 4 2 1 2 4 4 4 2 3 1 2 0 3 3 2 2 1 2 2 1 2 2 4 2 3 4 14-15 5 4 3 3 2 3 1 1 3 3 2 2 1 3 0 2 3 1 2 3 1 2 1 3 3 3 2 3 24-16 7 6 5 5 4 3 3 1 1 5 4 4 3 3 2 0 3 1 2 1 3 2 1 3 1 3 2 3 24-17 4 3 4 2 3 4 4 4 4 2 3 1 2 2 3 3 0 2 1 2 2 1 2 2 4 2 3 4 34-18 6 5 4 4 3 2 2 2 2 4 3 3 2 2 1 1 2 0 1 2 2 1 2 2 2 2 1 2 14-19 5 4 5 3 2 3 3 3 3 3 4 2 3 1 2 2 1 1 0 1 3 2 3 1 3 1 2 3 24-20 6 5 6 4 3 4 4 2 2 4 5 3 4 2 3 1 2 2 1 0 4 3 2 2 2 2 1 2 34-21 4 3 2 2 3 4 2 2 4 2 1 1 2 2 1 3 2 2 3 4 0 1 2 2 4 2 3 4 34-22 5 4 3 3 2 3 3 3 3 3 2 2 1 1 2 2 1 1 2 3 1 0 1 1 3 1 2 3 24-23 6 5 4 4 3 4 2 2 2 4 3 3 2 2 1 1 2 2 3 2 2 1 0 2 2 2 1 2 34-24 6 5 4 4 3 2 4 4 4 4 3 3 2 2 3 3 2 2 1 2 2 1 2 0 2 2 1 2 14-25 8 7 6 6 5 4 4 2 2 6 5 5 4 4 3 1 4 2 3 2 4 3 2 2 0 2 1 2 34-26 6 5 4 4 3 4 4 4 4 4 3 3 2 2 3 3 2 2 1 2 2 1 2 2 2 0 1 2 34-27 7 6 5 5 4 3 3 3 3 5 4 4 3 3 2 2 3 1 2 1 3 2 1 1 1 1 0 1 24-28 8 7 6 6 5 4 4 4 4 6 5 5 4 4 3 3 4 2 3 2 4 3 2 2 2 2 1 0 34-29 5 4 3 3 2 1 3 3 3 3 2 2 1 1 2 2 3 1 2 3 3 2 3 1 3 3 2 3 0

example 23. Optimal offsets for tetrachords (as a chart).






4-10123

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example 24. Optimal offsets for tetrachords (as the map of a voice-leading space).




representation, one needs to imagine the 220 set classes hov-ering in a multi-dimensional space and linked together bythe bonds of parsimonious voice leading.

Within such a space, optimal offset numbers provide agood measure of the degree of proximity or distance betweenany two set classes. The set-class geography that emerges

from a focus on voice leading has interesting points of con-tact with that produced by two rather different theoreticalenterprises: similarity relations and set complexes. Similarityrelations have traditionally been concerned with assessingthe resemblance between two set classes, usually based onshared intervallic or subset content.64 Theories of set com-plexes have attempted to group set classes into families,often based on some combination of inclusion and comple-mentation.65 Yet despite the variety of approaches to similar-ity and set complexes, they offer strikingly similar maps ofthe set-class terrain. That is, there is broad agreement as towhich set classes are relatively similar that transcends indi-vidual methodologies for showing such similarity.66 What ismore, the voice-leading spaces described here show strikingresemblances to the topographies produced by similarity re-lations and set complexes. As Quinn observes, “the system ofpc set classes has a natural topography that presents itself asthe elusive intuitions of similarity-relation creators and that


64 For good, critical summaries of earlier work on pitch-class set similarityand important original contributions, see Buchler 2000, Scott &Isaacson 1998, Castren 1994, and Isaacson 1990. Ariza 2000 has overtwenty similarity measures available for comparison to each other, andfor comparison with the atonal voice-leading measures discussed in thispaper. Similarity measures are usually based on intervallic or subsetcontent.

65 For discussions of set complexes and their latter-day descendants,pitch-class set genera, see Forte 1973 and 1989, Morris 1997, Kaplan1990, and Doerksen 1999.

66 This is the central contention of Quinn 2001: “One of the more signifi-cant contributions of the present research is the observation and char-acterization of the heretofore unnoticed connections between similarityrelations advertised as widely divergent . . . It should be quite clear bynow that all aspects of similarity, together with many topics of pc set-class theory, are deeply and inextricably interrelated. It is not a weak-ness, but a strength of these similarity relations that (when we listen tothem!) they speak with a single extensional voice, regardless of whatthey profess to measure” (155). See also Scott & Isaacson 1998, 117–18, for a discussion of the high degree of correlation among a variety ofsimilarity measures.

3-1 3-2 3-3 3-4 3-5 3-6 3-7 3-8 3-9 3-10 3-11 3-124-1 1 1 2 3 4 2 3 4 5 4 5 64-2 2 1 1 2 3 1 2 3 4 3 4 54-3 3 1 1 3 4 2 2 4 5 3 4 64-4 3 2 1 1 2 2 1 2 3 2 3 44-5 4 3 2 1 1 3 2 1 2 2 2 34-6 5 4 3 2 1 4 3 2 1 3 2 44-7 5 3 1 1 3 2 2 2 4 3 3 44-8 7 5 3 1 1 4 2 2 2 3 3 44-9 9 7 5 3 1 6 4 2 2 3 3 44-10 3 2 2 3 4 2 1 2 3 2 3 44-11 4 2 2 2 3 1 1 3 4 2 3 54-12 4 3 2 2 3 3 2 1 2 1 2 34-13 5 3 3 2 2 2 1 2 3 1 2 44-14 5 4 3 2 2 4 3 2 1 2 1 24-15 6 4 2 2 2 3 1 1 3 2 2 34-16 8 6 4 2 2 5 3 1 1 2 2 34-17 5 4 3 3 4 4 3 2 3 2 1 24-18 7 5 3 3 3 4 2 2 2 1 1 24-19 6 5 4 3 3 5 3 3 2 2 1 14-20 7 6 5 3 3 6 4 2 2 3 1 24-21 5 3 3 3 3 2 2 2 3 2 3 44-22 6 4 4 3 3 3 2 3 2 2 2 34-23 7 5 3 3 3 4 2 2 2 3 3 44-24 7 5 5 4 3 4 3 2 3 3 2 24-25 9 7 5 3 3 6 4 2 2 3 3 44-26 7 5 5 4 4 4 3 4 3 3 2 24-27 8 6 4 4 4 5 3 3 3 2 2 34-28 9 7 5 5 5 6 4 4 4 3 3 44-29 6 4 4 3 2 3 2 1 2 2 1 3

example 25. Optimal offsets for trichords in relation to tetra-chords (as a chart).




lies behind many of the tools of pc set-class theory.”67 Fromdifferent points of view, then, both the voice-leading ori-ented methodology presented here and the harmony-oriented methodologies of similarity relations and set com-plexes appear to be engaged in mapping the same terrain.

What this suggests is the deep connection between har-mony and voice leading in atonal music.68 I previously ob-served that harmony constrains voice leading—that is, thatthe internal structure of a set shapes the kinds of voice lead-ing connections it can create with other sets. And the reverseis also true: Voice leading constrains harmony in the sensethat the kinds of voice leading connections a set is capable ofmaking with other sets will go a long way toward defining itsinternal structure. Similarity relations and set-complex rela-tions ask: How much like set-class X is set-class Y? Thevoice-leading model presented here asks: How smooth, uni-form, or balanced is the path from X to Y? How much effortis needed to get from X to Y? It turns out that the answers tothese questions are more or less the same.

In addition to the impressive work on similarity and set-complexes, a great deal of recent work on musical transfor-mation testifies to a longstanding desire among theorists tofind effective ways of relating sets belonging to different setclasses.69 The optimal offset maps presented here place the

set-classes in spatial relationship, with each set class at ameasurable distance from the others. And the distances inquestion are determined not by internal resemblance but byvoice leading from set to set: sets are understood as closer ifit is possible to move from one to the other in a relativelysmooth, uniform, or balanced way.

As a musical composition unfolds in pitch space, thesevoice-leading distances can be traversed in a variety of waysand with a variety of effects. Some progressions will movesmoothly and easily from place to place, following relativelyshort, direct paths. Others will take more circuitous routes,with wide leaps to distant locations.

ix. analyses

The progression in Example 26 involves five arpeggiatedtetrachords representing five different tetrachord types.70 Asshown in Example 26(a), the first two moves involve increas-ing levels of voice-leading offset, (2) then (3), while the con-cluding two moves involve decreasing levels of offset, (2)then (1). The relative restricted portion of the tetrachordalmap traversed by the progression is reprinted at (b). Thefinal chord in the progression (4-21[0246]) lies in the mid-dle of the map sector just explored and within an offset of atmost (2) in relation to the preceding four chords. One canthink of the progression as a gradual purging of semitones


1998), “near-transposition and near-inversion” (Straus 1997), “pseudo-transposition” (Lewin 1998), “Q and X transformations” (Gollin 1998),and “FOLDSIM” (Morris 1995a). O’Donnell 1998 observes: “The de-velopment of all these new transformations is a reaction to the exclusivenature of the traditional transposition and inversion operators, whichallow at most twenty-four relations among the 4,096 possible pitch-class sets. The exclusivity of traditional set classes not only leads towarda fragmented analytical approach that traces independent paths foreach prominent set class within a given musical work, but often alsoleaves large temporal gaps between two ‘adjacent’ members of the sameset class.” (61).

67 Quinn 2001, 153.68 The relationship between harmony and voice leading in atonal music

obviously merits extensive research, which has had to await significantprior progress in understanding harmony and voice leading on more orless their own terms. Quinn 2003 takes a significant step and underliesthe assertions of this paragraph.

69 The most important recent effort in this direction has involved Klump-enhouwer networks. Relevant literature includes Klumpenhouwer 1991and 1998, Lewin 1994, Lewin 1990, and O’Donnell 1998. Lewin 1990points out that one of the primary advantages of these networks is their ability “to interpret chords of different set classes with iso-graphic networks” (84). In addition to K-nets, theorists have adduced avariety of ad hoc transformations to connect sets belonging to differentset classes, including “split transformations” (O’Donnell 1997 and





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from the chords: the first two chords contain two semitones;the third and fourth chords contain one semitone; the lastchord contains no semitones. In this way, the intervallicstructure of the chords changes even as the progression con-sists of a few relatively smooth moves.

Webern’s op. 7, no. 1 pursues a contrasting strategy ofgradually increasing levels of offset, as shown in Example27.71 A progression of seven trichords, representing four dif-ferent trichord types, is heard in the alto register of the righthand piano part (Example 27[a]). The first two moves are bycrisp-Tn, connecting three forms of 3-4[015]. Thereafter, themoves involve increasing levels of offset, with the registrallines confirming the voice leading in every case.72 The finalleg of the progression involves the greatest amount of offset,dramatizing both the conclusion of the movement and therelative anomaly of a major triad in this context. The triad isanomalous not only because of its intrinsic sound but alsobecause of its voice leading distance from the harmonies thathave previously characterized the progression. The relevantportion of the trichordal map is provided in Example 27(c).

The opening of Sessions’s Piano Sonata, shown in Ex-ample 28, pursues the opposite strategy. As seen at (a), itsfive trichords, representing four different trichord types, in-volve decreasing amounts of offset, leading to a repeat of the initial trichord, {F, A �, A} at the end. These motions canalso be traced on the portion of the trichordal map repro-duced at (b).

In the passage from Schoenberg’s Piano Piece op. 11, no.2 shown in Example 29, the degree of offset is kept level

throughout the progression.73 The first chords produce thehighest level of offset as the first phrase concludes. There-after, even as the size of the sets increases and, therefore, soalso the range of possible offset values, the offsets remain ex-tremely low.74 Amid the variety of chords, with their obviousdifferences in size and structure, Schoenberg maintains ahigh degree of voice-leading smoothness (to within transpo-sition or inversion).

In this progression, as in the others discussed here, dis-parate chords are linked by fuzzy-Tn or fuzzy-In. The offsetnumbers measure the amount of work it would take to makethe chords equivalent. To put it another way, the offset num-bers measure the number of steps that separate the chords onthe appropriate voice-leading map. They thus describe thedistances traversed by the chords as the progression movesthrough the voice-leading space, and give a sense of the difficulty of the voyage—now smooth, close and easy; nowrough, distant, and strenuous.

x. motivic transformation

The methodology presented here provides a useful way ofdescribing the process of motivic transformation that haslong been understood as central to the structure of atonalmusic, but which has been difficult to describe with preci-sion. For traditional set theory, two motives either are or arenot members of the same set class, and even the slightestmodification in intervallic or pitch-class content changes


73 Lewin 1994 has written about this passage using Klumpenhouwer net-works. See also O’Donnell 1998, which uses Klumpenhouwer networksand dual transpositions.

74 Transformations other than those shown also produce these optimaloffsets. The transformations shown on the example are those that pro-duce voices that conform most closely to the registral lines. Tracing theactual voice leading paths is left as an exercise for the reader.

70 See Straus 1995 for a related discussion of this passage.71 See Lerdahl 2001 for an interesting account of linear continuity in this

piece on a substantially different basis.72 Example 27 describes the motion from chord 5 to chord 6 as *T11 with

an offset of (3); that is the most uniform way of connecting the twochords. The same motion coule be interpreted as *I6 with an offset ofonly (2). The *T11 interpretation has the advantage of creating a seriesof four equivalent transformations (all fuzzy-*T11) with gradually in-creasing offset.





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set-class membership.75 But the methodology presented hereis well suited to describing motivic variation within the tra-ditional framework of pitch-class transposition and inver-sion. It permits us to observe that two motive-forms are re-

lated by fuzzy-transposition or fuzzy-inversion with a cer-tain measurable offset and thus to recognize and specify boththe nature and the extent of the motivic variation.

Example 30 examines a familiar passage.76 Ethan Haimoargues that this passage, the work from which it is drawn,and Schoenberg’s atonal music generally are best analyzedusing Schoenberg’s own concept of “developing variation.”77


76 Lewin 1998 discusses aspects of this passage in terms of “pseudo-transposition.” Straus 1997a discusses it in terms of “near-transposition.”

77 Haimo 1996, 197–8, contends: “In op. 11, no. 1, a limited number offundamental motives provides the source material for the entire move-ment. These basic motives are transformed into a remarkably varied arsenal of musical events by a flexible process of developing variation

75 Theorists have often used contour theory to deal with the musical pos-sibility of changing intervallic content while retaining contour or viceversa. See Marvin & Laprade 1987, Marvin 1991, Morris 1987 and 1993. Inaddition, theorists have proposed a variety of ad hoc transformations to tracemotivic development. See, for example, Lewin 1987 (FLIPEND and FLIP-START); Straus 1995 (EXP and PE in the music of Ruth Crawford Seeger);Morris 1994 (ADDTAIL, BEHEAD, and INSERT in the music of Bartók);and Gingerich 1986 (INSERT, EXPAND, CONTRACT, EXCHANGE, andREORDER in the music of Ives).

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Haimo identifies the first three notes (B–G �–G) as motivea1 and traces its varied development over the course of thepassage through ten subsequent a-forms—these are circledand labeled in the Example.78

As we see in Example 30(a), the move from a1 to a2 in-volves the minimum offset of (1), as the minor third in

a1 expands to the major third in a2.79 The motion from a2 to a3 has the same minimum offset, and involves a registralrearrangement—as Haimo observes, “the G � is pulled out ofregister and placed in the lowest voice” (195). At this point,the offsets begin to increase in size, in what Haimo calls “anorderly process of intervallic expansion” (196). The offsetnumbers measure precisely the extent of the expansion. Thisprocess continues until the arrival of a7 and a5, which signala return to the intervals of a1 and, in the case of a5, its pitchclasses as well, documented in Examples 30(b) and (c). Atthis point, the musical focus shifts from 3-3[014] to 3-8[026], but the process of developing variation continues to


. . . Given the transformations (notably interval expansion and octavedisplacement) that are applied to a limited number of basic motives, itis possible to understand every note in the composition as derived fromthe opening phrase by an easily followed path of developmental logic.”For relevant discussions of developing variation see Boss 1992 and1994, Haimo 1996, and the numerous primary documents cited inthose articles.

78 I have changed Haimo’s order slightly, placing a5 after a6 and a7. Also,I have maintained my usual practice of writing sets in registral order,from highest to lowest. For motivic analyses, which are normally con-cerned primarily with melodic statements, it will often be better towrite the sets in temporal order, from first to last. But the voice-leadingconnections and the offset values will be the same in either case.

79 As Haimo 1996 observes “The changes from a1 to a2 are very nearlythe minimum possible. The opening interval of the trichord, which hadbeen three semitones, is expanded by the minimum possible distance tofour semitones. The closing interval, a descending semitone, is heldfixed. This careful limitation on the pace of the transformations en-hances the significance of the connections between the motives” (193).

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example 29. Schoenberg, Piano Piece, op. 11, no. 2, mm. 10–13.





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be traced by the fuzzy-Tn that connects the a-forms. At thehighest level, shown at (d), one can feel the connection be-tween a1/a5 and the concluding a11—they are connected by*T11 with an offset of (2). That gesture replicates the descentfrom G � to G within a1, and indeed maps G� onto G overthe course of the passage as a whole.

One of the problems with motivic analysis, whether prac-ticed by Schoenberg or Haimo, has been the difficulty in es-tablishing reliable, systematic criteria for identifying varia-tion. When is Motive Y a variant of Motive X and when is it something new? The methodology presented here offersan answer: when the offset numbers are small, it is reason-able to assert a motivic relationship; when the offsets arelarge, that assertion becomes harder to maintain. The offsetnumbers thus function as a threshold for asserting motivicrelationships.80

Example 31 plots Haimo’s a-forms onto the relevant por-tion of the trichordal voice-leading map. There are three de-grees of separation between the progenitor a1 (representing3-3[014]) and a6 (representing 3-11[037]). All of the othera-forms are offset no more than (2) from each other. Thissuggests that Haimo’s a6 may not be a legitimate a-form.

In a previous study, I have analyzed the motivic organiza-tion of melodies by Ruth Crawford Seeger using a variety ofad hoc transformations.81 Example 32 takes one such melodyand reanalyzes it in terms of fuzzy-T and fuzzy-I.82 Bracketsunderneath the score identify all of the melodic trichordsthat involve an internal change of direction—“twist neumes,”in the terminology of Seeger 1994, with which Crawfordwas intimately familiar (the apparent exception, #5, involves

an octave displacement of D–F �). The opening and conclud-ing motives are members of 3-2[013], as shown at (a). In be-tween, the music moves through other set classes that lienearby on the trichordal voice-leading map, shown at (b).The methodology presented here is well suited to tracing themotivic ebb and flow of music such as this, which is “held together not by a sense of directed motion back to a startingpoint, or a sense of ineluctable motion toward any particulargoal, but rather by the cogency of each step along the way.”83

xi. conclusion

Transposition and inversion, those traditional sources ofharmonic coherence in atonal music, can also be understoodto shape atonal voice leading. These operations underpinstandard theories of atonal music, and their power to createmusical development and musical coherence has long been


80 Boss 1992 and 1994 develop a taxonomy of motivic relations forSchoenberg’s music. In general, his method of determining the degreeof motivic variation conforms well with the offset measure employed inthis study.

81 Straus 1995.82 The pcs in each motive are listed in temporal rather than registral

order. That is, within each motive, the first note is written at the bot-tom, the second note in the middle, and the third note on top. 83 Straus 1995, 40.

a1, a4, a5, a7, a93-3[014]

a23-4[015]

a3 a8, a10, a113-5[016] 3-8[026]

a63-11[037]

example 31. Haimo’s motivic a-forms in relation to the tri-chordal voice-leading map.




acknowledged. Their impact on the linear organization ofatonal music, however, has not received comparable recogni-tion. When set X is transposed or inverted onto set Y, eachnote in X maps onto some corresponding note in Y, andthose mappings comprise a counterpoint of pitch classes, atransformational voice leading.

But to realize their full voice-leading potential, transposi-tion and inversion need to be understood more broadly thanusual, along a continuum of musical effects, with traditional,crisp transposition and inversion at one end of a wide spec-trum. When transposition and inversion are generalized asuniformity and balance, it becomes possible to measure thedegree of transposition or inversion between any two sets.There need no longer be insurmountable barriers separatingthe 220 set classes; instead, we can move in a measurably

transposition-like or inversion-like way, with varying degreesof smoothness, from one set to any other. And each time wemove in this way, we produce a hearable, measurable, theo-retically secure atonal voice leading.

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1 23

4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

E E� G G� F � G� F G F �C� E E� B� D E� A � E GD C� E A � F G G� F E

I5 *I7 *I1 *I0 T1 *I0 T11 I11(0) (1) (1) (1) (0) (1) (0) (0)

3-2[013] 3-6[024]

3-3[014]

example 32. Crawford, Diaphonic Suite No. 4 ( for oboe and cello), third movement, mm. 1–5, cello part.

(a)

(b)

3-2[013] 3-2[013] 3-3[014] 3-6[024] 3-3[014] 3-3[014] 3-2[013] 3-2[013] 3-2[013]




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