A Geometry of Music

Journal of Music Theory

Dmitri Tymoczko A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. New York: Oxford University Press, 2011: xviii+450 pp. ($39.95 cloth)

John Roeder

In this impressive work, Dmitri Tymoczko, a talented composer and theorist, seeks to comprehend a thousand years of diverse Western composition through a modern mathematical model. He ranges widely and inventively from the respectably familiar to the irreverently original, writing with a pedagogical flair that persuades the reader to ponder every definition and musical example carefully. As I did, I found myself taking issue with various aspects of his work, but it did not diminish my admiration for his ambitious vision. Many essays could be written on its rhetoric, sources, and intellectual context, but since other reviewers1 have begun those efforts, I focus my assessment on its internal logic and the nature of the evidence it presents. The book begins with an informal exposition of five “features” that music may exhibit: conjunct melodic motion, acoustic consonance, harmonic consistency, underlying pitch-class collections of limited size (“macroharmonies” with fewer than nine notes), and centricity. A confluence of these, Tymoczko believes, defines an “extended common practice of tonality” that has endured from the earliest notated examples of Western polyphony through to recent jazz and minimalism. One might demur that melody, harmony, counterpoint, scales, and centricity are all rich concepts whose meanings and correlations vary in different cultures, eras, styles, and even individual works. Yet he sees the repertoire as consistently exploiting certain underlying and interacting interrelationships among these features, which he characterizes by four general claims: 1. 2. 3. 4.

Harmony and counterpoint constrain each other. Scale, macroharmony, and centricity are independent. Modulation involves voice leading. Music can be understood geometrically. (12–21)

Two interlocking types of discourse develop these ideas. One, substantiating the fourth claim, is theoretical and instrumental: it demonstrates a geometric model through which the other claims may be represented and explored, and which can serve composers seeking ways to organize their own 1  See Hook 2011, Whittall 2011, and Headlam 2012a (as well as the author’s response [Tymoczko 2012] and the reviewer’s rejoinder [Headlam 2012b]).

Journal of Music Theory  57:1, Spring 2013 DOI 10.1215/00222909-2017151  © 2013 by Yale University

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music. On its own terms, I find this model persuasive and poietically suggestive, as the author intends. The other is specifically historical, asserting that development of Western polyphony can be understood as progressively more elaborate attempts to realize all five features, whose mutual dependency induced composers to find similar solutions. Evidence for this thesis is adduced from analyses of individual passages, references to recent (rarely historical) theory, and statistical characterizations of many related works. Its persuasiveness may be judged by how well the model fits the diverse music, what processes it can explain that other theories do not, whether it makes convincing connections between passages that other theories would treat as distinct, and how it comports with other theories of tonality that are accepted as broadly applicable and analytically productive. The first five chapters expose the basic concepts needed to characterize the musical features within the geometric model. Much of this material has been presented with technical precision in earlier articles,2 so Tymoczko’s main task is to make it accessible to a less mathematically elite readership. His strategy is to employ concepts and terms with which college freshmen would be acquainted, and he also adopts a subjective stance from which he launches ethical appeals to such a readership—a narrative (xvii) of idealism, betrayal by established authority, and a consequent desire to rebuild an understanding of music from the ground up, informed by putatively scientific methods. This rhetoric occasionally spawns unnecessary critiques of supposedly received wisdom, usually unattributed and oversimplified into easily debunkable caricatures.3 More regrettably, the strategy hinders a self-critical exposition of the model, blurring some of its own constraints and simplifications that make it exquisitely powerful but also limit its applicability. In the following synopsis, then, I try to set the strategy aside and focus on the essential technical features of the model, preparatory to evaluating the evidence for his claims. Chord spaces

All five features of tonality concern pitch, which Tymoczko models as the continuous set of real numbers, consistent with “atonal” pitch-class set theory 2  The two primary sources I would point to are Tymoczko 2006 and Callender, Quinn, and Tymoczko 2008. 3  For a few examples, see 258–67 (on Schenkerian analysis), 220 and 268 (on the explanational inadequacy of the “Romantic Composer’s Whim”), and 29 (on traditional pedagogues’ failure to conceive of voice leading in his abstract, collective way). Certainly, faults in others’ theories may provide a legitimate motivation for new theory, but such straw men leave the unfortunate impression that the author is not confident that his model can stand on its own merits. And this rhetoric is especially grating when it is based on mistaken information, such as the claim that “‘mode mixture’ is not even included in the section on chromaticism”

(218 n. 26) in Edward Aldwell and Carl Schachter’s Harmony and Voice Leading. A glance at that textbook’s discussion of voice leading (e.g., 1989, 69–76) will show a nuanced and generally applicable set of aesthetic considerations—not quantitative rules easily reducible to formal statement—that Tymoczko neither admits nor can match in his model. His characterization of the pedagogy of music since 1900 is also subjective and dated. Contemporary textbooks (e.g., those by Straus and Roig-Francoli) do include substantial treatments of tonality, scales, and voice leading, and jazz compositions have been included in popular anthologies (e.g., Wennerstrom’s) for decades.

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and with the representation in MIDI (Musical Instrument Digital Interface) format that is the lingua franca of many student musicians but also capable of representing non-equal-tempered and nondodecaphonic tone systems. A basic musical object (henceforth BMO, an acronym Tymoczko does not employ) is defined as a series of pitches in which ordering has a significance that I will discuss below. By construing each order position as a dimension, one can plot any n-member BMO as a point in an n-dimensional Euclidean space that is geometric, in the sense that it allows one to define and measure distances between points.4 A succession from one BMO to another can then be conceived as a motion between corresponding points along some path in the space. This space remains inchoate until one decides that some of the objects it represents are equivalent under certain operations, or “symmetries,” for certain purposes. Tymoczko considers an assortment of symmetries: O transposes one member of a BMO by an octave; P permutes an entire BMO; T and I apply a transposition or inversion, respectively, to every member of a entire BMO; and C changes the cardinality of a BMO by adding an order position somewhere and inserting a pitch that duplicates a pitch in another order position. Although heterogeneous, they correspond to plausibly musical equivalence classes and (except C) can be interpreted as “gluing together” (70) points to form a reduced, non-Euclidean space, known technically as an “orbi­ fold,” in which each point represents a class of BMOs, as described below. Tymoczko focuses on equivalence under octave transposition, permu­ tation, and (sporadically) note duplication, implicitly adopting assumptions of pitch-class set theories that associate different pitch series with the same pitch-class content. As a compact representation of the OP equivalence classes, which he calls “chords,” he takes a portion of pitch-series space that has one instance of every set (including multisets on some edges); Examples 1 and 2 show his representation of dyad space and an outline of trichord space. (Just as octave equivalence changes a line of pitches into a circle of pitch classes, the edges of each space are connected in a complicated way, forming a Möbius strip and its higher-dimensional analogue, which he explains with clear diagrams and narrative.) Each point in pitch-class set space represents a multitude of pitch series, and such spaces provide a consistent way to represent both the smaller pitch-class sets used as simultaneities and larger sets that are the source collections, or macroharmonies, for longer time spans. Each space can further be understood as organized around a line of maximally even chords running through its center—in Example 1, the tritones; in Example 2, the augmented trichords, represented as a string of cubes. Surrounding each of these are nearly maximally even chords, the special properties of which are the subject of considerable attention later in the 4  The decision to model pitch combinations, despite the fact that “musicians are primarily sensitive to the distances between notes” (33), reflects the compositional orientation of the book: what should be written on the

page? It is possible to construct interval spaces in which evenness and voice-leading relations can be modeled, but they cannot accommodate many of the relationships Tymoczko is interested in.

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Example 1.  A Geometry of Music, figure 3.5.1b

Example 2.  A Geometry of Music, figure 3.8.2

book. One such property is mentioned early on, since it seems to impinge upon a tonal “feature”: nearly even dyads and trichords are the most acoustically consonant. In order to relate his chords to the structures of tonal harmony, Tymoczko supplements his model with informal concepts. To his formal definition “a C major chord is . . . [an] equivalence class [of BMOs such as (E4, G4, C5), (G3, G4, C5, E4), [etc.]” he adds, “or in other words, to contain all and only the three pitch classes C, E, and G. We can therefore represent the C major chord as the unordered set of pitch classes {C, E, G}” (36). This statement proves momentous for two reasons: it reifies the notion of chord, suggesting that a point in a chord space has a meaning that is independent of the various BMOs it represents, 5 and it allows him to adopt a familiar chord-naming convention (root and quality) that he never formally defines.

5 Indeed, Tymoczko treats pitch-class sets as anthro­ pomorphic agents, such as a “phantasmic and elusive chord [that] . . . exerts its influence on twelve-tone equal-­ tempered music” (60), or a “symmetrical chord [that] passes it symmetry to nearby chords” (58). The distinct nature of this second definition of chords leads him to propose an alternative geometric representation (112): a pitch-

class circle on which one can plot two chords simply with differently shaded dots and show their (potential) voice leading with arrows. As he explains, the benefits of simplicity that this representation offers are offset by its inadequacy for comparing among a large number of sets and for evaluating overall voice-leading efficiency.

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Intervals and voice leading

As elegant as the chord spaces are, they seem inimical to the concept of conjunct voice leading that is supposed to be a feature of tonality. Without order, sets have no voices, and the idea of distance between chroma—between two categories of pitch—is difficult to grasp. To be able to model both harmony and counterpoint, Tymoczko needs first to be able to say how distances in the abstract geometries correspond to the real intervals in the voices of BMOs. His approach is to remember the pitches of events that are represented as pitch classes. For example, in order to conceive of a path from C to E, we must identify the specific pitches in a passage that belong to those classes, say, C4 and E5; then we know the path, in this case involving a clockwise journey of sixteen semitones around the pitch-class clock. In other words, distances between pitchclass events are represented as pitch intervals. These ideas of chord and interval underwrite modeling of the first claimed feature of tonality, conjunct motion, in terms of “voice leading.” For the concept of voice itself, Tymoczko provides inconsistent descriptions and no formal definition.6 This vagueness reflects a curious feature of the model: voice leading is conceived essentially as the collective property of all the pitches in a BMO progression. The formal definition—“equivalence classes of pairs of basic musical objects . . . generated by uniform applications of the permutation symmetry” (42 n. 18)—mentions melody not at all; rather, it can be understood to represent voice leading as all the pairs of pitches that occupy corresponding order positions in two BMOs, pairings that remain the same when the BMOs are permuted the same way. This in turn requires us conceive of voices jointly as an unordered collection of pitch interval paths, without omitting, differentiating, or singling out particular order positions (42) such as highest or lowest. In his figure 2.5.2a (Example 3a), for instance, the voice leading consists of the unordered collection of ordered pitch pairs (each pair involving a path in pitch space) {(F4, E4), (D4, C4), (B3, C4), (G3, G3), (G2, C3)}. “Voice leadings [not individual voices] are like the atomic constituents of musical scores, the basic building blocks of polyphonic music” (42). If one were to concentrate solely on BMOs, this indifference to voice ordering would have little practical significance, because pitches possess an intrinsic registral order that no formal permutation (say, by instrument) can disrupt. But chords, not BMOs, are the main focus of the model, and right 6  A voice is variously characterized as a perceived auditory stream (5), a written melody (13), and a performed instrumental part (36, 85); a feature of chord progressions (17), but something that chord progressions do not have (44); a fixed order position in a BMO (47) but also a member of a BMO whose order position can be changed (39); and the interval from a pitch in one BMO to a pitch in another (41).

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(b)

   

(c)

 



  

Example 3.  A Geometry of Music, figure 2.5.2

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hiddenClearly, octaves

  

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here Tymoczko makes the critical conceptual leap, buttressing his explana withauthoritative    tion adverbs:

spacing unnecessary theugly voice leading in Figure leap 2.5.2b is closely related to that in Figure octave in which some voices appear. We can 2.5.2.a; all that has changed is the       represent what to them by writing (G, G, B, D, F)  5,0,1,–2,–1  (C, G,  is common  C, C, E). This indicates that one of the voices containing G, whatever octave it may be in, moves up five semitones to C; the other voice containing G is held over in to the next chord; the B moves up by semitone to C, and so on. I will refer to such octave-free voice leadings as voice leadings between pitch-class sets or pitch-class(b) voice leadings. (42) –5, –3 (c) –10, –10 7, 9 –22, 14

  “octave-free,” “pitch-class   sets” could be The words and “pitch-class   voice,”    misunderstood   to say that the voice leading connecting two pitch series X and              as  same Y is the the voice leading connecting any member of the pitch-class

set-class of X (under permutation and octave transfer) to any member of the pitch-class set-class of Y. But Tymoczko does not mean that. A footnote offers Roeder, Ex. 7: a clarifying definition: “Voice-leadings in pitch-class space are equivalence classes of progressions [of BMOs, presumably] generated by uniform applications of the octave and permutation symmetries” (42 n. 20). That is, the pitch isonly  leading  voice   if X class   the same  and Y are reordered in the same way,    and if the pitch in every voice in Y has undergone the same octave transfer as the pitch in the corresponding voice in X. Again, there is no explicit men   ofmelodic  a voice    intervals; is a way of understanding different BMO  leading    tion    progressions as equivalent under operations that remove distinctions of order (except with respect to each other) and register. Yet what the equivalent progressions have in common is the way they associate their pitch classes into pairs, each pair defining an interval, so the set of those intervals is indeed a property of each such “voice-leading” equivalence class. Such a class is but a small subset of the all the possible associations of the pitch classes of X and the pitch classes of Y. Tymoczko generalizes so perfunctorily from one specific comparison to this general definition that readers may not grasp the implications. If

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hidden octaves

  





 



 

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unnecessary leap



 

4.  Other pitch-class voice leadings identical to Roeder, Ex.Example 5: Examples 3a and 3b

(a) 2, 2

 

(b) –5, –3

(c) –10, –10

7, 9

–22, 14



 pitch-class Examples voice leading, then so must     3a and 3b havethe same        the BMO  shown in Example progressions   4, which  simply shiftthe register of     some of the pitch voices. They involve hidden octaves, ugly spacing, or odd voice behavior. Tymoczko does not consider the propriety of such entailed equivalents. However, he does give one example of nonequivalence: “Figure Roeder, Ex.2.5.2c 7: is not an instance of this voice leading [the one common to 3a and 3b], since G moves to C by seven descending semitones rather than five ascending semitones (The matters!)” (42–43). It is puzzling that he reem path   specific        by contrasting two  phasizes    this key feature of his model counterintuitively, contrapuntal situations (3a and 3c) that many musicians would regard as  essentially why a composer or listener would     identical.   He does not explain   make such a distinction; indeed,   (205) he himself will treat the bass  later in such textures as mere doubling, exempt from the imperative of conjunct motion. In any event, it is clear that (as with BMOs) pitch-class voice leading involves the entire collection of pitch-interval voice paths, not considering any voice separately from the others. Given two BMOs, the voice leading between the two corresponding chords is represented by a single vector between points in the appropriate chord space. Example 1 provides a clear illustration. It shows two arrows from the point labeled CF to the point labeled DG, one shorter and rightward directed and the other longer, leftward directed, and wrapping from left to right around the space, in a complicated way determined by how the edges of the space are associated. Respectively, these vectors denote the distinct voice 2,2 –5,–3 leadings (C, F)    (D, G) and (C, F)    (G, D), such as appears in all the BMO progressions in Example 5a, which are thereby equivalent, and in the likewise mutually equivalent BMO progressions of Example 5b. Many other, longer voice-leading paths, not graphed in the space, are conceivable between those dyads ((–10, –10), (7, 9), etc.), each corresponding to BMOs with different relative registration of the two possible pitch-class pairings in the voices, as in Example 5c. This illustrates an important aspect of the model: one cannot speak of “the” voice leading between chords (pitch-class sets) without knowing what pitch series the chords represent, because that is the only way to know  

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Roeder, Ex. 5: (a) 2, 2

 

  

   

 

 

(b) –5, –3

 

(c) –10, –10

  

   

   

7, 9

  

–22, 14

   

Example 5.  Pitch-series pairs that realize various pitch-class voice leadings from {C, F} to {D, G}

Roeder, Ex. 7:

what pitch-interval paths describe the pitch-class successions in the chord   blurs  see,Tymoczko  critical   point.   this voices. As we shall      While this model makes it possible to express formally the first feature of tonality,“melodies tend to move by short distances from note to note” (4),      constrained way. The plural  “melodies”  it does so in  a very must be taken to    mean not tunes from a variety of compositions but all concurrent voices in a homophonic two-chord progression representing actual pitches in a score. Moreover, the tendency to move by short distances must be characterized as a collective or overall “efficiency,” not as the property of just some voices in a texture. Efficient voice leadings between BMOs are represented as short vectors between close points in the chord space; thus, in Example 1, the rightdirected arrow from CF to DG denotes a voice leading (2, 2) that is more efficient than the left-directed arrow, (–5, –3). This also shows that the proximity of two chord points does not necessarily mean that the corresponding BMOs manifest efficient voice leading. This crucial point is easy to forget when “chord” is used informally to refer to both pitch-class sets and BMOs. The book’s mentions of “efficient” are often modified with various adverbs—“maximally,” “relatively,” “fairly,” “reasonably”—that may be confusing because their meanings are not self-evident and vary with the context. Obtusely, if we understand “efficient” to be “involving the least motion,” then “maximally efficient voice leading” would involve no change in any voice. Tymoczko, however, always requires some change. In a diatonic triadic context, voice leading is “maximally efficient” when only one voice changes by step; in a chromatic triadic context, on the other hand, it is when two voices change by a semitone. Exact comparison of voice leading size depends on how it is gauged; he does not fuss with those technicalities, however, claiming that for his purposes many reasonable measures will produce the same results. Formalizing the claims

Exploiting all the concepts developed so far, Tymoczko ties together two of the features of tonality in a formal expression of his first claim: Musically, the most important consequence of all this geometry is the following: since a nearly even n-note chord occupies the same cross section of the space as its transposition by 12/n semitones, such chords can be linked by particularly efficient voice leading. Thus, nearly even two-note chords [perfect

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fourths/fifths, that is, Tn class 05] are very close to their six-semitone transpositions, nearly even three-note chords [such as major triads—Tn class 047] are very close [to] their four- and eight-semitone transpositions, nearly even fournote chords [such as dominant sevenths] are very close to their three-, six- [sic], and nine-semitone transpositions, and so on. (97)

Below I evaluate some of the many applications he makes of this statement, but first it is important to understand how he is blending the two different conceptions of chords that I brought out above.7 It is true, for example, that point CF is near point BF ♯ in dyad space (see Example 1), and that {B, F ♯} as an unordered set of pitch classes is a six-semitone (pitch-class interval) transposition of the unordered set {C, F}. But an ordered BMO belonging to chord class CF is not close in pitch space to its six-semitone pitch-interval transposition belonging to chord class BF ♯ (since both voices change by 6), so as representatives of that BMO progression, the chords CF and BF ♯ are not near. Put another –1,1 6,6 way, both (C, F)    (B, F ♯) and (C, F)   (F ♯ , B) are voice leadings between chords CF and BF ♯ , but only the first is “particularly efficient,” while the second more clearly involves a “six-semitone transposition.” This is why Tymoczko says, in the first sentence, that the chords “can be,” not “are,” linked by efficient voice leading. If we conceive of a chord as a set of pitch classes, it makes sense to think about its potential to voice lead efficiently, but when we conceive of it more abstractly as the class of a specific BMO, this potential— the proximity in chord space—says nothing about the actual voice leading, which, Tymoczko emphasizes, is “always represented by specific pitches” (43).8 The larger the chord, the less relevant this potential efficient voice leading is (or, one could say, the more special it is when it actually occurs), because it is just one out of a number of possible voice leadings that increases as the factorial of the chord’s cardinality. Against this objection, however, stands the substantial body of work by many theorists, whom Tymoczko cites, demonstrating some systematic advantages to thinking of efficient pitch-class set content changes as “abstract schemas” (43) through which some inefficient BMO progressions are understood as reregistrations of minimal perturbations of close-position BMOs, particularly when those BMOs are the same as those treated by tonal harmony. Although the space is easiest to visualize for two, three, and four dimensions, corresponding to polyphonies of dyads, trichords, and tetrachords, its structure generalizes readily to higher dimensions, modeling larger collections

7 The “[sic]” inserted into the block quote indicates an invalid inference in Tymoczko’s claim. Dominant seventh chords can voice lead efficiently to their tritone transpositions (in the sense I explain next), but that does not follow from the algebra: for n = 4, 12/n = 3, not 6. Nor does it follow from the fact that such chords are close to their minor-third transpositions, for voice-leading size is not a transitive relation.

8 It is possible to relate actual voice leading to abstract potential voice leading if one considers exactly how the BMOs relate permutationally and transpositionally to referential (e.g., prime) forms of the sets. But that (excruciating) detail is not necessary for modeling Tymoczko’s claims.

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that can be construed either as scales (that is, as means of measuring distance by possibly uneven step sizes) or simply as macroharmonies (the total pitch class content of a passage).9 This modeling occupies chapter 4. In this context, it seems logical to consider analogies with lower-dimensional structures. For instance, which large collections are nearly maximally even, like the tertian triads and seventh chords? Tymoczko shows they are the diatonic, acoustic, harmonic minor, and “harmonic major” (harmonic minor with raised 3ˆ) collections, which not only are familiar but also have some other musically significant features.10 And what does it mean for scales to be near each other in these spaces? For smaller chords, Tymoczko has called upon intuitions of counterpoint, correlating proximity in chord space with the potential for efficient voice leading. So he pushes his analogy even further to achieve a statement of his third claim in terms of his model: just as “chord progressions use efficient voice leading to link structurally similar chords . . . modulations use efficient voice leading to link structurally similar scales” (17). As examples, he describes a modulation from C major to G major as the pitch-class set voice leading in which F moves to F ♯ (129), and the change from F ♯ mixolydian to E acoustic as the pitch-class set voice leading in which D moves to D ♯ (137). But in the latter case, as with many modulations, the pitches articulating the collection change are actually in different octaves, so such voice leading (measured, as required elsewhere, by pitch interval) is actually not efficient at all. Indeed, it is difficult to regard scales and macroharmonies as representing large BMOs because they typically are presented in a few voices across many moments of pitch change, not as a one-change homophonic succession of many voices; that is, their pitch classes typically do not form independent voices.11 To my mind, this invalidates the association between feature 1 and feature 4. To interpret scales’ geometric proximity as potential voice-leading efficiency requires setting aside the premises of the model—that the chords represent voice-ordered BMOs and that distance is measured by pitch intervals—and treating “voice leading” even more abstractly than in the “schemas” advocated for harmonic progression. 9  Since, “typically, macroharmonies are also scales” (15), the distinction between the two is often blurred when Tymoczko speaks informally (as with “chord” for BMO). The distinction is important, though, at the end of chapter 4, which shows how voice leading can be modeled as the combination of chromatic transposition with “interscalar transposition” (the step transposition of chord within the scale that contains it). 10 These four seven-note scales, as well as the wholetone, octatonic, hexatonic, and pentatonic, contain every chord that does not itself contain the chromatic trichord, and they can all be constructed by stacking major and minor thirds, so they include the nearly maximally even trichords and seventh chords that can connect by efficient

voice leading. Some of nondiatonic ones, though, have the problem that some leaps are the same absolute size as steps (127). 11 If we conceive of the pitch classes in scales ordered traditionally as scale degree, it is hard to understand the semitone change in a modulation (say, C major to G major) as occurring in a single order position: F is in order position 4 of C major, but F ♯ is in order position 7 of G major, so the latter (like the other members of G major) would be a “voice-leading interval” of 7 from the order-corresponding pitch class, B, in the C-major scale. To hear F and F ♯ in the same order, position we would need to conceive of G major as C lydian.

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This is not to say, however, that the concept of single-semitone collection change is not valid. Here and in earlier writings, Tymoczko uses it effectively to analyze passages by Debussy, Ravel, and others. A pragmatic difficulty arises, though: the larger the collection, the harder it is to imagine its space and its near neighbors. He shows a clever way of representing nearly even collections in two- or three-dimensional lattices, where each edge represents a voice leading by a single semitone (or step) in a single voice. Each lattice extracts just a few points from the full space and simplifies the geometry, not only helping the reader to visualize special relations among limited chords but also connecting Tymoczko’s representations with other theorists’ graphs of similar collection relationships. The final theoretical chapter explores the mutual constraints of macroharmonies and the smaller collections they contain, in support of claim 2. Geometry plays little role here; only its basic conceptions of collection are employed to define certain quantitative measures of macroharmonic circulation, in an attempt to distinguished different styles. Charitably, one might concur with the author that these measures are simplistic,12 but they do affirm some sensible distinctions, for example, between Max Reger’s triadic but constantly modulating music and Arnold Schoenberg’s equally rapidly circulating but chordally more diverse “atonality.” One might wonder, especially since macroharmonies are identified as keys by tonic and mode (172 n16), whether centricity derives from the structure of a macroharmony, but the final section opines that centricity is mostly achieved contextually, through rhythm and accent. Perhaps in his enthusiasm to regard larger collections as exactly analogous to smaller collections in other respects, Tymoczko generalizes the correlation of consonance and near-evenness to higher dimensions: “A larger chord will be consonant when it contains a preponderance of consonant intervals, and this in turn requires that its notes be relatively evenly distributed in pitch-class space. . . . Highly consonant chords always divide the octave relatively evenly” (62–63). One problem, though, arises from the conflation of pitch-class sets and pitch series: any nearly even collection of pitch classes can be realized as a variety of pitch series, some of which exhibit more acoustic consonance than others. (I assume he does not believe that root-position and third-inversion dominant seventh chords are equally acoustically con­ sonant.) Moreover, his examples are suspect. He identifies three tetrachord 12  Footnotes 5 (155) and 7 (164) outline the “simplistic but hopefully unbiased” procedures employed to generate the statistics of this section; they suffer from some of the faults that I identify below for another statistical study. For instance, his sweeping claim that “Reger’s highly chromatic tonal music often exhibits more harmonic consistency than Schoenberg’s atonal music” (182) is attested by the selective comparison of only one Reger piano work for each of the three pieces of Schoenberg’s op. 11. Each com-

parison contrasts the tallies of different transpositional trichord types (i.e., counting inversionally related chords separately, such as major and minor triads) that the two pieces feature. Not only is there no account of how these trichords were determined from the highly fluctuating textures, but the tallies inexplicably show two different counts for every inversionally symmetric Tn/TnI trichord type (012, 024, 027, 036, 048).

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types as “highly consonant” but includes one (the major-major seventh chord) that is distributed in pitch-class space less evenly than the unlisted French augmented-sixth chord; and he does not list two tetrachords 0347 and 0148 that have more consonant interval classes than the added-sixth chord. For still larger chords, the claim is untenable.13 Tymoczko tries to argue that such distinctions are significant by analyzing Debussy’s “Voiles” as a “macroharmonic transition from dissonance to consonance and back again” (155), but to the extent that “consonance” connotes stability, such a design seems counterintuitively unstable and nonclosural. Reservations aside, I admire these chapters as an imposing, even visionary achievement. They achieve a remarkable integration of various concepts of musical structures through a single geometrical model. They flesh out the four claims, explaining their mathematical representations thoroughly in accessible language and detailed, vivid illustrations. To the extent one accepts its constraints, the model makes it possible to address fundamental questions about how various choices—of counterpoint, chord types, and underlying scales—may affect each other, providing a “unified perspective” to help composers “think about the full spectrum of musical possibilities, ranging from traditional tonality to out-right atonality . . . whether by making creative use of nearly symmetrical chords, exploring the voice-leading spaces of chapter 3, manipulating interesting scales in novel ways, . . . devising new combinations of scale, macroharmony, and centricity, or by doing something else entirely” (190–91). Tymoczko offers a short excerpt of his own music as a sample (178). Applications: A symptomatic example

The second half of the book is intended to show that composers in the past have been interested in combining the five features, to point out how some constraints that are evident in the model manifest in some music, and even to reformulate some accepted theories in light of that evidence. There are many worthwhile insights here, but I must say that I found some of the analytical and historical discussion unconvincing, even exasperating. My frustration stems (as hinted above) from the way that Tymoczko slips between distinct conceptions of chords, the occasionally tenuous relation of his model to my habits of hearing, and his tendency toward hasty generalization on the basis of flimsy or questionable evidence. Consequently, I find some of his methods helpful, as I shall demonstrate, but I am not willing to accept all his conclusions. 13 Evaluating the claim is made difficult by the vague terms “preponderance” and “relatively evenly,” but under a reasonable interpretation it seems fallacious. Tymoczko lists only two “highly consonant” heptachords, the diatonic scale and the ascending minor scale. But the former (ic vector [254361]) has fewer consonant interval classes

3, 4, and 5 than does the much less even 0124589 (ic ­vector [424641]), and the very uneven 0134568 (ic vector [444441]) has the same number of consonant interval classes as the latter (ic vector [254442]). So there are two heptachords that are at least as “highly consonant” as the two Tymoczko cites but do not divide the octave as evenly.

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In order to streamline the rest of my review, I think it would be useful to give a preview of these sorts of problems. They are evident in section 3.10, Tymoczko’s first argument that the constraints that efficient voice leading and harmonic consistency exert upon each other in theory are manifest in actual historical practice. This is a crucial passage because its conclusions motivate much of the subsequent reconceptualization of tonal harmony and chromaticism. The reading I give is very detailed, but I think worth the effort, because it helps to clarify the author’s way of thinking—which other readers may be more willing to accept than I—as well as the problems I allege. After Tymoczko explains, in the passage quoted at the beginning of the previous section of this review, how the model shows that major-third root progressions of major triads can be realized with the most efficient (nonzero) voice leading (as well as a similar observation about minor-third progressions of dominant seventh chords), he launches the application as follows: “Suppose, then, that a composer is interested in exploiting efficient voice leadings between two major triads or two dominant seventh chords. Theoretically, we should expect an abundance of major-third relations between the triads and minor-third relations between the dominant sevenths” (97). This may be read innocuously as advice to current composers, deriving clearly from the established geometry, as discussed above: if you want efficient voice leading between major triads (resp., dominant-seventh chords), use major-third (resp., minor-third) progressions. The possibility of such a strategy says nothing about whether past composers actually wanted to achieve efficient voice leading with such progressions or others. Yet the apparent advice morphs into a hypothesis that past composers did indeed share a common interest in efficient voice leading, and that their interest can be inferred from their favoring of particular root progressions of chords: “And when we look at actual music, we see that this is indeed the case. . . . This suggests that composers’ harmonic choices are indeed guided by the voice-leading relationships we have been exploring” (97–98). Below I evaluate the evidence that Tymoczko musters, but whether or not one finds it convincing, the argument is insufficient. As I have emphasized in my summary of the geometric model, just because certain pitch-class set pairs can manifest efficient voice leading does not mean that the corresponding pitch series (BMOs) must or always do. Therefore, an “abundance” of them would never establish that composers chose them to achieve efficient voice leading, unless the corresponding BMO progressions actually manifested that efficient voice leading, and not just the favorable root relations. (And even if they did, composers may have had other reasons to choose them.) Revealingly, Tymoczko does not undertake such a direct demonstration. His argument is indirect, at best, and relies on some rhetorical sleightof-hand. It begins by changing the hypothesis: “Direct juxtapositions of minor-third-related dominant seventh chords already occur in baroque and classical music, while direct juxtapositions of major-third-related seventh chords are very rare” (97). Suddenly, he is not investigating whether minor-

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third progressions of dominant seventh chords are more “abundant” than all other progressions, but only whether they occur more frequently than majorthird progressions. It is not immediately clear why he has reduced the scope so severely; since the model is purely chromatic, we are given no basis for considering a root progression by 4 (“major third”), the magic number fixed by the formula 12/n, to be anything like a progression by 3 (“minor third”) any more than a progression by 2 is. In any event, finding that such progressions “occur” does not prove that they are “abundant.” Moreover, a footnote (97 n. 28) shows that even this existence claim is limited: “baroque” means only some Bach chorales, and “classical” means Mozart’s piano sonatas. No justification is given for regarding these as a representative subset of their respective styles, exhibiting the same statistical properties. The fact that only “several” minor-third progressions of dominant sevenths appear in the Mozart corpus (as the footnote reports) would seem to also make them “rare,” too. If these composers were writing dominant-seventh chord progressions to achieve efficient voice leading, would not one expect there to be many by minor third? Further questions dog the next piece of evidence: “Similarly, majorthird-related triads sometimes occur across phrase boundaries, as when V/vi moves directly to I, while there is no analogous convention associating minorthird-related triads” (97). This claim is even more limited: it says only that there is no “convention” that minor-third-related triads “occur across phrase boundaries.” It also assumes, without justification, that it is meaningful to conceive of voice leading across phrase boundaries—that is, at moments of discontinuity when conjunct motion would be counterproductive; it seems odd to focus on those moments rather than on moments of continuity within phrases. Indeed, there one can find minor-third progressions of triads—for instance, I–V/ii—arguably more frequent than the major-third succession Tymoczko favors. (An instance from a Mozart sonata will be cited below.) All the special pleading suggests weakness in his hypothesis. Note also that he has dropped the modifier “major” from “triad”; this seems to be merely a convenient synecdoche, but it prefigures later unwarranted generalization, as we shall see. Taken literally, his assertion is con­ tradicted by the existence of conventions involving minor-third progressions across phrase boundaries between different-quality triads, for instance, the return to minor tonic after a cadence in the relative major. Regardless, he forges ahead: “Figure 3.10.3 indicates that this asymmetry also characterizes the more chromatic music of the nineteenth century. In Schubert, major-third-related triads occur about 1.5 times more often than minor-third-related triads, while minor-third-related dominant seventh chords occur about fourteen times more often than major-third-related dominant seventh chords. In Chopin [the proportional disparity is 1.6 and 2.2, respectively]” (97–98). The figure (reproduced as Example 6) tabulates “nondiatonic chord progressions in a large number of pieces by Schubert and

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Schubert

  Roeder on Tymoczko

Chopin

Root Motion

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Example 6.  A Geometry of Music, figure 3.10.3

Chopin . . . show[ing] that major triads [sic, not any “triad,” as the preceding text says] are more likely to be connected by major third than by minor third, while the reverse is true of dominant sevenths” (99). As above, Tymoczko is addressing the relative, not absolute, abundance of only two root progressions. He does not observe whether the tallied progressions actually manifest efficient voice leading, nor does he provide or justify any null or competing hypothesis against which these statistics could be interpreted (e.g., that all third progressions of triads are equally likely). Indeed, he imposes a “nondiatonic” condition that excludes the numerous fifth progressions, such as V–I, as well as such common major-second progressions as IV–V, that are fundamental to this musical style. He offers no explanation, but I suspect that including them would have greatly reduced the relative frequencies of the root motions Example 6 does show, making the putative asymmetry less significant. Even with these maneuvers, Tymoczko seems bothered by the frequency of putatively unfavorable minor-third progressions of major triads, for he rationalizes, “Since minor-third-related triads can be connected by reasonably efficient voice leading, we would still expect these progressions to appear periodically” (97 n. 29). But this is a self-defeating observation. Fifth and minor-third progressions (of major triads) involve exactly the same voiceleading intervals, and since semitone progressions involve the same taxicab (or smaller Euclidean) distances, and since fifth, minor-third, and minorsecond progressions vastly outnumber the major-third progressions, one might reasonably conclude from the data that Schubert and Chopin prefer less than maximally efficient voice leading.

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The statistics, such as they are, also argue in other ways against efficient voice leading as an important determinant of root progression. For dominant seventh chords, tritone progressions involve exactly the same voice leading (two voices moving in contrary motion by semitone) as minor-third progressions. According to Example 6, however, Schubert and Chopin write the latter but almost never the former. Nor do the statistics bear out the inherent symmetries of counterpoint as Tymoczko has defined it. Since any chord progression can be retrograded without affecting the size of the voice leading, there should be no difference in the number of progressions that ascend and descend by the same interval. But the data show that for many intervals there are differences. Similarly, since any chord progression can be pitchinverted without affecting the size of the voice leading, Tymoczko’s assertion about the connections between major triads must also be true about minor triads, but he does not seem to have conducted such potentially confirmative investigations. Methodological flaws that are hazards of corpus studies further cripple the demonstration. In the earlier examples, by examining all the Mozart piano sonatas, Tymoczko implicitly assured us that the data sets were, in some sense, comprehensive and integral. But about Schubert and Chopin he says only that the sample was “large” (and from Bach chorales he considered only seventy), leaving open the possibility of selection bias, a problem he acknowledges elsewhere (160) but not here. Harmonic language varies even within a single composer’s repertoire, so including, say, the écossaises but not the mazurkas could skew the results. Even less reassuringly: “The methodology here was crude but, hopefully, unbiased: I programmed a computer to look through MIDI files for simultaneously sounding chords that were either [major?] triads or [dominant?] seventh chords. I then tallied up the root progressions of each type” (97 n. 29). For one of the central claims of the book, I expect better proof than a “crude” method whose lack of bias is only hoped, not demonstrated. How reliable are the MIDI encodings—are the files unproofed public domain, and how were they quantized? By considering only “simultaneously sounding chords,” does not the method neglect harmonies articulated by the Alberti bass figuration found throughout this repertoire? What are the criteria for succession in this method—if two simultaneities are separated by a nonchordal but harmonic texture, does the method count them as successive? Why should we assume that efficient voice leading, or any kind of musical continuity, for that matter, is constantly operative? Judging by the rest of the book, Tymoczko believes that his statistical investigations provide some objective test for his claims, but (particularly in light of recent misuses of statistics in other fields; see Siegfried 2010; Yong 2012), it is important to note, at least for the benefit of future researchers, some of their flaws: unclear hypotheses; no details about, or justification for, the data set; no critical evaluation of the reliability of data; and ad hoc data analytical methods, with no critical discussion of alternatives. Tymoczko admits to some of

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Example 7.  Mozart Piano Sonata, K. 281, iii, mm. 1–2

these weaknesses but seems to believe that they affect only the precision of his results, not their validity. Other methodological problems stem from the interpretative nature of harmonic analysis. The difficulty, familiar to all instructors of freshman-level theory classes, of determining roots in nonhomophonic but implicitly harmonic textures is subtler than one might gather from Tymoczko’s casual mention. The third movement of Mozart’s Piano Sonata, K. 281 (Example 7, considered only implicitly in the statistics), provides an example of how many heuristics are involved in reading the harmony underlying even a very simple texture. It begins with the attack of a single pitch F5. Is there a chord here? If so, what other pitches are implied? (In order to hear the first pitch as the consonant initiation of a linear progression in B ♭ major, and considering how its reprises are prepared by dominant harmony, I understand other pitches to be implied that belong to tonic harmony.) On the second attack are three pitches, but two are of the same pitch class. Is it an incomplete triad? If so, is it major or augmented? (If we hear it as G major, and we hear it preceded by tonic, then we hear a minor-third progression of major triads, which Tymoczko claims is relatively infrequent in these sonatas.) On the following downbeat, the notation shows a C-minor triad with its E ♭ delayed by an F grace note; in MIDI recordings (e.g., those on classicalmidiconnection.com), the chord on the beat is {C, F, G}. So it is not clear that Tymoczko’s algorithm would find any triads in this sort of passage. For any but the simplest textures, harmonic analysis is a hermeneutical exercise whose results may vary depending upon assumptions about large-scale tonal processes, dissonance, rhythm, phrase structure, and form. One might “hope” that the “more than thirty” uncredited theorists (229) who analyzed the Mozart sonatas were that consistent, but apparently they were not to be trusted with Schubert or Chopin. In short, the evidence Tymoczko presents does not prove that com­ posers were seeking efficient voice leading. The problems could have been avoided if he were not trying to prove such a restricted hypothesis. The theoretical distinctions the model makes could be useful if a composer were interested in composing purely major triads (or purely dominant sevenths) with smallest voice leading possible, but that seems too constrained a principle to manifest itself consistently in a broad repertoire, or even across a single piece. But it is easy to accept that the difference the model predicts between minor-

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third and major-third root progressions manifests in some passages of music, the distinctiveness of which cannot be appreciated as readily by other theories that do not discriminate so strongly between these two progressions. Happily, many of Tymoczko’s introductory examples are directed toward the appreciation and comparison of such special passages. For instance, directly following the statistical discussion just considered, he cites two brief chord series from quite different genres (98–99). The A–F–D7 progression repeated in Nirvana’s “Heart-Shaped Box” switches from major triads to a seventh chord as the root progression changes from major to minor third. Similarly, a Schumann piano piece moves chromatically from an F-minor 5/3 to an E ♭7 chord through an enharmonic A-dominant 4/3, the change from triads to sevenths occurring at the change from major-third to minor-third root motion. They both involve a sort of “double emploi” in which the middle chord in the progression is conceived both as a major triad (thus omitting a note in the second passage) and as a dominant seventh chord (thus adding a nonextant note to the first passage). When one considers the actual passages, however, rather than relying upon Tymoczko’s sketches of them, they prove less than ideal exemplars. In the first case, the actual guitar voice leading, typically for the grunge style, involves parallel fifths moving by leap, and the A-minor vocal melody contradicts the A-major chord. Tymoczko must resort to claiming that the progression evokes an underlying “very familiar schema” that is only potentially efficient. In the second case, the BMO voice leading is clearly efficient only if one selectively ignores doublings; even then, the triad is minor not major, and the seventh chords progress by tritone, not by minor third (recalling the problem with the Schubert statistics). Tymoczko claims there are “countless other examples of this phenomenon” (99), but when he cites one, the Beatles’ “Glass Onion,” he mischaracterizes the Am–F7 progression in its verse as “ juxta­ pose[d] major-third-related triads.” History

The strengths and weaknesses evident in these preliminary examples are amplified as we move into the second half of the book. Tymoczko embarks upon an ambitious program, inspired by Jared Diamond’s inquiry into how human history may have been molded by environment: Since the five components of tonality constrain each other in interesting and nonobvious ways, composers who wish to combine them have only a few alter­ natives at their disposal. The simple fact is that many composers from before Josquin to after John Coltrane have wanted to combine these features, and hence have struggled with a problem that has a relatively small number of solutions. We should therefore expect that when we dig deep enough, we will start to find interesting similarities among their approaches. (196)

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When they were introduced informally, the five features seemed pertinent to a wide range of polyphony. But the constraints on “solutions” have been shown only within a restrictive and distinctly modern model of them, in which “conjunct motion” means efficient collective voice leading and “harmonic consistency” means OPT equivalence indifferent to ordering and octave. Tymoczko’s daunting task is not only to establish the “simple fact” of composers’ intentions and “similarities” of their music but also to prove that their conceptions are so consistent with the model that it is the only explanation for the similarities.14 (Otherwise the “limited solutions,” such as they are, could be attributed to other, perhaps less systematic reasons.) I find some arguments more persuasive than others. Because the later examples serve as springboards for more extensive discussion in chapters 7–9, I incorporate my review of those chapters into the summaries of the corresponding examples in chapter 6. Early polyphony

The first is a simple two-part counterpoint, “Alleluia Justus ut Palma,”15 sourced from the eleventh-century didactic treatise Ad organum faciendum. The claim is that, “although it stands at the very beginning of the Western polyphonic tradition, the piece clearly exemplifies a number of our five features.” As Tymoczko runs down the checklist, though, the number turns out to be small and the clarity variable. Concerning conjunct motion, he generalizes that “the voices move by step, with contrary motion preferred to parallel motion” (197). However, inspection shows that more than half of the motions in the organal voice are by leap (forty-five of seventy-five, not counting crossphrase changes, which make the statistics more unfavorable), and more than half the relative motions are not contrary. Momentarily not in a quantitative mood, Tymoczko presents no precise tallies, but he tacitly admits the problem later by modifying his description to say that “the two voices move stepwise or near stepwise 70% of the time” (199)—quite a different matter, and one that does not establish the prevalence of collective voice-leading efficiency, which is how the first feature has been modeled. Concerning harmonic consistency, he acknowledges that it only exists crudely as a distinction between consonance and dissonance, not as the model’s equivalence relation. Macroharmony is constant but somewhat ambiguous, since hard and soft Bs both appear, and there is only a “weak sense of centricity.” 14  Across the book, Tymoczko seesaws between claiming and disclaiming knowledge of composers’ thoughts and intentions. Nowhere is the conflict more evident than in this section, for soon after the passage I have just quoted, he says that his goal is not “to explain how earlier composers actually thought” (196). Compare also his reading of

Brahms’s and Debussy’s thoughts about Tristan (47), and of Schubert’s and Chopin’s conceptions of chromaticism (219). 15  Tymoczko always misnames this piece, substituting “et” for “ut.”

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To examine, nevertheless, whether the geometric model is appropriate, Tymoczko plots the seventh phrase in dyad space as points connected by vectors indicating the voice leading between successive simultaneities (199). It is not a notably successful confirmation of his theory about the mutual constraints of harmony and counterpoint: only two chords voice-lead to their transpositions, neither maximally efficiently, and only one of eight voice leadings is by step in both voices. Finessing that point, Tymoczko demonstrates that a series of two voice leadings is reprised between different series of three chords. But what should be a clear geometric representation of this phenomenon—the appearance of identical two-vector wedges in different parts of the space—is obscured by the extra chords and vectors interpolated into the second series. Renaissance vocal polyphony

Apparently the music of the next half millennium is irrelevant to (or poses difficulties for) the historical thesis, for the second excerpt is the opening of the homophonic second part of a phrygian motet attributed to Josquin (the top two staves on each system of Example 8). One may wonder whether this special passage is truly typical of Renaissance music, but it does literally comprise multiple concurrent voices, often changing stepwise in a constant scale through a limited selection of consonant simultaneities. To represent the passage as a chord succession strictly with Tymoczko’s model, one would need to plot every simultaneity BMO in a four-dimensional space, connecting them by vectors representing the voice leading. Few of these vectors would be short, because nearly every voice leading involves a leap in at least one voice. Because of the various kinds of doublings, few of these BMOs would be “consistent” in the sense of belonging to the same OP or OPT class (under either scalar or chromatic transposition), and some have only two pitch classes. The prospects of showing the mutual constraint of harmony and counterpoint in such a representation seem dim. So Tymoczko proposes a further abstraction that renders the passage more tractable, as demonstrated on the lowest staves in the systems of Example 8. In brief: disregarding suspensions, passing tones, and BMOs with only two pitch classes, for BMO with three pitch classes, it is often possible, by disregarding one of the two voices sharing the same pitch class, and by disregarding octave leaps (as at progression 2), to understand their successions as stepwise motion connecting triads that are related by step transposition within the prevailing diatonic scale. In effect, he represents selected fourvoice BMOs as three-voice OP chords, in effect imposing an “anachronistic” (201) pitch-class set description. The geometry of this is quite complex, involving the projection of points in the middle of a four-dimensional space to its three-dimensional boundaries, and he does not explain it. But at this level of abstraction, he is able to make trenchant observations about the relation of

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Roeder, Ex. 8 (correx 1/12/13):

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not even clear how well the concept of voice leading, which was defined in terms of uniform transformations of different BMOs, applies. Moreover, their connection involves a voice crossing in pitch-class space, which Tymoczko will shortly inform us is less efficient than the “natural uncrossed

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contingent both upon the others’ (range, relative motion) and upon emer� gent of the� sonorities. ��� � �� These �� 34 �� properties ��considerations � �� are even more appar�� � � ��� �� ent later in� the same chorale, mm. 6–10 of which are�shown in Example 10. � motion seemingly affirms the principle � Frequent stepwise of efficiency, but closer inspection reveals many exceptions. At (a), overall efficiency could have been improved if alto and tenor both ascended by step; apparently the tendency of 4ˆ to resolve down, and a desire for an open-spaced 5/3 at the cadence, take precedence. At (b), all voices leap to mark a phrase boundary without harmonic change. At (c), the tenor’s sustained pitch across three chords, mutating from preparation to suspension during a bass-alto voice exchange, suggests that the counterpoint is conceived not simply as chord-tochord voice leading and that not all chords with tonic notes mean the same thing. At (d) the tenor future behavior is determined by its present dissonance, not simply by a desire to minimize its motion, and so the chord’s meaning is not simply a function of its unordered pitch-class content. At (e), the tenor foregoes small motion to 1ˆ in order to create a complete 5/3. So, inasmuch as stepwise voice leading is operative, it is as only one heuristic among many that frequently override it. Tymoczko does acknowledge such factors, dismissing them as “style-specific nuances” not pertinent to the “basic conditions under which polyphony itself is possible” (49). But he has established neither that overall efficiency is necessary nor that “harmonic meaning” must rest solely upon identity of unordered pitch-class content. The chorale phrase Tymoczko cites in chapter 6 (figure 6.4.1), however, is somewhat more suitable for the model. He simplifies it as he did the Renaissance excerpt, this time dropping bass voice to concentrate on the upper three voices. While this procedure flies in the face of the figured-bass conceptions of Bach’s era, the upper voices are clearly BMOs, and most of them

alternative” (50). Nor is the voice leading into the third BMO the most efficient possible (it would be if the bass leaped up to A ♭ 3). The final BMO is highly atypical of cadences in the style, and since it has more pitch classes

and is of a different class than the others, it raises questions of how one can be determine whether a harmony is “meaningful.”

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�

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Reviews 

Roeder, Ex. 10:

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  Roeder on Tymoczko

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Example 10.  Bach, Chorale no. 173, “O Herzensangst,” mm. 6–10

are complete triads that voice-lead efficiently within the diatonic scale (except at cadences). The two brief demonstrations in chapter 6 so far, of the mutual contingency of voice leading and chord structures, raise an important issue, which Tymoczko recognizes and addresses. He has established that nearly maximally even n-member chords—for n = 3 and n = 4, the very chords that are considered to have roots in tonal theory—are close in the orbifold to their own transpositions by 12/n, offering composers a way of connecting rooted chords smoothly. To what extent are such efficient progressions compatible with accepted theories of harmonic progression? One might expect the answer to be: not very. Triad progressions by major third have not been considered basic to harmonic tonality. There is also the inherent symmetry of the voice-leading model, a problem that arose implicitly in chapter 3’s statistical characterization of Schubert and Chopin chord progressions (see Example 6 above). Voice-leading size is the same for a chord progression and its retrograde, so if it were the governing principle of chord succession, one would expect to see V moving to ii just as often as ii moves to V. But one does not. Rameau’s harmonic theory provides a clear take on both issues: musical succession derives from an axiomatic, descending-fifth motion of a fundamental bass, from which all actual “voices” are generated. Nevertheless, Tymoczko makes a valiant attempt to reconceive the theory of harmony in terms of the voice-leading paradigms his geometric model favors, in which third, not fifth, progression is fundamental. It is understandable why he wants to do this: fifth progression can be decomposed into two third progressions, and third progressions facilitate relatively efficient collective voice leading. If achieving such voice leading, as he supposes, is a composer’s constant imperative, then third progressions could be understood as fundamental. And this would allow him to expand the connotation of “harmony” in claim 1, from a mere classification scheme for BMOs to a principle of root succession. He addresses the problem of symmetry by adding a new constraint on chord succession, in the form of directed graphs of roman numerals that express a harmonic grammar, that is, a representation of all syntactically

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Example 11.  A Geometry of Music, figure 7.1.1

valid progressions. (The graph for major keys is reproduced in Example 11.) He explains that “chords can move rightward by any number of steps along this chain from tonic to dominant; however, they can move leftward only along one of the labeled arrows” (226; that is, the dashes in the center of the graph should more properly be written with more right-pointing arrows). It is meant to show how third progressions, which the geometric model shows may be efficiently voice-led, relate to the fifth progressions more commonly understood to govern harmonic tonality: “The idea . . . is to portray these falling third progressions as the fundamental path from tonic to dominant,” asserting that “falling thirds are more fundamental than falling fifths, even though fallings fifths may be more common” (228). Strangely, the path marked by the series of single-step voice leadings along the chain of thirds from I to V (figure 7.3.1a, 239) makes those two chords quite distant, contrary to usual conceptions. Although Tymoczko insists on the distinction between permissibility and probability, perhaps to allay possible doubts about his demotion of fifth progressions, he blurs it by saying that the grammar shows “why some progressions [are] common while others are rare” (226), why certain kinds of motion are “more often permissible” than others, and that certain chords are “more likely” to move in certain ways (227). He omits mediant harmony entirely because it is “rare,” although later examples (7.1.4, 7.3.2c) and a table (7.1.6) do show progressions with iii are used, suggesting that it is “permis­ sible.” The reference to probabilities suggests that the model is constructed post hoc from existing practice, but contrarily, he speaks about it “predicting” certain patterns. As a first-order grammar, the graph permits unlikely strings, such as I–vi–vii–V–IV6 –ii–vii–I. To the eager young composer whose instructor would critique such a putatively valid progression as unstylistic, Tymoczko’s book offers little corrective guidance. He is mostly interested in two-chord successions, so he does not consider how well it models longer ones. (This narrow focus, he admits later [258–59], is a general weakness with his conception of tonality.) Even the chord-pair grammar is not very restrictive, as it allows

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twenty-one of the thirty possible successions of different harmonies. Therein lies a problem: although he calls it a “thirds-based” model, third progressions are actually the least prevalent in the graph (there are six, compared with eight step progressions and seven fifth progressions). This, and the fact that the layout of graphs is intrinsically arbitrary, suggests that a different interpretation of the graph might be equally plausible; indeed, it is easy to rewrite it as an “ascending second” model, or a “descending fifth” model—simply move the nodes around while maintaining their arrow relationships. Manifestly, then, this particular presentation of the graph is intended to highlight efficient third progressions and does not follow from any intrinsic structural priority they take in the grammar. Tymoczko’s evidence for the validity of his model is a table (230), referring to the same data sets of (selected) Bach chorales and Mozart piano sonatas, that shows, for each of the six non-iii roots, the relative probability of its progression to each of the other five. This is a bit of misdirection, because it obscures the fact that relative to all progressions, V–I and I–V are by far the most common, which might naturally sway one’s opinion about which progressions are basic. The tallies also omit fauxbourdon passages, fairly efficient by nature but involving stepwise root motion, and even sequences, although one would expect at least some of the latter to involve harmonic continuities (and indeed, later examples in the chapter treat them this way). Even so, in no case is a third the most common progression. It also seems self-defeating to tabulate Mozart analyses by humans who were likely indoctrinated in the very fifth-oriented harmonic theory Tymoczko is trying to supplant. In any event, the probabilities are irrelevant, because he has just asserted that they do not determine which progressions are fundamental; apparently, the mere presence of a progression, no matter how relatively common, is sufficient to be allowed in his model. But then the data are not decisive: one cannot determine that a grammar in which thirds are abstractly “fundamental” suits them better than does any one of a vast multitude of other grammars. Chromaticism

Chapter 6 proceeds next to examples of nineteenth-century compositions, where voice leading involves more semitones and triad successions do not remain in a single diatonic collection. Figure 6.6.2 (see Example 12) analyzes the “radical chromaticism” of Chopin’s E-major Prelude, in which “almost all the unusual moves . . . can be interpreted in light of two different voiceleading ‘systems’: one connecting triads whose roots relate by major third, and the other connecting seventh chords whose roots relate by minor third or tritone” (218). The analysis seems intended to indicate that the passage manifests an especially tight relation of harmonic consistency and efficient voice leading. But on closer inspection, such an interpretation is tenable only

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Journal of Music Theory 57.1: Music Examples, p. 17

Roeder, J o uEx. r n a12 l (correx o f M u s i3/8/13): c Theory

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minor third system

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Example 12.  A Geometry of Music, figure 6.6.2

if one accepts readings that are inconsistent, unjustified, or contradictory to some of the premises of the model. The sole instance of the “major-third system” is a single pair of chords; Tymoczko must implicitly invoke the “3+1” abstraction here to disregard the bass voice and its leap (which is shown with a wrong pitch). The labeling of the second chord in this progression—as a pure G-major triad—is inconsistent with the analysis of its transposition one measure later, labeled as a C dominant seventh. In the score, this chord progresses to an A-major triad, but Tymoczko adds a note to the latter to make it a dominant seventh. In violation of his definition of voice leading, he also performs “individual permutations” on the following diminished-seventh chord, so that the A7 chord’s A moves to B ♭ rather than to the actual G—and on the last chord of the measure—eliminating a voice exchange. These emendations raise methodological questions. If one is allowed to read any major triad as an incomplete dominant seventh, then any apparently anomalous minor third progression of major triads can be read as a supposedly normative progression of dominant sevenths. (Similarly, any anomalous major third progression of dominant sevenths can be read as a normative progression of major triads with extra notes that one may ignore.) And if one may individually permute the chords, the efficiency of inconveniently large voice leadings can be improved and possibly even minimized.17 The idea of voice leading as the actual pitch connections between BMOs and the distinction between the possible and the actual become vague in such analyses. 17  Further, by interpolating an untextual F ♯ into the penul­ timate chord, Tymoczko contrives a final instance of the minor-third system, but because that chord is a minor, not dominant, seventh, it raises the same questions that I did with respect to Example 6. One could just as well argue that there is no such system for seventh chords in general,

because some seventh-chord progressions by nonminor thirds intervals (such as C7 to B7) have the same voiceleading size as seventh-chord progressions by minor third (C7 to E ♭Ø7); that is, minor-third progressions of seventh chords in general do not exhibit distinctively efficient voice leading.

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These issues persist even in the best parts of the second section. The analyses of chromatic and scalar music in chapters 7–9 display Tymoczko’s strengths, supported by his model—the ability to conceive of all possible instances of particular voice-leading patterns, to organize them systematically, and to generalize them. For example, from familiar notions of chromatic embellishments of diatonic progression, he presents ideas of “generalized augmented-sixth” progressions, in which the voice-leading characteristic of true augmented sixths is deployed between other types of chords. He draws links among some special passages of Chopin’s music that are well suited for modeling in the four-dimensional chord space, because they involve literal semitone voice leading between nearly even tetrachord types: “This cubic geometry . . . encapsulates in a single image rules that would otherwise require careful verbal specification—presenting a kind of music ‘game board’ whose interval structure mirrors Chopin’s presumed compositional process” (287). The climax of chapter 8, a Cook’s tour of Tristan und Isolde, shows both the virtues of his theory and its limitations. Beginning abstractly as a composer might, by cataloging all efficient (noncrossing) voice leadings between half-diminished and dominant seventh chords, he builds a case that Wagner demonstrat[es] a sophisticated understanding of four-dimensional chord space: utilizing all of the most efficient voice-leading possibilities from half-diminished to dominant seventh . . . , substituting one half-diminished chord for another . . . , moving between chords by way of their chromatic intermediaries . . . , reusing the same basic contrapuntal schema with different sonorities . . . , and even reproducing the open-ended quasi-sequences of Chopin’s E minor prelude. (301)

I find these insights and demonstrations compelling, but they also elicit a candid and telling (294) acknowledgment of the problem that has been obvious but rarely explicit in many previous examples as well: “We [!] are dismayed to find that [the actual] voice leadings are not maximally efficient.” Although he admits that this might suggest that “all our work of efficient voice leadings has been wasted,” he asks the reader to regard them instead as “embellishment[s] of an efficient ‘background’ voice leading” (295)—that is, to regard potentially efficient pitch-class set progressions as the sources of actual BMO progressions that are not actually efficient. This reverses the ontology of his main geometric model, in which chords are derived from BMOs, and jettisons a significant feature of his theory, that “the specific path matters!” (43), in order to regard the opera as a protoserially exhaustive investigation of abstract possibilities. Twentieth-century tonality and jazz

As it happens, such an attitude serves him well in the last chapters as he turns to post-1900 repertoires that involve nearly maximally even chords and scales. He distinguishes three approaches to the coordination of these two types of

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structures, exemplifying them with excerpts by a variety of composers, including Scriabin, Ravel, Debussy, Grieg, Nyman, Prokofiev, Shostakovich, Janácˇek, Reich, and Stravinsky (plus Lennon and McCartney, Pete Townshend, and Miles Davis). In one approach, chords are each elaborated with scales to which they belong. In another, the scales themselves change, often by semitone, and the chords are simply drawn from them. Lastly, the succession of scales sometimes seems directed by the intent to hold a subset of pitch classes constant across the change.18 The examples vary in their level of abstraction and the degree to which they instantiate claims about modulation as efficient voice leading. In some, the voice leading is literally semitonal, but in others (319) only schematically so, as in Tristan. The identification of scales can be speculative when not all pitch classes are present, and Tymoczko labels them by tonic and mode somewhat haphazardly and inconsistently, such as in figures 9.3.1 and 9.3.2 (324– 25), when “D dorian” on the score is shown as “C dia” on a scale lattice. (This indifference stems from the same aspects of the theory that prevent the distinction between chord inversions: only pitch-class content is important.) Even in these presumably exemplary passages, scales are quite variegated, and there are often gaps in the scores with no obvious scale. Tymoczko often plots them on lattices that make them appear to be close as a group, but the actual successions in the music usually change the collections by more than one semitone (325, 329–30, 335, 339). The music of Steve Reich receives special attention, inspired partly by the work of Tymoczko’s erstwhile collaborator, Ian Quinn. In many of these minimalist textures, large, nearly maximally even collections appear as BMOs, and changes from one to another help articulate the formal design. Although these collections can be represented as forming a compact lattice, Tymoczko’s analyses acknowledge that their successions are not always efficient; indeed, to my ear, common-tone retention seems more important than minimal voice leading.19 The culminating chapter 10 seeks to break down “irrational barriers” that prevent the appreciation of “the profound connection between jazz and other musical styles” (390) in order to hold up jazz as the “modernist synthesis” (387–88) of the chordal and scalar techniques in much post-1900 concert music. To anyone acquainted with jazz history and theory, this will not be as big a surprise as Tymoczko imagines. Jazz musicians have been quite conversant with early twentieth-century concert music, and chord-scale theory, taught to all jazz novices and based on traditional harmonic notions of chords and

18 It is interesting that Tymoczko is willing to entertain common-tone retention as a principle of scale succession but does not consider others’ claims that the technique might explain chord progressions (303)—regrettable, because the sharing of subsets has an elegant geometric representation.

19  I am thinking especially of the opening and close of the first movement of Reich’s New York Counterpoint, in which the alternation of the two close scales Tymoczko identifies is accomplished through a subtle layered overlapping and nonefficient voice leading of smaller chords.

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scales as stacks of thirds, exactly matches (and, I suppose, inspired) one of techniques discussed in chapter 9. Tymoczko gives a beautifully concise, detailed summary of this theory, showing the links between such techniques as tritone substitution, avoid tones, and “side-stepping,” and his voice-leading concerns. For his final exhibit, he applies as many of the book’s concepts as he can to his transcription of an improvisation by Bill Evans. He shows that Evans employs clever chord substitutions that preserve close voice leading, and that he often arpeggiates chords that exhibit efficient voice leading either literally or abstractly. (Efficient voice leading is even more evident in Evans’s left-hand comping, which Tymoczko does not discuss here.) On the other hand, few changes are elaborated with their own scales, and the progressions are generally diatonic, not distinctively suitable for the geometric model. Indeed, much of Tymoczko’s discussion concerns motives and rhythm, which are fascinating but not directly related to the model, either. Conclusion

Considered as a whole, then, Tymoczko’s evidence does not convince me of his historical thesis. Although the model provides a useful way of conceiving and investigating certain possibilities, some of its principles and relations seem too constrained or abstract to constitute an “extended common practice.” To discern the five features and the interrelationship so evident in the model, one needs to model all music as consistent homophonic textures in which notes must be imagined or ignored to maintain constant cardinalities and chord types. For early music, this entails an anachronistic conception of vertical structures as pitch-class sets. For romantic music, the model works well for some passages of music, most concretely when clear voices move by small intervals to form successions of triads and seventh chords. The idea of an efficient “schema” as the basis for inefficient passages is also appealing, but it entails relinquishing some of the original premises about the nature of voice leading and distance. These objections do not mean that I find the theory unproductive. Tymoczko intends it principally to support new composition, and that may be the best way to take it. Readers may not accept that it describes what older composers were doing, and it may not help them to imitate them, but it is a creative way to think about basic local compositional problems involving the counterpoint of chord types within scales. To analysts of chromatic tonality and scalar music, moreover, it offers a bridge between pitch-class set theory and tonality, and I urge them to try to apply the geometrical model, in order to get an informed grasp of schemas and their explanatory potential. As a modest illustration, I submit Example 13. It presents sketches of two passages from Rachmaninoff’s grand Etude-Tableau in E ♭ minor, which rang in my brain as I studied this book.

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Example 13.  Analytic representations of passages from Rachmaninoff’s Etude-Tableau in E ♭ minor, op. 39/5: (a) and (b), mm. 18–21; (c) and (d), mm. 41–50

At m. 18, the etude abruptly departs the essentially diatonic world of its theme and seems to plunge into an alternative pitch universe, signaled rhythmically by a surprising change of beat subdivision. The harmonic succession, sketched in (a), presents a minor-third sequence of chord pairs alternating half-diminished sevenths with dominant sevenths. The diagram in (b) shows that they seem to be systematically exploring the tetrachord types nearest the B diminished-seventh chord. (It is adapted from Tymoczko’s description of Debussy’s “Fêtes,” abstractly associating these near-contemporary but rather different composers.) The last possible leg of the sequence does not appear; rather, its D ♭7 is represented by a D ♭ major triad that directs the music toward an arrival at B ♭ . Such a purely pitch-class description emphasizes collection

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Example 13 (continued).  Analytic representations of passages from Rachmaninoff’s Etude-Tableau in E ♭ minor, op. 39/5: (a) and (b), mm. 18–21; (c) and (d), mm. 41–50

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change while discounting the exact registration of the notes. But I hear register as significant: the outer voices encourage conceiving of the passage octatonically, and each leg of the sequences registrally emphasizes B ♮ , as indicated by the asterisks. Those emphases should not be dismissed, because they anticipate future events. A simple repercussion is evident immediately after the arrival in B ♭: a modulation through an ascending chromatic 5–6 progression, which one might characterize as efficient voice leading involving a major-third rootprogression of triads, takes us first to B minor. But B is most important at the climax of this central section, sketched in (c). A fortissimo B major leads into a sequence that alternates root motion by major third and tritone, suggesting Tymoczko’s two efficient voice-leading systems (although his theoretical distinction between the behaviors of triads and seventh chords is difficult to maintain here). It works through an entire whole-tone scale to achieve a B7 in m. 45. For this chord sequence, (d) suggests a possible underlying, efficiently voice-led schema. At m. 46 a rapid progression suddenly shifts us to B ♭ major, which alternates strikingly with a Bm7 collection, suggesting semitone voice leading in three voices. My delight at being able to invoke so many of Tymoczko’s concepts in this description is tempered by the realization that they contradict or are mute about aspects that are important for a satisfactory analysis of the passages. The hypothetical schema in (d) involves an inexorable descent of registerless collectives of voices, but the focus of the actual progression, as indicated by stemmed notes, is the chromatic ascent of the highest line from tonic to fifth of B major. (Measures 46–48 affirm that the texture is directed by a single voice, a chromatic ascending line, and that other systematic but inefficient voice leadings can support it.) The simple homophonic schema also omits the sequence’s inner-voice elaborations—the overlapping whole-tone scale fragments directed alternately toward the thirds and sevenths of the chords. Lastly, the focus on chord-to-chord progression obscures the larger tonal context, in which the most important dominant chord of the piece is approached prepared by a greatly expanded augmented-sixth harmony. Such large- and small-scale linear elaborative processes seem primal to this music, and I am reluctant to sacrifice my experiences of them at the ascetic altar of the orbifold (64).

Works Cited

Aldwell, Edward, and Carl Schachter. 1989. Harmony and Voice Leading, 2nd ed. New York: Harcourt Brace Jovanovich. Callender, Clifton, Ian Quinn, and Dmitri Tymoczko. 2008. “Generalized Voice Leading Spaces.” Science 320: 346–48. Headlam, Dave. 2012a. “The Shape of Things to Come? Seeking the Manifold Attractions of Tonality.” Music Theory Spectrum 34/1: 123–43.

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Headlam, Dave. 2012b. “The Reviewer Responds, but Not in Kind.” Music Theory Spectrum 34/1: 150. Hook, Julian. 2011. “Review of Dmitri Tymoczko, A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice (Oxford University Press, 2011).” Music Theory Online 17/3. www.mtosmt.org/issues/mto.11.17.3/mto.11.17.3.hook.html. Siegfried, Tom. 2010. “Odds Are, It’s Wrong: Science Fails to Face the Shortcomings of Statistics.” Science News 177/7: 26. Tymoczko, Dmitri. 2006. “The Geometry of Musical Chords.” Science 313: 72–74. ———. 2012. “ ‘Hey, Wait a Minute!’” Music Theory Spectrum 34/1: 144–49. Whittall, Arnold. 2011. Review of Steven Rings: Tonality and Transformation. Musical Times 152: 93–102. Yong, Ed. 2012. “Bad Copy.” Nature 485: 298–300.

John Roeder, a professor at the University of British Columbia School of Music, made early contributions to the study of musical geometry and voice leading in post-tonal music. His recent projects include analytical essays on the music of Bartók, Carter, and Saariaho, as well on the analysis of world music, including the Oxford collection of essays Analytical and Cross-Cultural Studies in World Music, which he coedited with Michael Tenzer. 

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