headlam review of tymoczko

7/29/2019 Headlam Review of Tymoczko

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A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. By Dmitri

Tymoczko. Oxford Studies in Music Theory. Series Editor: Richard Cohn. New York: Oxford University

Press, 2011, xviii + 450 pages.

REVIEWED BY DAVE HEADLAM

In his new book—part text, part continuing research agenda, part personal statement, part corrective to

“contemporary music theory” (3) wherein “tonality remains poorly understood” and chromaticism is still

“shrouded in mystery” (xviii), and part polemic against “the dissonant, cerebral music of Schoenberg and

his followers” (xvii) and advocacy for “tonal” music that “sound[s] good” and is “likeable” (xvii)—Dmitri

Tymoczko offers a comprehensive attempt “to understand tonality afresh” (xviii).1 In a “retell[ing of] the

history of Western music” (xviii), Tymoczko writes not only for “composers and music theorists” but for

“students and dedicated amateurs” (xviii), the latter a larger audience including “scientists and

mathematicians” (book jacket). The book thereby includes some definitions of the basic elements of

musical sound (“Sound consists of small fluctuations in air pressure,” 28) and music theory (“a basic

musical object [is] an ordered series of pitches, uncategorized and uninterpreted,” 36), but also includes

terms such as “Möbius strip” and “tesseract” (69ff, 284ff), statistical charts reminiscent of information

theory-based approaches from the 1960s (158–85), a methodology based on geometry and expressed

largely in terms of concepts from “vector graphics” (Cartesian points, lines, and curves, in

multidimensional spaces), and appendices with more formal presentations of selected topics, such as a

“metric” (measurement system) for voice-leading distance defined by way of the “submajorization partial

order” (essentially a preference for even distributions, 398).2 In the two large parts, labeled “Theory” and

“History and Analysis,” Tymoczko first presents an interpretation of the elements of tonality and lays the

framework for the geometrical apparatus used, and then demonstrates the theory with analyses of pieces

from a wide variety of musical styles, spanning Organum to an original composition.

Tymoczko’s agenda is ambitious: he wants to simultaneously expand our purview and streamline

our understanding of tonality. As part of his presentation, he also wants to promote tonality, particularly in

its manifestation as jazz in the twentieth century, as a suitable topic for study in the “post-tonal” world,

following David Lewin and others by using theory developed for non-tonal music in the effort.3 The book’s

subtitle refers to an “extended common practice” Tymoczko proposes, “stretching from the eleventh

century to the present day” (195), larger than the usual referent, as a logical consequence of his assertions

about the elements of tonality. These assertions are brought into focus by comparisons to the “unpleasant”



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state of a lack of tonality in atonal music (185) and are expressed in terms roughly equivalent to Winston

Churchill’s views on democracy: according to Tymoczko, while “typical Western listeners prefer” it,

tonality is not “better” (7), but “people dislike atonal music . . . because they think it sounds bad” (185),

and the presumably few who do like it have essentially developed a taste for something like “clam chowder

ice cream” (186).4 Tonality, Tymoczko asserts, has elements of “universality” to it, related to our

“biological inheritance” (7), and is a “living tradition” still viable in “today’s wide open, polystylistic,

multicultural, syncretistic, and postmodern musical culture” (225).

Although the book jacket uses adjectives “groundbreaking” and “revolutionary,” and Tymoczko

refers almost exclusively to his own earlier work on the subject (except for references to John Roeder,

Richard Cohn, Clifton Callender, and Ian Quinn, 65, Note 1), the application of geometry and its

formalization of spatial relationships to model musical elements and relationships is, of course, as old as

the study of music itself, and is found in every era of music theory.5 The most recent manifestations of this

tradition, all influential on Tymoczko’s views, are found in the “compositional spaces” promulgated most

prominently by Robert Morris, and in the “transformational” musical spaces from David Lewin adopted in

Neo-Riemannian Theory (by Richard Cohn), particularly in related writings from John Clough.6 In its

continuation of the geometrical tradition and adoption of aspects of these present-day interpretations of

musical spaces, as well as its “back to the future” sensibility in simultaneously reviving tonality as well as

visiting its history, this book is both of its time and a return to the past.7

What is laudable in Tymoczko’s approach is his many demonstrations of the ways in which tonal

processes and repertoire are circumscribed; the geometrical methodology is presented as essentially style

and time period neutral and allows comparisons ranging from Josquin to Miles Davis. This approach

constitutes real theory-building, in that it reduces large quantities of note patterns, styles, and rules of

composition to a few theoretical constructs (for instance, in the section where Tymoczko notes, “At first

blush, this might seem shocking, as if all the glories of the Renaissance could be reduced to just two basic

contrapuntal tricks,” 237). Aside from the theoretical observations based on geometrical elements, the most

compelling arguments Tymoczko offers, which stem from his earliest writings, concern the role of scales,

in particular the parallel roles of chords and scalar collections, where functional relations among the latter

are treated as slowed-down versions of analogous relations among the former. This parallel structure

suggests a theory of structural levels, or at least a “hierarchical self-similarity” (266), which Tymoczko

notes are akin to Schenkerian levels (18–19). Tymoczko’s scalar interpretations take elements from both

diatonic scale theory and jazz scale-based theory, and remind us of the parallels between Debussy, Ravel,

and Stravinsky and jazz, as well as the return to diatonicism in the roughly parallel emergence of



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minimalism and modal jazz.8 Tymoczko even posits a new “twentieth-century ‘common practice’” to

connect “impressionism, jazz, and postminimalism” (351) and offers analyses of related compositions by

Steve Reich, Bill Evans, and others to buttress the parallels.

Aside from the somewhat uncharitable view of the music of the Second Viennese School, which

Tymoczko compares misleadingly to random processes (184, explained below), it is refreshing to encounter

arguments for continuity, rather than further bifurcation, of the great musical tradition that we all work

within. Tymoczko’s ecumenical attitude toward this continuing tradition is reflected in analyses of

composers outside the usual canonic “Bs,” such as Grieg and Jańăcek, and his continual referencing of

popular music, such as rock and recent figures such as Kurt Cobain and John Lennon, along with the

emphasis on jazz composers and performers, but does not extend to women composers, who are absent

from his consideration.

Tymoczko anticipates many of the arguments and directions a review of his book might take,

largely by a series of “disclaimers” and rejoinders throughout. To the curmudgeonly charge that his new-

fangled theories are too complicated, and what we have now is perfectly adequate, Tymoczko responds

with how he has learned from colleagues at Princeton to deal with “outraged forefathers” (vii). To his

slighting of the facts of different historical periods and styles, Tymoczko offers no apologies for his

assertion of a larger continuity (196), although he follows the usual categories in his roughly chronological

analyses of Part II. To the complaint that his interpretations don’t resonate with the reader’s experience as

listener, Tymoczko counters that he is presenting an idealized composer’s point of view (4, 8, etc.), even

invoking, Schumann-like, compositional alter-egos, “Lyrico” and “Avanta” (12, 19, etc.). However, the

text is supplemented by references throughout to idealized “ears,” often personalized, that have superior

hearing/cognizing abilities (266, Note 45, which contrasts “hearing” with “hearing plus thinking”). To

justify the use of geometrical grids rather than musical notation in analysis, Tymoczko asserts that “our

visual system is optimized for perceiving geometrical shapes such as triangles, but not for perceiving

musical structures as expressed in standard music notation” (76). To the discounting or outright ignoring of

Schenkerian principles, except to present them ahistorically as a “challenge” to his own view (258–67,

omitted from the Index), Tymoczko opines that “[Schenker] ended up advocating a radical model of

musical organization according to which entire pieces were massively recursive structures, analogous to

unimaginably complex sentences. The complexity of these hierarchical structures far outstrips those found

in natural language, and seems incompatible with what we know about human cognitive limitations” (259).

To the focus on pitch alone in his explanations and analyses and omission of motive, theme, form, and

rhythm and phrasing, Tymoczko responds that his book is not a “hands-on guide to composition” which



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would include items like “octave, instrument, and register” and other musical items (196). Concerning his

omission of tuning considerations, he states that “standard tuning systems are in the grand scheme of things

reasonably similar, and can all be represented by very similar geometrical structures” (196). Finally, to the

charge that he “models chords as unordered collections of pitch classes, and counterpoint voice leadings in

pitch-class space,” ideas “more commonly associated with the twentieth century,” Tymoczko asserts that

his “purpose . . . is to provide more general theoretical tools for understanding tonality—and in this context,

an abstract approach is perfectly appropriate” (196).

There is, nonetheless, room for commentary and additional considerations on these and other

issues. This review consists of four parts. First, I will outline the main ideas in an overview. Second, I will

detail what I consider to be problems in the book. Third, I will discuss what I consider to be its strengths

and innovations. I will end with some considerations of the analyses in the second part of the book, in

comparison with other approaches. In the course of the review, I will refer to four authors relevant to the

context of a theory of tonality, particularly in Tymoczko’s emphasis on chromaticism and the geometrical

model for analysis: Matthew Brown, Howard Cinnamon, John Roeder, and Walter O’Connell. Also notable

as a comparison to Tymoczko’s text is Timothy Johnson’s Foundations of Diatonic Theory (2008); I will

also compare Tymoczko’s work with Steven Laitz’s textbook The Complete Musician (2011). Steven

Rings’s Tonality and Transformation (2011) is also relevant, but is not considered in depth here.

OVERVIEW

The two parts of A Geometry of Music, “Theory” and “History and Analysis,” are roughly of equal

length (191 and 203 pages respectively). Each part contains five chapters, “Five Components of Tonality,”

“Harmony and Voice Leading,” “A Geometry of Chords,” “Scales,” and “Macroharmony and Centricity”

in Part I and “The Extended Common Practice,” “Functional Harmony,” “Chromaticism,” “Scales in

Twentieth-Century Music,” and “Jazz” in Part II. Six appendices follow, five on various topics presented

more formally and a final “pedagogically-oriented” appendix with “Some Study Questions, Problems, and

Activities.” The initial chapters in each part lay out the premises and topics for the remaining chapters,

which largely flesh out the ideas introduced. In addition, most chapters begin and end with general

ruminations that add nuance and context to the arguments: music and language (Chapter 1), acoustic factors

(Chapter 2), the extent of compositional possibilities (Chapter 5), pedagogy (Chapter 6), approaches to

analysis (Chapter 8), the twentieth-century common practice (Chapter 9), and a general plea for wider

purview in scholarly writing (Chapter 10).



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Two related points are emphasized throughout to generate the narrative that tonality is, if not

innate, the closest thing to it in human music-making, and seemingly virtually inevitable, given our shared

physiology.9 Indeed, the first point, the assertion of the extended tonal period given in the subtitle, might be

deemed an argument from human evolutionary physiology, and it stems from the five features Tymoczko

argues are central to tonality (4–11): conjunct melodic motion (stemming in part from auditory and vocal

physiology); acoustic consonance (largely derived from the harmonic series, asserted as a preference for

“many listeners,” 6); harmonic consistency (supported by arguments from gestalt studies on grouping by

similarity); limited macroharmony (a term suggested by Ciro Scotto for collections of 5–8 notes, where

“scale” is reserved as a term to measure distance, 6, Note 8); and centricity (another “widespread feature”

of music, 7). The continuous presence of these five elements suggests that musicians a thousand years ago

and today are really not that different, a position Tymoczko supports with references to music cognition

and perception, many from David Huron (5, Note 3, on conjunct melodic motion, for instance).

The second point is an argument roughly from “Occam’s Razor”—the principle of avoiding

unnecessary plurality. In this view, there are relatively few “solutions” to the melodic and harmonic

implications of the elements of tonality, and these take the form precisely of the tonal musical materials—

the relatively few triads and seventh chords and their consonant intervals, largely stepwise voice leading,

two main scales, etc.—commonly found in Western art and popular music. In other words, what has been

had to be, and, outside of the anomaly of non-tonal and serial/twelve-tone music, pretty much has been.

Accordingly, our explanations should reflect this inevitability and its connecting power—what remains,

and what Tymoczko undertakes, is to show how it all connects.

These two points—the human natural affinity, and even that of “nonhuman animals” (30, Note 2),

for tonal expression and the limited solution-space for tonal compositions—are presented within

Tymoczko’s interpretation of tonal behavior, which itself has two main aspects. First, rather than the usual

tale of tonic note, chord, and key, Tymoczko proposes a “general theory of keys” (16) from “scale,

macroharmony, and centricity,” and so highlights the relationships between the central elements of chords

and scales: tonality in this view stems from harmonic functionality between chords on local levels and from

parallel functionality between scalar collections on larger levels. Every description of chords—their

collectional aspects, voice leading, and functions—has a parallel description in scalar collections. Second,

in his interpretation of tonal materials, Tymoczko includes two melodic/harmonic features of tonality, both

associated with neo-Riemannian theory (neither one listed in the Index, but they are defined on pages 49–

50 and 61–64, respectively). The first is “efficiency” in voice leading, for the shortest possible distance,

also categorized as “smooth(ness)” and “parsimony,” terms with a long pedigree in music theory.10 The



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second is near-maximal evenness of tonal chords with respect to the octave, where interval cycles are

maximally even: C–E–G–C compared to C–E–G*s–C, for instance.11 The important element is the “near”

in “near-maximal evenness;” this allows a triadic progression such as C–E–G / C–F–A / B–D–G / C–E–G

to occur by the close proximity of the chord tones (135/513/351/135 with 1 = root, 3 = third, 5 = fifth) due

to the near-even spacing. By contrast, progression of perfectly even chords, such as augmented triads,

allows for much less complex arrangements of voice leading, such as straight transposition in all voices.

The opposite of maximally even, that is maximally clustered, progressions in collections such as C–C*s–D,

lack the clear separation of chord tones from connecting notes and present problems in distinguishing

musical figure from ground discussed in Straus (1987).12 The implications and interrelation of these two

features—Tymoczko formalizes this relation by showing how one emerges from the other on page 60 (Note

34)—direct the inquiry through the rest of the book.13

In Chapter 1, the five features of tonality (given above) lead to four claims: 1) harmony and

counterpoint constrain one another; 2) scale, macroharmony, and centricity are independent; 3) modulation

involves voice leading; and 4) music can be understood geometrically. The first claim is familiar from any

text on counterpoint or part-writing, but Tymoczko enriches it by beginning his arguments on the parallels

between chord and scale/collection here (13–14). The second constitutes Tymoczko’s conception of the key

structure of tonality, including adoptions of scale step, textural separation of voices, and “modal”

distinctions between and within collections and centric notes. The third is part of Tymoczko’s assertion that

scales and chords act similarly on different levels (continuing some of the fallout from the first claim), and

the fourth is the starting point for the geometrical apparatus for modeling voice-leading and harmonic

spaces.

The discussion of “Harmony and Voice Leading” in the second chapter presents a “musical set

theory” (28) from a Lewin and Morris-influenced application of group theory-based post-tonal theoretical

concepts (see especially Morris 1987, Chapters 4 and 5), focused on the so-called “OPTIC” symmetry

operations (octave shift, permutation, transposition, inversion, and cardinality change). The third chapter

follows through on earlier implications of musical spaces to unveil Tymoczko’s multi-dimensional

geometrical grids.14 The two-dimensional (2D) grid is a vector space with 144 (12 × 12) points defined by

pitch-class pairs and distances by line segments; in a musical “sleight of hand” (following Tymoczko’s

evocative metaphor of music theory revealing the magician’s tricks, 22–23) the grid is a 45 degree rotation

of a Cartesian x, y alignment of pitch-class pairs (Examples 1(a) and 1(b)), in which distances traverse the

same differences/intervals on the horizontal axis and the same sums/inversional measures on the vertical

axis (Example 1(b)). (Example 1 also contains a similar arrangement of sums and differences from Table 1



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from Perle [1996, 17] and Berg’s alignment of the interval (difference) cycles from a letter of 1920 to

Schoenberg, reproduced in Perle [1977, 5]. I shall refer to these below.) The 2D space maps onto a

“Möbius strip,” conceptually a ribbon that is created with a half twist; an imaginary 1D creature—an ant, as

presented by Tymoczko in a discussion (69–70) that recalls the “Prelude-Ant-Fugue” chapter of Douglas

Hofstadter’s Gödel, Escher, Bach, a book that is the ancestor of the multivalent approach to music

explanation employed by Tymoczko—traversing the entire strip becomes an expert in navigating boundary

conditions as it careens off the “edge” in defining a path through pitch-class space. The 3D version

(Example 2, from Tymoczko’s Figure 3.8.2) is a prism shape with a similar twist, with twelve planar faces,

each one consisting of 30 or 31 (sums 0, 3, 6, 9) multiset trichords at the same sum (= 364 multiset

trichords, reduced from 1728 [12 × 12 × 12] by permutational equivalence). The three types of trichordal

multisets, in tonal terms the “complete” through “incomplete chords” (C–E–G, C–E–E, C–C–C,

proportionally about 60, 36, and 4%), move from the inner central core which connects the four maximally-

even (ME) augmented triads, through the surrounding area which constitutes the “nearly ME” major and

minor triads, to the edges, which contain the doubletons, then the singletons. A pyramid moving in time

represents a 4D space, at which point we lose our ability to conceive of higher dimensions of these spaces,

but in which nonetheless, similar principles are at work. Tymoczko summarizes these principles as follows

(96): 1) efficiency equals close proximity (the 1-cycle connections of differences and sums—for instance

047 and 048 in sum planes 11, 0); 2) “evenness” is distance from the center (most even at the center); 3)

“layers” and direction define voice-leading characteristics such as relative motion; and 4) boundary

conditions are mirror-like.

[Example 1 here]

[Example 2 here]

The group-theoretic cyclic component of the grids is the 2 × 0,6,3 × 0,4,8 and 4 × 0,3,6,9

behaviors of even and related nearly-even chords and their transpositional levels (97): Tymoczko makes the

point that chords of cardinality n may be arranged with efficient voice leading to their “transposition[s] by

12/n semitones.” He later notes that adding contrary motion provides for additional possibilities, principally

efficient voice leading between T5-related chords. Here a chart such as the one in Example 3 might have

been helpful, showing interval-cycle collections and their mappings, ranging from “clustered” (larger

intervals) to “near equal collections” (smaller intervals) and illustrating how the mappings tend to the

smaller intervals in this order. Tymoczko explores some tonal implications of these cyclic characteristics,



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noting the preponderance of triadic relations by major thirds and seventh chords by minor thirds in tonal

music. He adds that five-note chords can participate in these features: the more even of these collections

have efficient voice leading as well as fractional transpositions, that in the quantized twelve-note world

require an accompanying parallel motion to allow for contrary motion models (101–2). Thus interesting

voice leading, combining parallel, contrary, and oblique motions, emerges.

[Example 3 here]

The essential geometry from these group relationships is given in two types of lattices that

Tymoczko presents in chromatic and diatonic forms, in cardinalities of 2, 3, and 4 notes. Example 4 shows

a representation in post-tonal notation. The chromatic “[0167]” lattice (my name) shows interrelated

set-class [05], [01], and [06] dyads and voice leadings, with a central [06] producing [01]s by [05]-based

voice leading or producing [05]s by [01]-based voice leading. The diatonic version includes three dyads

and a voice-leading step, 03/04/05/S (for “thirds, fourths, fifths” and diatonic steps) and thereby requires a

more elaborate connecting apparatus by a line segment rather than a point vertex in Tymoczko’s

representation. Tymoczko adapts both of these to show fifth-progressions in tonal contexts (109-10). The

corresponding cardinality 3 and 4 systems are hexatonic, with the familiar major third related triads, and

octatonic, with the transforms of the central set-class [0369] into [0258]s, [0268]s, and (not shown by

Tymoczko) [0358]s.15 Some of the many figures these arrangements (and the diatonic four-note system, not

shown here) appear in are given; the later sections on Schubert and major thirds (280-84), on Chopin (284-

93), and on Wagner’s Tristan (293-302) are based on these lattices. Tymoczko expands on the use of their

use by moving from chords to scales as the central element (“x” in Example 4)—by showing that analogous

relationships can be created by voice leading among chords and scales, he fleshes out his main point about

the role of levels of chords and scales generating tonality (246-58).

[Example 4(a1)–(c1) here]

Chapters 4 and 5, and their follow-through to the twentieth century in Chapter 9 present related

topics centering on the functional equivalence of chords and scalar collections (which Tymoczko divides

into “scales” as units of measurement, and “macroharmonies” in collections of 5–8 notes) and the property

of centricity. Combining a post-tonal conception of the relation between large and small collections of

notes with a jazz theory-based understanding of the role of scales, Tymoczko essentially proposes a “scale-



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based compositional method” with Debussy as his flagship composer, and extends his interpretation into a

tonal “common practice” in the twentieth century, contrasted with atonality:

Diametrically opposed to atonality is the “scalar tradition” that makes extensive use of

familiar scales and modes. This tradition encompasses at least six major twentieth-century

movements—impressionism, neoclassicism, jazz, rock, minimalism/post-minimalism,

and neo-Romanticism—and a good deal of other music as well . . . there are enough

commonalities among twentieth-century composers to justify talk of a scalar “common

practice.” (186–87)

Part II consists of five chapters presenting analyses incorporating the geometrical methodology,

and preceding roughly chronologically from two-voice medieval counterpoint, through functional tonality,

tonal chromaticism, tonal features in the twentieth century, and ending with jazz. Tymoczko’s main point

about intervals, chords, and scales recurs throughout: it is not just the consonant intervals that define early

counterpoint, he argues, but the scalar collection that provides the boundaries for the note choices.

Tymoczko illustrates this with an “octatonic” counterpoint that sounds “wrong” despite using only

consonant intervals (200, Figure 6.2.4). Later, the same correlation between scale/collection and chord is

given as a defining attribute of the two types of chromaticism: the embellishment of diatonic harmony with

its source in scalar collections, such as V/V, allows for tonal function, but unmediated chromaticism,

outside of scales, locates function in the chord by chord succession alone. In this attribute, chromaticism

breaks down the tonality-forming parallels of the treatment of chord and scale, hence the loss of tonal

function with excessive use of accidentals.

The chapters on “Functional Harmony” and “Chromaticism” contain materials closest to a more

traditional text on tonal harmony. Tymoczko first defines tonal grammar in descending thirds, rather than

the traditional circle of fifths, insisting “that falling thirds are more fundamental than falling fifths, even

though falling fifths may be more common” (228); he does include fifths in progressions, however, as in the

section on “Fourth Progressions and Cadences” (207) in earlier music, and his harmonic statistics on chord

progression, drawn from Bach and Mozart, as well as a subsequent section on sequences, include fifth as

well as third motions.

The discussion of voice leading focuses on the familiar problem of fitting three-voice chords to

four-voice settings. Two solutions are presented: first, in what Tymoczko calls a “3 + 1” voice leading,

three voices maintain complete triads and the fourth doubles; and second, a “nonfactorizable voice leading”



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(here “4v”) in which the doublings change voices, thus all four voices are required for complete chords,

since the triad members appear uniquely in each voice (202–7, 236–38). Tymoczko shows the opening of

Josquin’s “Tu pauperum refugium,” in which most pairs are 3 + 1, with a couple of 4v pairs (chords 7–8,

12–13). Tymoczko states that it is remarkable that the 3 + 1 and 4v techniques, the latter defined as

“split/merge” voice leading as in 1513 / 3151 (split, merge), account for triadic progression from Dufay to

Bach (237, mentioned above).

The use of geometrical models in the chapter on “Functional Harmony” is focused in the section

on “Modulation and Key Distance” (246ff). The geometry of scales is a 3D “cube lattice” with minor-third

cycles of chromatic motions in the three directions: C–C*s, A–A*s, F*s–G, D*s–E on the x axis, F–F*s,

D–D*s, B–C, G*s–A on the y axis, and G–G*s, E–E*s, C*s–D, A*s–B on the z axis, so that C major

moves to G major (+y axis, add F*s or ^*s4), D melodic minor (+x axis, add C*s or ^*s1), and A harmonic

minor (+z axis, add G*s or ^*s5, Example 5). I find it easier to think of scale degrees for this section, where

the scales map as shown in the example.

[Example 5 here]

The chapter on “Chromaticism” includes sections on “Decorative Chromaticism;” “Generalized

Augmented Sixths;” a neat (although unacknowledged) updating of “Brahms the Progressive” in the

section “Brahms and Schoenberg;” a review of the now-familiar harmonic relations by thirds in “Schubert

and the Major-Third System;” a study of Chopin using a geometrical model in “Chopin’s Tesseract;”16

and

finally a long section on the Tristan Prelude, modeling the ubiquitous half-diminished seventh chord to

dominant seventh progression in this opera.

The final chapter on “Jazz” considers topics found in most textbooks on jazz, including “Basic

Jazz Voicings,” “From Thirds to Fourths,” ”Tritone Substitution,” “Altered Chords and Scales,” “Bass and

Upper Voice Tritone Substitutions,” “Polytonality, Sidestepping, and ‘Playing Out,’” and “Jazz as

Modernist Synthesis,” and presents an analysis of Bill Evans’s solo to “Oleo.” This chapter is a synthesis of

references to jazz sprinkled throughout, including Tymoczko’s self-described “Fundamental Theorem of

Jazz” (156), which “states that you can never be more than a semitone wrong.” This tongue-in-cheek

statement stems from the near-evenness property of macroharmonies, as always providing an escape route a

semitone away. Tymoczko’s ending chapter on jazz is fitting, as his scale-based theory largely derives from

the scale-based pedagogy common in jazz, and the principle of semitone sliding in improvisation reflects

his system of scalar modulation.



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PROBLEMS

The first problem I find in Tymoczko’s book is the lack of references, background, and

acknowledgement of other sources for many of the ideas. Notable in this context is the omission of the rich

history of geometrical metaphors and models in music theory: names such as Pythagoras, Calcidius, who,

like Tymoczko advocated for geometry over harmonics, and Zarlino, who debated Galilei over the

judgment of reason versus the practicalities of experience (somewhat like Tymoczko’s admonishments to

music theory texts, see 270–71 on the augmented-sixth chord), and terms like the Quadrivium, the

“sounding body” and the connection of the wave equation with a number of scientific and musical

developments and more recent fractals. (All this information is easily available online.) A charitable view is

that Tymoczko adopts a Descartian attitude of personal contemplation and self-discovery alongside the

somewhat more relaxed attitude toward attribution of ideas in the textbook format, allowing him to largely

eschew the usual scholarly practice of footnotes in the text, and to limit his references, which, aside from

himself (15 items) and Cohn (11 items), and collaborators Quinn and Callender, are largely restricted to

two or three items per author.

But virtually every page is in need of some references or a more realistic view of the literature: for

instance, the one reference to Charles Smith to the effect that he and Tymoczko have similar views on

chromaticism (269) hardly reflects the engagement that Smith has had with the whole notion of chromatic

functionality in relation to the Schenkerian view (as given in the original exchange by David Beach).17 The

explanation of maximally-even sets (61–64) contains no direct reference to the work of Douthett, Clampitt,

or Clampitt and Carey, authors who preceded Tymoczko in defining the categories and characteristics of

these systems (these authors do appear in other contexts in notes). But in a footnote (64, Note 41)

Tymoczko refers to Agmon and Cohn, noting that he generalizes many of their notions to many more

scales and to a larger collection of consonant sonorities than merely the triadic: “Ultimately, [there is] a

non-obvious connection between efficient voice leading, harmonic consistency, and acoustic consonance—

a connection that we now understand as a simple consequence of the hidden symmetries of circular pitch-

class space.” However, as I have mentioned, even Anonymous 2 was well aware of this connection, and the

step-wise progression of consonant chords has been the main focus of part-writing and counterpoint texts

for a millennium.

For another example of uncited predecessors, the structure of the 3D geometrical grid, previously

called an orbifold in Tymoczko’s writings, with its adjacent sectors either a difference of 1 intervals or

differences of l sums (047 and 158, or 047 and e48, etc.), indeed reflects a distinction between near-even



13

deriding of “contemporary music theory” and music theory texts in general leaves a bad taste by the end,

perhaps something on the order of clam chowder ice cream.26

My second criticism relates to the dismissal of Schenkerian theory, which is treated somewhat like

an unsubstantiated, peripheral theory rather than one of the mainstays of our understanding of tonality. It is

hardly necessary for me to support this assertion; instead I will comment on Schenkerian approaches to the

problems Tymoczko confronts. In his book Explaining Tonality: Schenkerian Theory and Beyond , Matthew

Brown neatly parses views of tonality in his explication of Schenkerian Theory as a model for explaining

tonality, analyzing tonal pieces, and formalizing a theory of tonality, as opposed to other interpretations of

the theory as a theory of musical structure, organic coherence, structural levels, or voice leading. Brown’s

chapters follow from his six criteria for evaluating theories, which are accuracy, scope, predictive power,

consistency, simplicity (Occam’s razor), and general compatibility,27 but his thesis emerges from “two

basic claims” from “Schenker’s thought:” “1) the laws of tonal voice leading are transformations of the

laws of strict counterpoint and are related to certain laws of functional harmony; and 2) complex tonal

progressions can be explained as transformations of simple tonal prototypes.”28 In the discussion of this

first point, it becomes clear that Brown contrasts with Tymoczko’s assertions of historical continuity, as for

Brown, although they share elements in common, there are three distinct compositional environments based

on intervals, triads, and functional tonality, with correspondingly different behaviors in each.29 While

Tymoczko alludes to this argument (212–13) and even makes a potentially useful distinction between tonal

and earlier music by the presence of hierarchic self-similarity in the former (212), his central premise

largely papers over these distinctions.

In his brief section on Schenker, labeled somewhat deceptively from a chronological standpoint as

“A Challenge from Schenker” (258–67), Tymoczko accuses the theory of abandoning functional harmonic

relationships; earlier (213), he states that Schenkerian theorists “seem to deny that functional tonality

involves purely harmonic conventions”—this is, of course, not true as the dominant–tonic motion

maintained in Schenkerian theory is the main harmonic convention of tonality. Tymoczko also rejects

large-scale tonal plans, likening them to massively recursive long and complex sentences. The latter point

ignores the whole connection between Schenkerian and Schoenbergian theories of composing out material

and their associated formal structures, and as well ignores one of the central explanatory models of the later

nineteenth century, namely the extended instrumental composition as literary novel. We are quite able to

comprehend and follow the central theme of Dickens’s A Tale of Two Cities from its seeming contradictory

opening “It was the best of times, it was the worst of times,” because, as in long musical compositions such

as Mahler Symphonies, there are formal divisions and structures on many different levels, to accommodate



14

and allow for our comprehension. As mentioned, form is largely absent from Tymoczko’s discussions; his

linguistic analogy may apply to the later James Joyce in literature and perhaps to the compositions of Max

Reger or Hans Pfitzner, but seems hardly applicable to the tonal canon he wants to explain.

The bulk of Tymoczko’s Schenker rebuttal outlines four positions, ranging from extreme to

ecumenical, that he finds in Schenkerian writing: 1) nihilism, which allows for no harmonic theory; 2)

monism, in which all harmony can be explained by counterpoint; 3) holism, in which harmony and

counterpoint are inseparable; and 4) pluralism, which sees Schenkerian theory as a complement to

“traditional harmonic theory” (261). This latter position is adopted by Tymoczko, but he replaces the

melodic/contrapuntal position with a geometrical model. While he recognizes that some aspects of his

approach are commensurate with a Schenkerian view—for instance, in his assertion that “chord

progressions use efficient voice leading to link structurally similar chords, and modulations use efficient

voice leading to link structurally similar scales”, resulting in “tonal music [being] both self-similar and

hierarchical”—he clearly adopts the notion of structural levels (17). However, Tymoczko abandons

prolongation and any systematic way of distinguishing consonance from dissonance, chord tone from non-

chord tone, or functional chord from embellishing chord except for the problematic assertion of acoustic-

based explanations or the Procrustean bed of “tradition.”

Tymoczko’s harmonically-focused position, which omits any defined structural melodic motion

save for “efficiency,” also leads to some strange results and queries. For instance, in the chapter on

“Functional Harmony,” his insistence on third motions over fifth motions is contradicted by his own

statistics; drawn from Bach and Mozart, they show clearly that fifths, allowing for starting anywhere in the

circle (I–vi–ii–V–I, or I–ii–V–I, etc.), third substitutes (I–(iii, I6)–(vi, IV)–(IV, ii)–(vii, V)–(I, vi)), and the

embellishing motion of a falling fifth like I–IV–I, rather than thirds, are the basis for tonal progression.30 As

well, it is commonplace knowledge that basic harmonic questions have melodic rationales. The reason that

IV–ii occurs more than ii–IV, for instance, is found in the melodic motion of phrases, which favors scale

degree 2 at cadences; similarly, Aldwell and Schachter conflate vi and IV6, not in an adoption of

Riemannian function theory, but because of the contrapuntal 5–6 motion, which groups a chord and its

lower third harmony in first inversion: IV–ii6, etc. (232, Note 13).31 This motion is altered to a retained

note, now a contrapuntal 5–7 in IV–ii6/5, which explains the contrapuntal origins (as suspensions) of

seventh-chord progressions, rather than the thirds progression-based rationale of Tymoczko (235). His

directional-motion tendencies similarly stem from the falling-fifths model (which therefore also

encompasses thirds rather than deriving from them). The apparently remarkable similarity between the

minor and major modes, compared to the other modes (229), arises from the simple fact that composers



15

alter the minor to make it more like the major; in that sense there is comparatively little “pure” minor

music. The pull of the major key, however, makes the ascending third progression i–III–V much more

common, on different structural levels, in minor.

To consider Tymoczko’s views on chromaticism, a large area of focus which he claims is little

understood and in which he rejects explanations based on mixtures and tonicization, we turn to the writings

of Howard Cinnamon, starting with a brief mention of an analysis by Edwin Hantz. In 1982, both wrote on

Liszt’s song “Blume und Duft.”32 Hantz points out the features of what Tymoczko calls a “tesseract” in his

Chopin analysis—the first four chords, major-minor sevenths A*f 7 –F7, B7 –A*f 7 share three notes with

diminished-seventh A–C–E*f–G*f, and Hantz notes the cyclic nature of this progression here and in the

later Liszt piece Un sospiro, where the T3-cycle is completed. Cinnamon then interprets the song in a tonal

context. (In the following I have recast his points slightly to add elements from the 1984 dissertation.) First,

the motivic and tonal structure has far-reaching implications for our understanding of tonality—through its

adaptation and reconfiguration of tonal features, this short song calls attention to the most fundamental

features of tonality. Second, two of the features of tonality—the hierarchical recursive relation of the large

and the small, and the role of the dominant—are largely minimized, and structural third motions, including

a “back-relating third” take on the prominence of structural fifth motions in a role reversal. Third, the tonal

universe that Liszt creates combines traditional voice leading by common tone or the smallest intervals

with unconventional harmonies and procedures derived from interval cycles. Fourth, the structural assertion

of the tonic, A*f, is not by a traditional Ursatz, or even an unfolding of the augmented triad A*f–C–E, but

by a neighboring motion A*f–C–A*f; the piece is thus somewhat like a partial tonal structure and the

interpretation shows that interesting voice leading is not precluded by underlying interval cycles.

Cinnamon’s final paragraph lays out his understanding in a passage that foreshadows by almost twenty

years Tymoczko’s initial questions about tonality and many of the latter’s assertions, but, notably, adds the

crucial element of a large-scale tonal structure to the interpretation. This structure is what guides us through

the apparent maze—the “massive recursion” Tymoczko refers to does not confuse the issue, it provides our

sense of form on many levels that we require for comprehension. To Cinnamon’s final question concerning

harmony and counterpoint (below), I answer in the affirmative, with Tymoczko’s book as my first witness.

I have shown in this piece how various harmonic and voice-leading structures and the

relationships they form on various levels serve to build a sophisticated and complex tonal

system. I have also noted that despite the unorthodox nature of some of the harmonic

relationships, the voice-leading and prolongational procedures are often quite analogous



16

to those recognizable from our more traditional tonal system. Further, I have shown that it

is, in fact, the familiar voice-leading procedures and the interrelationships between

structures on several different levels that make the harmonic structures work; the

harmonic relationships, in themselves, are not sufficient to provide structural significance

to given musical events. This leaves us with an interesting question to consider: to what

extent might the same be true in our traditional tonal system? Have theorists

overemphasized the role of harmony and underemphasized the role of counterpoint in our

explanations of the relationships which are at the root of our traditional tonal system? I

have no answer to offer at this point; only the question for your consideration.33

Cinnamon has in these and other writings defined a Schenkerian interpretation of chromatic

harmonies and progressions, and Brown has similarly written on a Schenkerian view of diatonic and

chromatic structures in tonality.34 There are also many textbooks, such as Steven Laitz’s text (discussed

below), that deal comprehensively with chromaticism. Thus Tymoczko’s repeated comments throughout to

the effect that “theorists have sometimes depicted chromaticism as involving whimsical aberrations,

departures from compositional good sense, rather than as the systematic exploration of a complex but

coherent terrain” (21) have the flavor of writers like Daniel Gottlob Türk (1750–1813) rather than his

contemporaries. Tymoczko’s assertion that in his chapter on chromaticism he intends “to present

chromaticism as an orderly phenomenon rather than an unsystematic exercise in compositional rule

breaking” (268), is wholly unwarranted by a review of the literature. Indeed, if we compare Tymoczko’s

approach with that of Cinnamon, the lack of a large-scale supporting structure in the former is obvious. In

his discussion of the opening of Liszt’s Sonetto 104 del Patrarca, Cinnamon retains both the local chord

successions and the larger tonal context (in the passage, a prolonged V/V is effected by a minor-third

division of the octave, with stepwise, mostly chromatic, voice motions connecting diminished seventh

chords with triadic-based seventh chords); both the intricacies of the local connections and the implications

of the larger tonal motions are maintained. If we compare Tymoczko’s Figure 8.5.9, from Chopin’s A-

minor Mazurka, we find no tonal plan shown and no leveled positioning of the harmonies; rather the chord

labels suggest no pattern for the larger motion, only local connections. We are trapped, snorkel in hand, on

the surface.35

My third criticism of Tymoczko concerns the presentation of the geometrical model that is at the

heart of his approach, but for which no background is given. Spatial metaphors are used more than any

other in musical explanation and they are rooted in our daily lives. Although there is no “higher” in higher



17

notes, heightened tension, higher intensity, etc., these metaphors all stem from our experience with facts of

our lives such as gravity, which requires effort to overcome (higher) and associates relaxation with

succumbing to it (lower).36 There is also no “left to right” in time, but this directional orientation works for

us in its spatial flow, and there is no smaller and larger in intervals—they all get larger in absolute terms,

but we perceive pitch and loudness in varying modular and logarithmic terms. Given that experiential

metaphors are at the core of the geometrical models Tymoczko uses, we should probably ask our models to

maintain these useful directional musical analogies.

An earlier use of geometrical models by Roeder considers questions such as the one above

carefully in its explication of voice leading in pitch, pitch-class, and also interval space. Roeder is

mentioned as an inspiration by Tymoczko (65, Note 1), but more specifically, Roeder’s geometrical

system, using interval models instead of pitch-class pairs, is in fact very close to Tymoczko’s.37 Example 6

shows several examples of Roeder’s geometrically-based equivalences, which follow his careful

acknowledgments of earlier pitch, pitch-class, harmonic, and voice-leading spaces, in particular as concerns

the lineage from twelve-tone matrices to invariance matrices (Bo Alphonce) to “com” (comparison)

matrices (Robert Morris) to Roeder’s geometrical representations.38 Roeder extrapolates from both

algebraic and geometric implications of pitch and interval relationships, including voice leading, distance,

and similarity between collections derived from the models used. While Tymoczko is interested in tonal

music, many of his demonstrations of geometrical models are subsets within Roeder’s larger explanation of

any possible intervallic combinations, associated with non-tonal music. But even prior to Roeder dates

Walter O’Connell’s representation of the six interval-classes as 6D space and his accompanying discussion

of the two all-interval-class tetrachords in terms of M-spaces.39 This fascinating and little-known article

lays a framework for spatial and geometric models that underlies much of the following decades of work.

[Example 6 here]

To be sure, the format of Tymoczko’s book precludes an extended exegesis of earlier

interpretations of geometrical models for musical explanation, but some mention of these earlier writings

would seem to be in order. Instead, we find only a brief engagement with Fred Lerdahl’s Tonal Pitch Space

(Appendix E) and a statement at the beginning of Appendix C that “In recent decades, music theorists have

produced a number of graphs representing voice-leading relationships,” followed by the usual in-house

references (Cohn, Quinn, Douthett, Callender).40 This Appendix continues, however, chastising an

unnamed number of efforts to model voice leading for not satisfying criteria such as those that require that



18

every edge should be able to represent voice leading by the smallest intervals within the system, that chords

should be adjacent in the model space to all of their transpositions across collectional space, and others.

This list confusingly mixes in criteria for voice leading with criteria for chord progression and key

relationships, and in fact shows the problems with Tymoczko’s model of collectional modulation as a

slowed-down version of chord modulation. The Tonnetze, although they do show the voice leading for PLR

relations in neo-Riemannian study, display them as harmonic, not melodic, relationships, and thus they are

not required to display all possible versions of the latter. Nor can any model display all the theoretical

possibilities of elements such as transpositions; models, like quotient spaces and equivalence classes, must

be simpler than the phenomena they model, and be designed to show single characteristics of the system at

hand.

Tymoczko generally reserves commentary on the advantages and justification of his geometrical

models for the Appendices. But even here, we search in vain for clarity along with nuance. The opening of

Appendix A states “My view is that measures of voice leading should depend only on the distance moved

by each voice” (397). Not on the resulting intervals with other voices? This seems to negate one of his

guiding principles on the mutual effects that harmony and voice leading have on each other. Tymoczko

does not advance any one metric for voice leading, stating only that any choice should follow dictums like

the triangle inequality, etc. (399). Logically, such metrics are required to support Tymoczko’s earlier

assertions on voice leading among near-maximally even collections, and other voice-leading characteristics

associated with neo-Riemannian studies, but musically, it is a commonplace that distance can be deceptive,

and that the most obtuse “purple patches” can be only a chromatic shift away, while a seemingly simple

move to the dominant can take an exposition. Appendix B makes explicit the modularizing of musical

space, and his Figure B3 (405) indicates the extent to which Tymoczko’s motions in his spaces are indebted

to vector graphics. The closest thing to an explanation of the raison d’être for the geometrical approach

appears at the end of Appendix B:

The fundamental idea here—and it is both simple and profound—is that ordinary numbers

provide a natural and musically meaningful set of geometrical coordinates, with points

representing chords and line segments representing voice leadings. Any sequence of

numbers can be understood as an ordered list of pitches, while any pair of (equal-length)

sequences can be understood as a voice leading in pitch space. When we disregard octave

and order information, we are restricting our attention to a region of Cartesian space . . .

This involves moving arbitrary points and line segments into our region. If we do this



19

carefully and thoughtfully, we realize that the boundaries of this region have special

properties. In other words, we make the transition from regions of ordinary Euclidean space

to quotient spaces proper. (411)

After this statement, reminiscent of the writing in the journal Die Reihe which similarly invokes

formalized mathematics in musical explanation, the following footnote again refers to bouncing off

boundaries and disappearing off the edge of the figure which are foreign to our experience of musical

continuity. We are still left with the problem of explaining why C–E–G to C*s–E*s–G*s is such a vast

space harmonically and such a short distance melodically—central to any notion of voice leading in a

system that combines harmony and voice leading in equal measure such as tonality. In this and many other

respects, Tymoczko’s criteria are often only applicable to post-tonal constructs, as in the permutational

equivalences which is the basis of the orbifold compression of compositional space.41 This is clear from his

final statement in Chapter 1, which shows his interest not in modeling the past accurately, but in generating

new musical situations: “My goal is to describe conceptual structures that can be used to create musical

works, rather than those involved in perceiving music” (22). This situation, and the essentially

compositional generative, rather than the analytically synthetic, intent it represents, also reflects the aims

and goals of early music theorists, usually composer-theorists, explored and responded to by Brown and

Dempster in their exegesis of the scientific image of music theory. Tymoczko’s book represents a more

recent example of the same problems (from the point of view of music theory, and of course, in my

opinion).42

My fourth criticism concerns Tymoczko’s view of non-tonal music as an aberration, associated

with eating “disgusting food” or causing “pain” (185). The view of atonal and serial music as somehow

unnatural is just silly and wholly unnecessary at this point in time, akin to long forgotten arguments about

Country and Western music in popular music circles, or to confusion about human and dinosaur interaction.

Rather than look at actual pieces, Tymoczko uses the row from Schoenberg’s Op. 25 in an invented

sequence of segmental trichords and a three-part counterpoint, stating that he does not hear any structure

and comparing his sequences to random processes (10–11, Fig. 1.2.2).43 It is hardly necessary to point out

that this is an absurd methodology and proves nothing. Later, Tymoczko makes the potentially interesting

distinction between Debussy’s “scalar” compositions, which “explore a much wider range of scales and

modes” than the whitewash of Schoenberg’s centerless “chromatic” compositions (16); as it happens,

Schoenberg himself addresses the exact issues that Tymoczko raises, in “Composition with Twelve Tones

(I)”, particularly in section iii, in Style and Idea.44 Briefly (from a long and involved argument Schoenberg



20

presents), the distinction Tymoczko attempts to assert is based on functionality and comprehensibility—

essentially the driving forces behind Schoenberg’s adoption of twelve-tone techniques, rather than an

aimless result of the statistics of pitch-class circulation.

I won’t go further at this point on Tymoczko’s negative depiction of non-tonal music, but will only

comment on his statistics showing that Schoenberg’s Op. 11, No. 1 is essentially random in its pitch-class

presentation (183–84). Using software that derives pitch and interval collections from midi files, we find

the statistics shown for this piece in Example 7: a roughly equal pitch-class distribution, as Tymoczko

notes, but highly nuanced interval unfoldings, with a melodic distribution favoring directed intervals 1 and

4, and with harmonic intervals favoring interval 4. It is the latter which shapes our experience of this and

other post-tonal pieces. The interpretation of non-tonal music in terms of intervals rather than pitch-class

content is, of course, basic to our understanding of this music, as is clear from the literature, and this shift

from the “end points” of pitches and pitch-classes to the transformational paths that lead between them, as

generalized “intervals” is, again obviously, basic to the Lewinian view. Tymoczko’s presentation of this

music as random based on pitch-class circulation flies in the face of the writings of many distinguished

authors, including my own mentor Andrew Mead and his teacher, the first of four people Tymoczko claims

had a profound impact on his musical life: Milton Babbitt.

[Example 7 here]

SUCCESSES AND INNOVATION

In any field, attempts to create something like a “unified field theory” to explain all behaviors and

structures under one model is a difficult task. Tymoczko’s approach to such a theory for tonality, writ large,

is such an attempt, and, despite the shortcomings detailed above, does offer some useful advances in our

understanding of this type of musical structure. I have mentioned above the significant elements of the

geometrical approach and the presentations of related scales and chords as explanatory models. In a more

general sense, the features in Tymoczko’s approach that lead to advances in tonal understanding stem from

his somewhat Socratic method of asking questions and pursuing, step by step, interesting interrelationships

among the materials he posits. The questions include the following: What are the connections between the

five features of tonality given, does one imply another, and which are most salient? (4, 9) How can we

“depict the voice-leading possibilities between all the triads in the chromatic scale?” (20) (This leads to the

40 chords in the center of the trichordal grid: four augmented triads, and twelve each of the diminished,



21

major, and minor triads.) At the conclusion of the first part of the book, Tymoczko offers a concentrated

and helpful set of such questions, on both local and global levels to orient analysis (191).

The method adopted by Tymoczko equally helpfully starts with the simplest cases, then adds

complications, for instance in his demonstration of how a scale is necessarily created from chords (12–15).

Here Tymoczko presents his fundamental distinction between a triad and a chromatic cluster: creating a

scale from a nearly maximally-even triad results in a regular alternation of chord tones and passing tones;

creating a scale from a chromatic cluster results in a long string of passing tones which looses coherence.45

Tymoczko elaborates on the two situations to show how the triad’s structure “overdetermines” its role in

tonality. This simple, Feynmann-like comparison (and other “pull the O-ring from the ice cubes in the

water glass” type explanations) lays the groundwork for the theory.

One important feature throughout the book is the focus on the structure of surface chords, and

progressions like the half-diminished seventh moving to the dominant seventh chord, or the common

diminished-seventh notes in the T3-cycle of dominant sevenths, that have been underreported in studies of

nineteenth-century music. We might categorize this feature as an “antidote” to reduction when presented

with a Schenkerian-like graph. By incorporating these chord progressions into a theory of the behavior of

all tonal chords, Tymoczko allows us to connect behaviors across a wide spectrum of music, and allows us

to theorize about a feature of tonal music which we all sense: that “Dido’s Lament” of Purcell, the

Chromatic Fantasy and Fugue of Bach, the Dissonance Quartet of Mozart, Haydn’s Introduction to The

Creation, Beethoven’s Grosse Fuge, and Wagner’s Tristan Prelude are all tonal in similar ways, and share

common features.

Finally, it is useful to compare Tymoczko’s book with other texts. Laitz, in The Complete

Musician , starts with an implicit criticism of theory teaching and materials, noting that students often

“suffer” through activities they find “arcane and antiquated” and don’t find the experience “relevant.”46 His

approach attempts to solve these problems by demonstrating the “same simple processes” through “all tonal

music” that show “how the harmony of a given passage emerges from the combination of melodic lines,”

but including “motivic relationships that make a given work unique.”47 The repertoire spans Wipo to

Chicago and the approach integrates musical skills with writing and analysis, and includes very few

references. At the other end of the text continuum is Timothy Johnson’s Foundations of Diatonic Theory: A

Mathematically Based Approach to Music Fundamentals (2003, 2008), which covers much of the same

ground as Tymoczko’s book, including maximally even scales and a geometric approach, but also adds

more context, both theoretical and historical, with clear references to the ideas and concepts presented, as

well as more integrated exercises and teaching materials. Chapters include topics such as “Spatial Relations



22

and Musical Structures,” “Musical Structures from Geometric Figures,” and “Maximally Even Triads and

Seventh Chords.” Examples in the Laitz book are expressed wholly in musical notation, and Johnson uses a

mix but incorporates less intricate geometries than Tymoczko. Johnson leads with the kind of Socratic

inquiry that drives discussion in Tymoczko’s text (“Why are there black and white notes,” etc.),48 while

Laitz takes the tonal universe as a given, to be explored and appreciated, but not to ask the “why”

questions. Tymoczko’s book has text-like qualities, but if it is contemplated as a course text, I would

recommend Johnson’s book as a useful and more pedagogically designed version; and, of course, Laitz’s

text integrates every aspect of an undergraduate curriculum, but is more traditional in its approach.

ANALYSES

In this final section, I will consider some analyses presented by Tymoczko and compare them with

some other approaches. One of the strengths of the book is not only the wide repertoire under

consideration, comparable to Salzer’s Structural Hearing, but also the attempts to show continuity by

comparing disparate styles. For instance, a progression from Clementi’s Piano Sonata in D major, Op. 25,

No. 6 is shown (18–19) in which a G*s is interpreted as creating a V7/V chord and effecting a “smooth”

modulation from the D-major to the A-major scales (G–G*s). It is the scalar connection, more than the

chords, that creates the modulation and the long-range harmonic progression for Tymoczko. A similar

motion is compared from Debussy’s “Le vent dans la plaine,” where, in an E*f natural minor scale, B*f–

B*f*f creates a scale equivalent to an F*s melodic minor ascending scale, presumably F*s–G*s–A–B–C*s–

D*s–E*s–F*s as G*f–A*f–B*f*f–C*f–D*f–E*f–F–G*f. The comparison ends there, without a parallel

long-range progression identified in the Debussy piece, and, strangely, no reference to an analysis in

Tymoczko’s own 2004 article on scales in Debussy or to analysis by David Lewin.49 Lewin posits a series

of chromatic motions around pentatonic and diatonic collections emphasizing E*f or B*f; Tymoczko posits

a succession of scales in the piece, with motions between scales effected by the chromatic motions. Here is

a representation of his analysis using pitch-class numbers, pointing out the similarity between Tymoczko’s

approach and an application of Fortian set theory: both methods use labels for collections—Tymoczko uses

labels from scale theory, such as G acoustic scale, or the “third mode” of a scale, meaning that the third

note of the scale is a focal point—and both relate collections based on shared subsets. Lacking any

functional relationships, however, except for ones that he recalls from tonality (B*f or E*f as tonic, etc.),

Tymoczko’s scale theory is, as stated in another context by Pieter van den Toorn, somewhat of an exercise

in labeling.50 The one relationship Tymoczko posits outside of a few tonal references that depend less on

scales than on repetition and registral placement, is an “inversion:” in Note 66 he states that I10 (F/F, F*s/E,



23

G/E*f, A*f/D, A/D*f, B*f/C, B/B) maps C diatonic to G*f diatonic, B to G acoustic, and C*s whole-tone to

itself. But in the absence of a context in which inversion of scales or I10 has any meaning for a tonal

interpretation, this is again a label. Clearly, in Debussy’s “Le vent” not every note is structural or part of a

structural event such as a scale. Very few pieces before twelve-tone music have this quality; Debussy’s

Voiles from the first book of Preludes for piano is one example (with only the chromatic motions in m. 31

outside the whole-tone/pentatonic collections that pervade the piece). But this model is too simplistic for

“Le vent,” where we have to reincorporate notions of motives and chord versus non-chord tones.

The second analysis we will consider is that of Brahms’s Intermezzo, Op. 116, No. 5; Tymoczko’s

figures are given in Example 8, and the main paragraph is given here:

We begin with four voice leadings from the opening of Brahms’ Intermezzo, Op. 116,

No. 5. Figure 3.5.2b graphs the voice leadings in two-note chord space, representing each

measure as a pair of line segments forming an open angle. We can imagine sliding the

pair {X1, X2} so that X1 nearly coincides with Y1, and X2 nearly coincides with Y2.

This represents the most obvious analysis of the passage, according to which voice

leading Y1 is a slight variation of X1, and Y2 is a slight variation of X2. (That is, X1

moves its two voices by semitonal contrary motion, whereas Y1 moves by slightly

skewed contrary motion; X2 moves in a skewed fashion, while Y2 moves in pure contrary

motion.) Geometrically, however, it is clear that (Y1, Y2) is also the mirror image of (X1,

X2). Hence we can move the pair (X1, X2) off the left edge so that it exactly coincides

with (Y1, Y2), as in Figure 3.5.2c. … On this interpretation, Y2 is exactly equivalent to

X1, and Y1 is exactly equivalent to X2. Figure 3.5.2d represents this musically,

heightening the comparison by switching hands and reordering dyads. Now both pairs

begin with perfect contrary motion and move to less perfectly balanced motion, with

melodies in each staff being transpositionally related.” (77)

The discussion continues, to the effect that it is difficult to see the relationships in the notation; it

is asserted that a geometrical representation makes the point immediately obvious.

[Example 8 here]



24

Although this analysis suffers from a similar lack of context and lack of attention to the large-scale

motion found throughout the book, Tymoczko’s presentation isolates and focuses attention on the essence

of this piece; namely the role of counterpoint (his mirroring of X1, X2, Y1, Y2 as abba) in the underlying

harmonic progression (his parallel of X1, X2 in Y1, Y2). First, we must note earlier observations on the

role of counterpoint in this piece. Malcolm MacDonald describes “the E minor Intermezzo, [Op. 116] no. 5,

whose hesitant rhythm of two quavers, accented on the weak beat, plus quaver rest, with the left hand an

almost exact mirror inversion of the right in motion and intervals, creates an impression of twentieth-

century rigour.”51 He footnotes Michael Musgrave, who writes “[Op. 116, No. 5] is perhaps the most

‘progressive’ of the late pieces in terms of its appearance in the score. The strictly mirror appearances of

the hands in the outer parts and extreme consistency of figuration in the middle section suggest the Webern

of the Piano Variations. Thus there exists a connection between the instinct for contrary motion of outer

parts so strong from the earliest Brahms, to the strict symmetries of the twentieth century. So strong is the

feature that it seems of itself to generate the harmony, creating unusual dissonances, as at bars 6-10.”52 By

comparing this piece to Webern’s Variations for piano Op. 27 in its rigor of intervallic alignment,

McDonald thus anticipates Tymoczko’s comments on the “love for invertible counterpoint and other forms

of compositional trickery” (79), traits Brahms shares with Webern.

But the more interesting point is this: Brahms is working with inversion within tonality—a much

more difficult to control technique: in E minor the X scale degrees are 5–6–5–*s4 over 1–*s7–1–2; the Y

notes, if interpreted in E minor, seem to be a nonsensical (*f)6–*f7 over 2–*s1 and 6–5 over 2–3. But, more

clarity is obtained with a larger purview. A comparison with the score shows that Tymoczko’s figures X

and Y are not actually adjacent in the music; instead, they are at the beginning of the first two motivic

groupings in the 12-bar phrase, which divides into a sequential progression that gradually reduces in

number of attacks from 8 to 2 in a Schoenbergian “liquidation.” In this sequential context, the local scale

degrees in the second subphrase, beginning with Y1, Y2 do in fact sound parallel to the opening E minor,

but in a modal alteration that might take a scalar interpretation (E minor to F*s locrian) pace Tymoczko.

The locrian implication also allows for the inverse parallel of scale degrees 5–6, 5–*s4 in minor with 1–

*s7, 1–2 in locrian, and 1–*s7, 1–2 in minor and (*f)5–6, (*f)5–4 in locrian (E*s–F*s–G–A–B–C–D / C–

B–A*s–G–F*s–E–D*s or *s7–1–2–3–4–5–6 in locrian / 6–5–*s4–3–2–1–*s7 in minor). But this rather

forced scalar interpretation begs the question as to what is more relevant to the piece: the parallelism or the

mirroring between X and Y. The piece features a basic ascent in the first section to a static plateau over V

in the middle, followed by a descent in the final phrase; a representation in two staves, divided into the two

moving voices over the static, doubling voices is given in Example 9. In the larger context, the ascending



25

motion controls the large-scale progression, while the mirroring is part of the embellishing details at each

stage. Again, Tymoczko has pointed out the essence of the piece, but lacking a context, we find it difficult

to ascribe any meaning other than the purely local to his interpretation.

[Example 9 here]

We will end with a consideration of Chopin’s F-minor Mazurka, which Tymoczko claims is a

“virtual reworking” of the Prelude Op. 28, No. 4, a piece which he interprets as “a four-voice”

“descending-fifths sequence” (287, Note 21). Tymoczko references a study by Marciej Golab as being

“similar” to his interpretation (ibid.); and he refers to a Kallberg study (284, Note 18) but does not mention

that Kallberg compares mm. 25–40 of Op. 62, No. 2 with mm. 24–39 of Op. 64, No. 4.53 Both passages are

unstable tonally, but relatively diatonic, compared to the preceding passages of both which are chromatic

but relatively stable. Kallberg makes this interesting but somewhat paradoxical juxtaposition the basis of

his comparison. Tymoczko, by contrast, finds the opening chromatic passage “blurred” and not

“articulating a clear tonal center” (284). The lack of a larger purview for understanding tonal processes is

evident in Tymoczko’s Figure 8.5.1 which diagrams the chords in the opening phrase (mm. 1–8) without

including the opening chord—thus missing the bass arpeggiation of A*f–F–C–F which clearly moves

through an F-minor triad space to the dominant and back to the tonic (bass notes: A*f–G–G*f–F, then F–

E–E–F, then G–C, then F; in the second phrase the F–E–E–F part is recast as F–F*f = E to lead to A).

Tymoczko then explores the interesting idea of regarding the piece as a controlled passage of improvisation

and looks for patterns among the descending major-minor sevenths of the opening, and finds one in a

“tesseract” of four such seventh chords in a T3-cycle, connected to a diminished seventh chord by common

tones. He then helpfully finds instances of this procedure in other pieces by Chopin, although his

interpretation of a circle of fifths in the Prelude Op. 28, No. 4 (which he contrasts with unattributed

implications that the harmonic content is “insignificant” [287, Note 21]), turning it into an “Autumn

Leaves”-like version, is unconvincing. Ultimately, however, these readings suffer from the same affliction

of Tymoczko’s analysis throughout: without a larger-scale understanding of tonal forces, we are left to the

vagaries and delights of the surface, but we are unable to connect the larger dots that explain how tonality

works on large scales to unify and control musical structures on the scale of symphonies.

CONCLUSION

Throughout history (and herstory), humanity has been limited by our physical bodies and senses,

but arguably around the turn of the twentieth century, most people in the world began to have access to



26

machines and technological advances that allowed them to go beyond human capacities for travel, storing

information, and so forth. Of course in music, for hundreds of years, we have had instruments that allow us

to leap by larger intervals than we can sing. While it may be the case that our physiology—from our vocal

chords to the size of our hands—strongly prefers singing and shaping chords in small melodic intervals and

that our ear canals, like all physical systems, favor acoustic consonance, just as we now drive cars and fly

planes to get places we could never walk to, or use computers to remember amounts of data that our brains

cannot access by themselves, or create machines that can lift weights far beyond our human capabilities, we

can now imagine and create music that challenges our senses and allows us to go beyond our evolutionary

boundaries. Despite his preference for tonality, Tymoczko’s geometrical approach actually allows for this.

If we can just get beyond the model provided by the triad, the semitone difference between major and

minor, and the emphasis on thirds and fifths from the harmonic formation, etc., we can use geometrical

models for other intervallic/harmonic systems. The “limiting” factors of tonality are interesting, but are not

mandatory.

Within his chosen area of inquiry, Tymoczko asks the often obvious but complex questions at

each stage about “why” things are the way they are and then attempts to find out. These are questions that,

if we allowed them, many of our students would ask (or do ask now): why are there seven notes in a major

scale? What is a “functional progression” and “why is it difficult to interpret the key structures in late

nineteenth-century music?” Arguably, it is always useful to break down complex structures into basic

elements. But the utility of Tymoczko’s approach requires a more comprehensive and comprehensible

mechanism for connecting the surface to the abstraction, the local to the global. One obvious cause of this

requirement in this regard is that Tymoczko, with his focus on the endpoint pitches and pitch-classes of his

defined musical relationships rather than the intervals which connect them, denies the full implications of

the Lewinian arrow.54 In this model there are no fixed endpoints: the magnitude and direction define the

relationships, ones based on interval, which can be dislodged from their moorings at a tonal dock, and

allowed to move freely through the chromatic ocean, defined now by direction, now by distance. Indeed,

armed with the fruit of approximately 60 years of insights—with Lewin’s laconic definitions of intervals,

Morris’s multiple-dimensional spaces, Babbitt’s basic permutational views, Mead’s Mosaics, O’Connell

and Roeder’s rotating grids, Straus’s slinking voice-leading connectors, Perle’s omnipresent sums and

difference alignments, and Forte’s infamous sets—there is no reason to limit our analysis and

interpretations to tonality; indeed, we would be the poorer for it.



27

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1The book builds on Tymoczko’s earlier writings and his collaborations primarily with Clifton Callender,

Rachel Hall, and Ian Quinn. It is accompanied by a website with audio files for the musical examples. The

References include a series of articles by Tymoczko and collaborators in the journal Science, a first for

music theory in the modern age. 2Here and elsewhere in this review, the emphasis is Tymoczko’s. The Index of the book does not contain

“Möbius strip” or the other terms mentioned, as well as important terms such as “efficient” and “(near)-

maximally even” (see below), and seems to be more of a “selected” Index. The term “tesseract” has another

connection; in addition to its geometrical proclivities, it is the name of a scientific project wherein time and

space are folded in on one another, somewhat like a wormhole, in the book A Wrinkle in Time by

Madeleine L’Engle (1962). Tymoczko’s later discussion of an ant walking on his 2D pitch space grid

shares elements with the “wrinkles” in the fabric of space-time which enable the characters to travel in

space and time in this book. I am indebted to Lisa Behrens for pointing out this connection.

3Tymoczko supplies statistics to show that writings on post-tonalists Schoenberg and Cage and topics like

serial music outpace those on jazz musicians such as Ellington and Coltrane and jazz topics by a ratio of

roughly 6:1 in academic journals (13,903 to 2,262 “hits,” 390).

4“Many forms of Government have been tried, and will be tried in this world of sin and woe. No one

pretends that democracy is perfect or all-wise. Indeed, it has been said that democracy is the worst form of

Government except all those other forms that have been tried from time to time.” From a speech to the

British House of Commons, November 11, 1947 (Churchill 1974, 7566).

5Outside of a few references to the first manifestations of Tonnetze (Gottfried Weber, 246–48) there is

virtually no mention of the history of the use of geometry to explain music. The References include few

writings prior to 1900, and only two titles apart from Tymoczko’s own writings with “geometry” in the

title, both recent.

6See Morris (1987, 1998), Lewin (1987), Cohn (1998), Clough (1985, 1991).

7Many of Tymoczko’s topics can be found in literature that stems from Lewin and neo-Riemannian ideas.

If, however, we compare articles such as Bush (1946), Crocker (1962), and Schubert (2002) we see similar

topics and musical issues to those confronted by Tymoczko, from the beginnings of the larger musical

period he proposes.

8Concepts central to Tymoczko’s arguments on scales and collections, such as well-formed and maximally

even scales, are found in writings by Clough (1985, 1991) and Norman Carey and David Clampitt (1989),

among others. Clampitt’s (1997) unmentioned “Q” relation is a generalized function for mappings such as

{01267} to {01567}, where all but one pitch class is maintained and the other “slides,” without crossing



30

other notes; this mapping can be made from a set to a transposition or inversion, or even to a different set.

This relation is the basis for the “Cohn function” and for Tymoczko’s relations between scales. Clampitt

(1997), mentioned in Cohn (1998), also defines a broader “system modulation” from Lewin, which

underlies Tymoczko’s mapping of scales in his concept of “scalar modulation.” In connection with jazz,

Tymoczko cites Mark Levine in jazz theory but omits bebop scales and other topics from writers such as

David Baker (although he cites Baker in an earlier article), as well as any references to writers such as

Steve Larson, Henry Martin, or Keith Waters on jazz topics.

9Tymoczko backpedals a bit from a position of inevitability for tonality and the materials of tonal music in

later chapters, suggesting rather than a “deterministic or Hegelian view,” an image of the natural tendency

of mountaineers to follow a similar path up a mountain, given that the “structure of the rock will naturally

suggest certain routes” (211).

10The term “efficiency” is apparently from Agmon (1991), indirectly noted by Tymoczko (14, Note 19);

Jack Douthett’s influence in regard to this term is also mentioned by Cohn (1996, 1998). See also Straus

(2003) and Morris (1998). Agmon defines a “linear transformation” which is a one-to-one mapping with

intervals of a diatonic step. He notes that his “efficiency constraint” of moving the smallest possible

distance is similar to that given in many texts (22, note 3). Indeed, the long pedigree is indicated by a

reference to the writings of Anonymous 2 by Klaus-Jürgen Sachs in section 2 of his Counterpoint article in

The New Grove Dictionary: “A2 writes that three consonance sequences (3rd–unison, 3rd–5th and 6th–

octave) have particular advantages: close melodic connections through conjunct motion, independent part-

writing through contrary motion, and change in sound through the transition from imperfect to perfect

consonance” (Sachs).

11See Note 8.

12Proctor (1978) is cited in the case of transpositional relations in the modern “seconda prattica” of

chromaticism; we shall see, however, some interesting possibilities for voice leading in incomplete chords

in Cinnamon’s writings below. The problems of interpreting “cluster”-chord voice leading are similar to

those presented in Straus (1987).

13

On page 60 (Notes 33, 34), summarizing a section describing transpositional, inversional, and

permutational symmetry (based on C to C,C,C, etc.) and “nearby” chords (B,C,D*f to C,C,C, to C,D*f,B,

then B,C,D*f to C,D*f,B, the example essentially restates Forte’s T0I based definition of inversion, hence

the symmetry), Tymoczko formalizes the connection between efficiency and maximal evenness. Efficient

voice leading implies near-symmetrical chords, and the converse is also asserted, near-symmetrical chords

imply efficient voice leading. Slightly recast, Tymoczko notes that if chords X and F(X) have efficient



31

voice leading EVL, and EVL has two parts, vl and evl, where vl rearranges x in X and evl maps the results

of vl to F(X), since EVL is efficient, so must vl and evl be approximately equal (or they wouldn’t stay

within a small ambitus), with mostly equal-length motions. He also argues that if chord X leads to F-

symmetrical chord Y, the voice leading can be efficient. A more typical mathematical procedure might be

to prove the impossibility of non-efficient voice leadings for near-even collections. This requires, however,

a clear definition of the voice-crossing prohibition. This topic, and the problem of determining a “voice,”

hovers around Tymoczko’s arguments, which largely stem from music that is vocally influenced in its

relatively clear voices. There is, of course, a lot of instrumental and keyboard music in which the voices are

difficult to discern. See Straus (2003) on this question. In teaching Schenkerian principles, for instance,

defining the voices is an important first step. The question of discerning voices is also an important one in

considering non-tonal music (serial/twelve-tone music in fact solves precisely this problem in “atonal”

music). Tymoczko hints at this problem (183), but spends more time on “pitch-class circulation graphs” in

his criticism of post-tonal music. As discussed below, this is misleading because the paradigm shifts from

pitch to interval in the move from tonal to atonal music, and “interval circulation graphs” (as opposed to

Tymoczko’s essentially flat “pitch-class circulation graphs”) are highly nuanced in this music and are more

suitable for explaining structure.

14These geometrical grids and shapes were formerly described as orbifolds, a term from topology not used

here. See my discussion of the connections between the sums and differences used to define orbifolds with

George Perle’s theories in Headlam 2008.

15These lattices all appear in earlier writings, not cited by Tymoczko; for a thorough overview of the cycles,

group structures, voice leading, symmetries, and geometry used here, with references, see Douthett (2008).

16A tesseract is a “four-dimensional cube, lying at the center of four-note chromatic chord space” (288) to

model the four possible shifts, with minimum displacement, of dominant seventh chords to diminished

seventh chords.

17Smith (1986), Beach (1987).

18Perle (1996, 19, Example 4(a)).

19

Perle (1996, 19, Example 4(b)). 20Cinnamon (1986, 2 and 9).

21On pages 240–41, Tymoczko mentions Fétis in his distinction between “harmonic tonality” in chords

related by thirds and sequential tonality, wherein the tonic and dominant seem to lose some of their power.

22Levenson (1984).

23Roeder (1984).



32

24Everett (2001, 296–99).

25Lewin (1987–88).

26For instance, Chapter 1 begins with some larger questions concerning the term “tonal,” including the

assertion “Faced with these questions, contemporary music theory stares at its feet in awkward silence” (3).

However, even a glance at Bryan Hyer’s excellent article on tonality in The New Grove Dictionary reveals

that Tymoczko’s presentation style invents a straw man (Hyer).

Despite the lack of references, Tymoczko does have some “interesting” notes. In Note 4 (5), he

writes that David Wessel apparently correlates the spectral centroid with the perception of pitch; this

reference follows the assertion in the text that the eardrum is one-dimensional. The eardrum is referred to

as an “area” in acoustics literature, so is 2D, and Wessel, whose article concerns “timbre space,” correlates

the spectral centroid with timbre (as many other writers have), not pitch. Note 20 (15) reports that

Tymoczko, in the first person, uses the term “C major” where the white notes imply a tonal center on C,

and again “I call this ‘the I symmetry’” (33, Note 7) where pitch-class inversion is involved; he might add

“I and everyone else.” Notes 18–20 on page 42 cover a lot of ground, but the assertion that previous authors

conflate what Rahn (1980) called “directed pitch-class intervals” 1–11 with interval-classes 1–6 is not the

case. On page 16 Tymoczko reports, in an understatement, that the presence of ficta “may complicate

matters somewhat” in regard to finding diatonic scales, with no reference to the literature. On page 47,

Note 24, transposition is defined as addition and inversion as subtraction from a constant, both well-known

and long existing concepts, with the only reference being to “Tymoczko 2008b.”

27Brown (2007, 18ff).

28Ibid. (xvi).

29Ibid. (26).

30Tymoczko states here that only two harmonic statistical studies exist, both done by him, but—to mention

one article among many—Bret Aarden and Paul T. von Hippel (2004) cite a study of triadic progressions

from Bach (2,643) and Mozart and Haydn (960); they conclude that, in addition to moving the smallest

possible distance, a single rule about not doubling unstable, tendency tones is virtually sufficient to explain

part-writing. 31Aldwell and Schachter (2002).

32Hantz (1982), Cinnamon (1982).

33Cinnamon (1982, 24).

34Cinnamon (1984), Brown (1986).

35Meyer (2000, 262).



33

36Lakeoff and Johnson (1980).

37Roeder (1984, 1989, 1994).

38Alphonce (1974), Morris (1998).

39

O’Connell (1968 [1962]). We can include here Straus’s many representations of what might be called

“set-class” voice leading, and his central notions of uniformity, balance, and smoothness in voice leading,

not to mention his earlier seminal criteria for establishing prolongation—a central technique for relating the

vertical to the horizontal (as developed historically in many writings). The incorporation of inversional

operations on a level comparable to the importance of transposition to tonality, roughly from Tristan on, as

explained initially by Benjamin Boretz (1972), is part of the extensive compositional system created by

George Perle in response to his studies of the music of Alban Berg. That many of the “orbifolds” in

Tymoczko’s models are organized by sum was pointed out in Headlam (2008).

40Lerdahl (2001).

41Brown and Headlam (2007) point out that the orbifolds do not model tonal space, due to the fact that

permutational equivalence does not hold in tonal contexts, and that distinctions and equivalences among

chords such as CCC, CCE, CEEE, are more prescient than CEG, EGC, GCE, etc.

42Brown and Dempster (1989).

43A far more inventive example poking fun at twelve-tone techniques is provided by Berg’s setting of

Alwa’s panicked assertion that “None at the newspaper office knows what to write,” set, as noted by Perle,

in a rotated retrograde inversion. See Perle (1959, 191).

44Schoenberg (1975, 216–218).

45Straus (1987).

46Laitz (2011, xvii).

47Ibid. (xvii, xix).

48Johnson (2008).

49Tymoczko (2004), Lewin (1987–88).

50Van den Toorn (2003).

51

MacDonald (1990, 357). 52Musgrave (1985, 258–59).

53Kallberg (1985).

54In another place (Tymoczko 2009), Tymoczko reveals that this denial of the full implications of Lewin’s

GIS is intentional; here he directly confronts Lewin’s notion of the generalized arrow, and finds Lewin’s

system lacking in several important respects, which Tymoczko’s geometrical system “solves.” However, I



34

find (I benefit greatly from discussion with Tuukka Ilomaki on Lewin here, although my errors are my

own) that Tymoczko’s arguments result from a misreading of Lewin. Briefly, first, Tymoczko assumes that

his own representations of specific attributes as abstract equivalences are also adopted by Lewin, when that

is not the case; and second, Tymoczko presents “paths of pitch-class groupings” as a remedy for his

perceived lack of such multiple paths in Lewin’s GIS/transformational networks, when such paths are

precisely the point in Lewinian analysis. For the first point, the issues may be clarified by Lewin’s own

reply to Edward Cone on the nature of theory and analysis (Lewin 1969). Lewin is, in this context, an

analyst, and it is the specific mapping in a GIS, the conduit from the group of operations in IVLS to the

particular elements being acted upon in the Space, that is crucial in Lewin’s ordered triple. Although Lewin

calls his system a “Generalized Interval System,” it is based on specific mappings of specific events—

Lewin is almost alone in rejecting equivalence classes of any type—his generalizations really result from

his analytical reorientations of the particular spaces he is working within rather than the equivalences

associated with set theory. This is the point of his label-free systems (and the distinction, not recounted here

by Tymoczko, between a transformational graph and network). In Tymoczko’s terms, it is not that Aunt

Abigail will fall off the earth—but that her train(s) are really the same trains (at least, it looks that way to

her!) but are given different functions, or roles, etc. in a differently oriented/defined space. When she

bought her train ticket, she set a context for which way the train was going, which allows her to understand

which way to the dining car, etc. This choice is crucial—I’m reminded of a reply by Robert Morris to

Joseph Straus in an SMT session, on which T or I of an [048] to use—it is a contextual/analytical question

that cannot be answered in a theoretical context. Thus Tymoczko is, I believe, erroneous in much of his use

of “interval-class space,” “pitch-class space,” etc. There is no interval-class space: we imagine smallest

possible distances among all possible realizations as a convenience—Lewin is one of the few theorists who

maintained the distinctions between specific elements rather than erroneously resorting to abstractions

(which is not always easy to do)—and Tymoczko’s definition of an interval as an equivalence class of

motions is entirely contrary to the Lewin practice of carefully defining each contextual space in which he

analyzes. There is much consideration of these issues in the literature, despite Tymoczko’s assertion to the

contrary. For the second point above: in a reply to Michael Buchler, Henry Klumpenhouwer points out that,

for Lewin, a specific path is not the desired result of an analysis—it is the sum of all possible paths

(Klumpenhouwer [2007]). These multiple paths require multiple GISs and lead to some perceptual issues,

as pointed out by Steven Rings (2011, section 1.2.5, 20-21) but this is again, in my opinion, the nature of

the Lewinian enterprise (i.e. Rings’s “apperceptive multiplicity”). In short, Lewin does provide for multiple

paths—in fact, that is an emphasis in his approach.

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