Mts.2012.34.1.123-143 (Rev) HEADLAM on TYMOCZKO a Geometry of Music

Society for Music Theory

The Shape of Things to Come? Seeking the Manifold Attractions of TonalityA Geometry of Music: Harmony and Counterpoint in the Extended Common Practice by DmitriTymoczkoReview by: Dave HeadlamMusic Theory Spectrum, Vol. 34, No. 1 (Spring 2012), pp. 123-143Published by: University of California Press on behalf of the Society for Music TheoryStable URL: http://www.jstor.org/stable/10.1525/mts.2012.34.1.123 .

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123

The Shape of Things to Come? Seeking the Manifold Attractions of Tonality

dave headlam

A review of: A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. By Dmitri Tymoczko. Oxford Studies in Music Theory.

New York: Oxford University Press, 2011, xviii + 450 pages.

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 po-lemic 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 of-fers 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 also for “students and dedicated amateurs” (xviii), the latter a larger audience including “scientists and mathematicians” (jacket copy). The book thereby includes some definitions of the basic elements of musical sound (“Sound consists of small fluc-tuations in air pressure” [28]) and music theory (“a basic musical object [is] an ordered series of pitches, uncategorized and unin-terpreted” [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 de-fined by way of the “submajorization partial order” (essentially a preference for even distributions [3,9,8]).2 In 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 geometric apparatus used, and then dem-onstrates 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 simultaneously to expand our purview and streamline our understanding of tonal-ity. As part of his presentation, he also wants to promote tonal-ity, 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” state of a lack of to-nality in atonal music (185) and are expressed in terms roughly equivalent to Winston Churchill’s views on democracy: accord-ing to Tymoczko, while “typical Western listeners prefer” it, to-nality 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 as-serts, has elements of “universality” to it, related to our “bio-logical inheritance” (7), and is a “living tradition” still viable in “today’s wide open, polystylistic, multicultural, syncretistic, and postmodern musical culture” (225).

Although the jacket copy uses the adjectives “groundbreak-ing” and “revolutionary,” and Tymoczko refers almost exclu-sively 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 ele-ments and relationships is, of course, as old as the study of

1 The book builds on Tymoczko’s earlier writings and his collaborations pri-marily with Clifton Callender, Rachel Hall, and Ian Quinn. It is accompa-nied 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.

2 Here 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, nor 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 geometric 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.

3 Tymoczko supplies statistics to show that writings on post-tonalists Schoenberg and Cage and topics such as 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 Govern-ment except all those other forms that have been tried from time to time” (from a speech to the British House of Commons, 11 November 1947 [Churchill {1974, 7566}]).



124 music theory spectrum 34 (2012)

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” pro-mulgated 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 geometric 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 dem-onstrations of the ways in which tonal processes and repertory are circumscribed; the geometric methodology is presented as essentially style- and time-period-neutral, thus allowing com-parisons ranging from Josquin to Miles Davis. This approach constitutes real theory building, in that it reduces large quanti-ties of note patterns, styles, and rules of composition to a few theoretical constructs (for instance, in the section in which 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 ob-servations based on geometric elements, the most compelling arguments Tymoczko offers, which stem from his earliest writ-ings, 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 rela-tions 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 re-mind us of the parallels between Debussy, Ravel, and Stravinsky and jazz, as well as the return to diatonicism in the roughly parallel emergence of 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 mis-leadingly to random processes (184, explained below), it is re-freshing 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 con-tinuing tradition is reflected in analyses of composers outside the usual canonic “B”s, such as Grieg and Jan ´a ˘cek, 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 “disclaim-ers” 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 “out-raged 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 do not reso-nate 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 supple-mented 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 geometric grids rather than mu-sical notation in analysis, Tymoczko asserts that “our visual sys-tem is optimized for perceiving geometric 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 ahis-torically as a “challenge” to his own view (258–67, omitted from the Index), Tymoczko opines that “[Schenker] ended up advo-cating a radical model of musical organization according to which entire pieces were massively recursive structures, analo-gous 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

5 Outside of a few references to the first manifestations of Tonnetze (Gott-fried 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 “geom-etry” in the title, both recent.

6 See Morris (1987) and (1998); Lewin (1987); Cohn (1998); Clough and Myerson (1985); Clough and Douthett (1991).

7 Many 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 topics and musical issues similar to those confronted by Tymoczko, from the begin-nings of the larger musical period he proposes.

8 Concepts central to Tymoczko’s arguments on scales and collections, such as well-formed and maximally even scales, are found in writings by Clough (Clough and Myerson [1985]; Clough and Douthett [1991]) and Norman Carey and David Clampitt (1989), among others. Clampitt’s (1997) un-mentioned “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 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 Lar-son, Henry Martin, or Keith Waters on jazz topics.



the shape of things to come? seeking the manifold attractions of tonality 125

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 would include items such as “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 col-lections 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 tonal-ity— 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 in-novations. I will end with some considerations of the analyses in the second part of the book, in comparison with other ap-proaches. 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 geometric 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;9 I will also compare Tymoczko’s work with Steven Laitz’s textbook The Complete Musician.10 Steven Rings’s Tonality and Transformation is also relevant, but is not consid-ered in depth here.11

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 “ped-agogically-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 a wider purview in scholarly writing (Chapter 10).

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.12 Indeed, the first point, the asser-tion 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 prefer-ence 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 five to eight 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 mo-tion, 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 har-monic 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 pre-sented 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,

9 Johnson (2008). 10 Laitz (2011). 11 Rings (2011).

12 Tymoczko backpedals a bit from a position of inevitability for tonality and the materials of tonal music in later chapters, suggesting rather than a “de-terministic or Hegelian view,” an image of the natural tendency of moun-taineers to follow a similar path up a mountain, given that the “structure of the rock will naturally suggest certain routes” (211).




and functions—has a parallel description in scalar collections. Second, in his interpretation of tonal materials, Tymoczko in-cludes two melodic/harmonic features of tonality, both associ-ated with Neo-Riemannian theory (neither one listed in the Index, but they are defined on pages 49–50 and 61–64, respec-tively). The first is “efficiency” in voice leading, for the shortest possible distance, also categorized as “smooth(ness)” and “parsi-mony,” terms with a long pedigree in music theory.13 The sec-ond 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#,C), for instance.14 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#,D), lack the clear separation of chord tones from connecting notes and pres-ent problems in distinguishing musical figure from ground dis-cussed by Straus.15 The implications and interrelation of these two features—Tymoczko formalizes this relation by showing how one emerges from the other (60, Note 34)—direct the in-quiry through the rest of the book.16

In Chapter 1, the five features of tonality (given above) lead to four claims: 1) harmony and counterpoint constrain each other; 2) scale, macroharmony, and centricity are independent; 3) modulation involves voice leading; and 4) music can be un-derstood 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, includ-ing 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 geometric apparatus for modeling voice leading and har-monic spaces.

The discussion of “Harmony and Voice Leading” in the sec-ond chapter presents a “musical set theory” (28) from a Lewin-and Morris-influenced application of group-theory-based post-tonal theoretical concepts,17 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 geometric grids.18 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 forty-five-degree rotation of a Cartesian x, y alignment of pitch-class pairs (Examples 1[a] and 1[b]), in which distances traverse the same differ-ences/intervals on the horizontal axis and the same sums/

13 The 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 Mor-ris (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 Sec-tion 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 and Dahlhaus [2007–12]).

14 See Note 8. 15 See Straus (1987). Proctor (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).

16 On page 60 (Notes 33, 34), summarizing a section describing transposi-tional, inversional, and permutational symmetry (based on C to C,C,C, etc.) and “nearby” chords (B,C,Db to C,C,C, to C,Db,B, then B,C,Db to C,Db,B; the example essentially restates Forte’s T0I-based definition of in-version, hence the symmetry), Tymoczko formalizes the connection be-tween efficiency and maximal evenness. Efficient voice leading implies near-symmetrical chords, and the converse is also asserted, near-symmet-rical chords imply efficient voice leading. Slightly recast, Tymoczko notes that if chords X and F(X) have efficient 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 would not 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 dis-cern. See Straus (2003) on this question. In teaching Schenkerian princi-ples, 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 cir-culation graphs” (as opposed to Tymoczko’s essentially flat “pitch-class cir-culation graphs”) are highly nuanced in this music and are more suitable for explaining structure.

17 See especially Morris (1987, Chapters 4 and 5). 18 These geometric 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).




inversional measures on the vertical axis (Example 1[b]). (Example 1 also contains a similar arrangement of sums and differences from Table 1 from George Perle,19 and Berg’s align-ment of the interval [difference] cycles from a letter of 1920 to Schoenberg, reproduced by Perle.20 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

19 Perle ([1977] 1996, 17). 20 Perle (1977, 5).

example 1. (a) Euclidean x, y alignment of pitch classes 0–11 (10=t, 11=e); (b) alignment of (a) rotated by 45 degrees for Tymoczko’s 2D grid, with sums on the verticals and difference intervals on the horizontals; (c) Perle’s Table 1 from Twelve-Tone Tonality showing same alignment of sums and intervals (differences), separated into even and odd sums; (d) Berg’s alignment of interval cycles, from a letter to

Schoenberg of 1920 (Perle [1977, 5])

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 consist-ing of thirty or thirty-one (sums 0, 3, 6, 9) multiset trichords at

(a) (b)

x/y+1 (vertical), x+1/y (horizontal) sums 0 1 2 3 4 5 6 7 8 9 t e diffs

0/e 1/e 2/e 3/e 4/e 5/e 6/e 0/0 1/1 2/2 3/3 4/4 5/5 0

0/t 1/t 2/t 3/t 4/t 5/t 6/t 1/0 2/1 3/2 4/3 5/4 6/5 1

0/9 1/9 2/9 3/9 4/9 5/9 6/9 1/e 2/0 3/1 4/2 5/3 6/4 2

0/8 1/8 2/8 3/8 4/8 5/8 6/8 2/e 3/0 4/1 5/2 6/3 7/4 3

0/7 1/7 2/7 3/7 4/7 5/7 6/7 2/t 3/e 4/0 5/1 6/2 7/3 4 0/6 1/6 2/6 3/6 4/6 5/6 6/6 3/t 4/e 5/0 6/1 7/2 8/3 5 0/5 1/5 2/5 3/5 4/5 5/5 6/5 3/9 4/t 5/e 6/0 7/1 8/2 6 0/4 1/4 2/4 3/4 4/4 5/4 6/4 4/9 5/t 6/e 7/0 8/1 9/2 7 0/3 1/3 2/3 3/3 4/3 5/3 6/3 4/8 5/9 6/t 6/e 8/0 9/1 8 0/2 1/2 2/2 3/2 4/2 5/2 6/2 5/8 6/9 7/t 8/e 9/0 t/1 9

0/1 1/1 2/1 3/1 4/1 5/1 6/1 5/7 6/8 7/9 6/t 9/e t/0 t 0/0 1/0 2/0 3/0 4/0 5/0 6/0 6/7 7/8 8/9 9/t t/e e/0 e

(c) Perle’s Table 1 (partial) (d) Berg’s Master Array

sums

Ints 0,1 2,3 4,5 6,7

diffs 1 2 3 4 5 6 7 8 9 t e 0 diffs

0,0 0/0 1/1 2/2 3/3

2,t 1/e 2/0 3/1 4/2 0 0 0 0 0 0 0 0 0 0 0 0 0

4,8 2/t 3/e 4/0 5/1 e t 9 8 7 6 5 4 3 2 1 0 e

6,6 3/9 4/t 5/e 6/0 t 8 6 4 2 0 t 8 6 4 2 0 t

8,4 4/8 5/9 6/t 7/e 9 6 3 0 9 6 3 0 9 6 3 0 9

t,2 5/7 6/8 7/9 8/t 8 4 0 8 4 0 8 4 0 8 4 0 8

0,0 6/6 7/7 8/8 9/9 7 2 9 4 e 6 1 8 3 t 5 0 7

6 0 6 0 6 0 6 0 6 0 6 0 6

1,e 1/0 2/1 3/2 4/3 5 t 3 8 1 6 e 4 9 2 7 0 5

3,9 2/e 3/0 4/1 5/2 4 8 0 4 8 0 4 8 0 4 8 0 4

5,7 3/t 4/e 5/0 6/1 3 6 9 0 3 6 9 0 3 6 9 0 3

7,5 4/9 5/9 6/e 7/0 2 4 6 8 t 0 2 4 6 8 t 0 2

9,3 5/8 6/8 7/t 8/e 1 2 3 4 5 6 7 8 9 t e 0 1

e,1 6/7 7/7 8/9 9/t 0 0 0 0 0 0 0 0 0 0 0 0 0




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, proportion-ally about 60, 36, and 4%), move from the inner central core which connects the four maximally-even (ME) augmented tri-ads, 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 summa-rizes 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 rela-tive motion; and 4) boundary conditions are mirror-like.

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 il-lustrating how the mappings tend to the smaller intervals in this order. Tymoczko explores some tonal implications of these cy-clic characteristics, 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, which in the quan-tized twelve-note world require an accompanying parallel mo-tion to allow for contrary-motion models (101–2). Thus interesting voice leading, combining parallel, contrary, and oblique motions, emerges.

The essential geometry from these group relationships is given in two types of lattices that Tymoczko presents in chro-matic 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 term) shows interrelated set-class [05], [01], and [06] dyads and voice leadings, with a cen-tral [06] producing [01]s by [05]-based voice leading or producing [05]s by [01]-based voice leading. The diatonic ver-sion includes three dyads and a voice-leading step, 03/04/05/S (for “thirds, fourths, fifths” and diatonic steps) and thereby re-quires 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 tri-ads, and octatonic, with the transforms of the central set-class

example 2. 3D orbifold (Tymoczko’s Figure 3.8.2)

Figure 3.8.2 A single “tile” of three-note chord space is a triangular prism. Minor triads are light spheres, major triads are dark spheres, and augmented triads are cubes. The dark spheres on the edges of the prism are triple unisons. The lines in the center of the space connect chords that can be linked by voice leading in which only a single voice moves, and it moves by only a single semitone.




[0369] into [0258]s, [0268]s, and (not shown by Tymoczko) [0358]s.21 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 their use by mov-ing 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 in generating tonality (246–58).

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-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 en-compasses at least six major twentieth-century movements—im-pressionism, neoc lassicism, 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 incorpo-rating the geometric 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 be-tween 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, al-lows 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 har-mony. 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 in-clude 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, in-clude fifth as well as third motions.

The discussion of voice leading focuses on the familiar prob-lem 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

example 3. Intended to illustrate Tymoczko’s points (97–103) about efficient voice leading in n-note chords in transpositions by 12/n semitones; most clustered to most even from top to bottom; closest mappings within successive columns shown at right; bottom row

(T5 /cm indicates a T5 by contrary motion) shows smooth voice leading to T5 using contrary motion

21 These lattices all appear in earlier writings, not cited by Tymoczko; for a thorough overview of the cycles, group structures, voice leading, symme-tries, and geometry used here, with references, see Douthett (2008).

Clustered

closest mapping

T0,T6 T0,T4,T8 T0,T3,T6,T9 (ics) 1-cycle 01 67 0123 4567 89te 012 345 678 9te 5,4,3 2-cycle 02 68 0246 468t 8t02 024 357 68t 9e1 4,4,3 3-cycle 03 69 0369 47t1 8e25 036 369 690 903 3,3,3 4-cycle 04 6t 048 480 804 048 37e 6t2 915 2,0,3 5-cycle 05 6e 05t3 4926 816t 05t 381 6e4 927 1,1,3 6-cycle 06 60 06 4t 82 06 39 60 93 0,2,3 T5/cm 05 e6 t5 047 e48 059 0368 e479 e581 1,1,1 Even




22 A tesseract is a “four-dimensional cube, lying at the center of four-note chromatic chord space” (288) to model the four possible shifts, with mini-mum displacement, of dominant seventh chords to diminished seventh chords.

fourth doubles; and second, a “nonfactorizable voice leading” (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 geometric 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 direc-tions: C–C#, A–A#, F#–G, D#–E on the x axis, F–F#, D–D#, B–C, G#–A on the y axis, and G–G#, E–E#, C#–D, A#–B on the z axis, so that C major moves to G major (+y axis, add F# or # 4 ), D melodic

minor (+x axis, add C# or #1), and A harmonic minor (+z axis, add G# or #5, Example 5). I find it easier to think of scale degrees for this section, where the scales map as shown in the example.

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 geometric model in “Chopin’s Tesseract,”22 and finally a long section on the Tristan Prelude, modeling the ubiquitous half-diminished sev-enth chord to dominant seventh progression in this opera.

pcs voice leading (z) set-class or diatonic types

(a1) 05 16 06 e6 07

0e 10 00

e0 01

y y x = [06] x y = [05] ([01]) y y z = [01] ([05])

(a2) 50 72 70 7e 90

-S0 0S 00

0-S S0

w w x = 04 x w = 05 y y y = 03, z = step

(b1) 148 158 049 048 047 038 e48

100 010 001 000

00e 0e0 e00

y y y x = [048] x y = [037] y y y z = 1

(b2) DEG CFG CEA

CEG

CEF CDG BEG

S00 0S0 00S 000

00-S 0-S0 -S00

w w y x = M, z = step x y = m v w y w = sus v = truncated

(c1) 1369 0469 0379 036t 0369 0368 0359 0269 e369 1379 046t 136t 0479 0369 0259 e368 0268 e359

1000 0100 0010 0001 0000 000e 00e0 0e00 e000 1010 0101 1001 0110 0000 0ee0 e00e 0e0e e0e0

yyyy x = [0369] x y = [0258] yyyy z = 1 yyww y = [0268] x x = [0369] wwyy w = [0358] z = 00/11

example 4. Tymoczko’s essential geometry in two types of lattices—chromatic and diatonic; (a1) chromatic [0167] lattice (Figures 3.10.2[a], 3.10.7, 3.11.1[a]); (a2) diatonic 03/04/05 (third-fourth-fifth) lattice (Figures 3.10.2[b], 3.11.8 with scales, 7.5.2);

(b1) chromatic [014589] hexatonic lattice (Figure 3.11.2[a]); (b2) diatonic M, m, A, sus, trunc (sus=suspended CDG, trunc=CEF, Figures 3.11.2[b], 3.11.9 with scales, 7.5.5); (c1) chromatic [0134679t] octatonic lattice (tesseract, Figure 3.11.3[a], Chopin analyses)




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 through-out, including Tymoczko’s self-described “Fundamental Theorem of Jazz” (156), which “states that you can never be more than a semi-tone 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.

problems

The first problem I find in Tymoczko’s book is its lack of references, background, and acknowledgment of other sources for many of the ideas. Notable in this context is the omission of the rich history of geometric 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 ad-monishments to music theory texts; see 270–71 on the aug-mented 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 re-cent fractals. (All this information is easily available online.) A charitable view is that Tymoczko adopts a Cartesian attitude of personal contemplation and self-discovery alongside the some-what 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 refer-ences, which, aside from himself (fifteen items) and Cohn (eleven 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 ex-change by David Beach).23 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 connec-tion 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 geometric grid, previously called an orbifold in Tymoczko’s writings, with its adjacent sectors either a difference of +/-1 for intervals or differences of +/-1 for sums (047 and 158, or 047 and e48, etc.), indeed reflects a distinction between near-even chords and efficient voice leading, but the structure of many possible or-bifolds at different sums and differences is inherent to Perle’s theories, shown in concise form in his opening Table 1 in Twelve-Tone Tonality, and realized in his “keyboard” exercise representa-tions of combined cyclic sets in arrays.24 For Perle, building on the music and writings of Alban Berg, it is precisely the kinship be-tween tonal concepts such as “consonance” and “voice leading” (but defined contextually in terms of the intervals and sums at hand) that creates the “tonality” in “Twelve-Tone Tonality.”25

Other topics are similarly treated; for instance, two topics referred to repeatedly by Tymoczko have significant precedents. In his chapters on functional harmony and chromaticism,

example 5. Scale mappings using scale degrees (SD); hm=harmonic minor, HM=harmonic major (C–D–E–F–G–A b–B–C), mm=melodic minor

23 Smith (1986); Beach (1987). 24 Perle ([1977] 1996, 19, Example 4[a]). 25 Ibid. (19, Example 4[b]).

Chromatic SD 1 shift type major 1–#1 1–2 mm (C major to D mm) 4–#4 1–5 major (C major to G major) 5–#5 1–6 hm (C major to A hm) harmonic minor 3–#3 1–1 HM (E hm to E HM) etc.




Tymoczko promotes relations by thirds over fifths, with no mention of an article on that very topic by Howard Cinnamon, wherein analyses of pieces by Liszt that detail structural me-lodic, motivic, and harmonic components show the reversal of the traditional roles of thirds and fifths, where thirds move from their earlier “partitioning functions” (i–iii–V, i–vi–IV–ii) to be-come “contrapuntal chords within prolongations of more back-ground harmonies” and “agents of prolongation.”26 Later, in his discussions of modulation and measuring distances between keys, Tymoczko quotes Gottfried Weber (in one of the few his-torical references, 246–48), and invokes a “smoothness” metric. In an article entitled “Smooth Moves: Schubert and Theories of Modulation in the Nineteenth Century,” Irene Montefiore Levenson begins with Weber, but as a conservative foil for the more progressive theorists Reicha, Marx, Fétis, Hauptmann, and Louis and Thuille, and proceeds to show multiple ap-proaches to modulation with examples from Schubert. She in-cludes categories such as Fétis’s “unitonic” (no modulation), “transitonic” (closely related keys), “pluritonic” (using mixture, with Mozart as the exemplar), and “omnitonic” (modulate freely in “transcendent enharmony”), and modulations between ma-jor-third related keys, keys a semitone apart, and others—in-cluding the role of augmented triads.27 Levenson’s article, which also compares the attitude of these theorists to the changing roles of thirds and fifths, certainly merits mention.28

With regard to the geometric approach, I will comment more fully below, particularly by invoking John Roeder, who is referenced as a general resource, but whose integer grid model with its geometric demonstrations of collectional characteristics is a direct model for Tymoczko’s work.29 In other contexts, even brief items such as the analytical comments on the Beatles song “Help” (346) lack any reference, in this case to the definitive discussions in Walter Everett’s Beatles volumes.30 (The Brahms analyses similarly lack any reference to the literature on this composer, to which I will return below.) Even analyses by David Lewin, such as his consideration of Debussy’s “Le vent dans la plaine,” are omitted, in this case in Tymoczko’s discussion of this very piece (18–19).31 The lack of references, along with the 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.32

My second criticism relates to the dismissal of Schenkerian theory, which is treated somewhat like an unsubstantiated, pe-ripheral theory rather than one of the mainstays of our under-standing 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 interpreta-tions of the theory as a theory of musical structure, organic co-herence, 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,33 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.”34 In the discussion of this first point, it becomes clear that Brown is at odds with Tymoczko’s assertions of historical continuity; as for Brown, al-though they share common elements, there are three distinct compositional environments based on intervals, triads, and functional tonality, with correspondingly different behaviors in each.35 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 decep-tively from a chronological standpoint as “A Challenge from Schenker” (258–67), Tymoczko accuses the theory of abandon-ing functional harmonic relationships; earlier (213), he states that Schenkerian theorists “seem to deny that functional

26 Cinnamon (1986, 2, 9). 27 On 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.

28 Levenson (1984). 29 Roeder (1987). 30 Everett (2001, 296–99). 31 Lewin (1987–88). 32 For 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). How-ever, 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 [2007–12]).

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 Tymoc-zko, 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 Ty-moczko reports, in an understatement, that the presence of ficta “may com-plicate 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.”

33 Brown (2007, 18ff.). 34 Ibid. (xvi). 35 Ibid. (26).




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 tonal-ity. 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 as-sociated formal structures, and also 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 seemingly contradictory opening “It was the best of times, it was the worst of times,” because, as in long musical compositions such as Mahler sym-phonies, there are formal divisions and structures on many dif-ferent levels, to accommodate and allow for our comprehension. As mentioned, form is largely absent from Tymoczko’s discus-sions; 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 seeks to explain.

The bulk of Tymoczko’s Schenker rebuttal outlines four po-sitions, 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 the-ory as a complement to “traditional harmonic theory” (261). This latter position is adopted by Tymoczko, but he replaces the melodic/contrapuntal position with a geometric model. While he recognizes that some aspects of his approach are commensu-rate 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 no-tion of structural levels (17). However, Tymoczko abandons prolongation and any systematic way of distinguishing conso-nance from dissonance, chord tone from non-chord tone, or functional chord from embellishing chord except for the prob-lematic 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 mo-tions over fifth motions is contradicted by his own statistics; drawn from Bach and Mozart, they show clearly that fifths, al-lowing 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.36

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).37 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 sev-enth-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 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.”38 Hantz points out the features of what Tymoczko calls a “tesseract” in his Chopin analysis—the first four chords, major-minor sevenths Ab7–F7, B7–Ab7 share three notes with diminished-seventh A–C–Eb–Gb, 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 his 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 hierarchi-cal recursive relation of the large and the small, and the role of the dominant—are largely minimized, and structural third mo-tions, 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, Ab, is not by a traditional Ursatz, or even an unfolding of the augmented triad Ab–C–E, but by a neighboring motion Ab–C–Ab; the piece is

36 Tymoczko 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.

37 Aldwell and Schachter (2002). 38 Hantz (1982); Cinnamon (1982).




thus somewhat like a partial tonal structure and Cinnamon’s interpretation shows that interesting voice leading is not pre-cluded by underlying interval cycles. Cinnamon’s final para-graph lays out his understanding in a passage that foreshadows by almost thirty years Tymoczko’s initial questions about tonal-ity and many of the latter’s assertions, but, notably, adds the crucial element of a large-scale tonal structure to the interpreta-tion. This structure is what guides us through the apparent maze—the “massive recursion” Tymoczko refers to does not confuse the issue; rather, 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 an-swer in the affirmative, with Tymoczko’s book as my first wit-ness.

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 rela-tionships, the voice-leading and prolongational procedures are often quite analogous 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 struc-tures 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.39

Cinnamon has in these and other writings defined a Schenkerian interpretation of chromatic harmonies and pro-gressions, and Brown has similarly written on a Schenkerian view of diatonic and chromatic structures in tonality.40 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 such as Daniel Gottlob Türk (1750–1813) rather than his contempo-raries. Tymoczko’s assertion that in his chapter on chromati-cism he intends “to present chromaticism as an orderly phenomenon rather than an unsystematic exercise in compo-sitional rule breaking” (268), is wholly unwarranted by a re-view of the literature. Indeed, if we compare Tymoczko’s approach with that of Cinnamon, the lack of a large-scale sup-porting structure in the former is obvious. In his discussion of the opening of Liszt’s Sonetto 104 del Petrarca, Cinnamon re-tains both the local chord successions and the larger tonal con-text (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 mo-tion, only local connections. We are trapped, snorkel in hand, on the surface.41

My third criticism of Tymoczko concerns the presentation of the geometric 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 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 relax-ation with succumbing to it (lower).42 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 loud-ness in varying modular and logarithmic terms. Given that ex-periential metaphors are at the core of the geometric models Tymoczko uses, we should probably ask our models to maintain these useful directional musical analogies.

An earlier use of geometric 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 geometric system, using in-terval models instead of pitch-class pairs, is in fact very close to Tymoczko’s.43 Example 6 shows several examples of Roeder’s geometrically based equivalences, which follow his careful ac-knowledgments 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 geo-metric representations.44 Roeder extrapolates from both alge-braic 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 demonstra-tions of geometric 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.45 This fascinating and

41 For such analysis with leveled positioning of harmonies, see Meyer (2000, 262).

42 Lakoff and Johnson (1980). 43 Roeder (1987), (1989), (1994). 44 Alphonce (1974); Morris (1998). 45 O’Connell ([1962] 1968). We can include here Straus’s many representa-

tions of what might be called “set-class” voice leading, and his central 39 Cinnamon (1982, 24). 40 Cinnamon (1984); Brown (1986).




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

To be sure, the format of Tymoczko’s book precludes an ex-tended exegesis of earlier interpretations of geometric models for musical explanation, but some mention of these earlier writings would seem to be in order. Instead, we find only a brief engage-ment 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 refer-ences (Cohn, Quinn, Douthett, Callender).46 This Appendix con-tinues, however, chastising an unnamed number of efforts to model voice leading for not satisfying criteria such as those that require that 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 crite-ria 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 re-quired 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 geometric 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 lead-ing 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 decep-tive, 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

example 6. Roeder (1987), geometric models for voice leading and equivalence classes

notions of uniformity, balance, and smoothness in voice leading, not to men-tion his earlier seminal criteria for establishing prolongation—a central tech-nique for relating the vertical to the horizontal (as developed historically in many writings). The incorporation of inversional operations on a level com-parable 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).

46 Lerdahl (2001).




the modularizing of musical space, and his Figure B3 (405) in-dicates the extent to which Tymoczko’s motions in his spaces are indebted to vector graphics. The closest thing to an explana-tion of the raison d’être for the geometric 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 geometric 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 care-fully 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 jour-nal 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, concepts foreign to our experience of musical continuity. We are still left with the problem of explaining why C–E–G to C#–E#–G# 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 con-structs, as in the permutational equivalences which are the basis of the orbifold compression of compositional space.47 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).48

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).49 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 “chro-matic” 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.50 Briefly (from a long and involved argument Schoenberg 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 cir-culation.

I will not go further at this point on Tymoczko’s negative de-piction of non-tonal music, but will only comment on his statis-tics 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 nu-anced 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 distin-guished authors, including my own mentor Andrew Mead and his teacher, the first of four people Tymoczko claims had a pro-found impact on his musical life: Milton Babbitt.

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 to-nality, writ large, is such an attempt, and, despite the shortcom-ings detailed above, does offer some useful advances in our understanding of this type of musical structure. I have men-tioned above the significant elements of the geometric approach and the presentations of related scales and chords as explanatory

49 A far more inventive example poking fun at twelve-tone techniques is pro-vided 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).

50 Schoenberg (1975, 216–18).

47 Brown 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 (C,C,C), (C,C,E), (C,E,E,E), are more prescient than (C,E,G), (E,G,C), (G,C,E), etc.

48 Brown and Dempster (1989).




models. In a more general sense, the features in Tymoczko’s ap-proach that lead to advances in tonal understanding stem from his somewhat Socratic method of asking questions and pursu-ing, step by step, interesting interrelationships among the mate-rials he posits. The questions include the following: What are the connections among the five features of tonality given, does one imply another, and which are most salient? (4, 9) and How can we “depict the voice-leading possibilities between all the triads in the chromatic scale?” (20). (This leads to the forty chords in the center of the trichordal grid: four augmented tri-ads, and twelve each of the diminished, major, and minor tri-ads.) 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, and then adds complications, for in-stance in his demonstration of how a scale is necessarily created from chords (12–15). Here Tymoczko presents his fundamen-tal 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 loses coherence.51 Tymoczko elaborates on the two situ-ations 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 such as the half-diminished seventh moving to the dominant seventh chord, or the common diminished-seventh notes in the T3-cycle of domi-nant sevenths, that have been underreported in studies of nine-teenth-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 theo-rize 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 do not find the experience “relevant.”52 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 combi-nation of melodic lines,” but including “motivic relationships that make a given work unique.”53 The repertory 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,54 which covers much of the same ground as Tymoczko’s book, including maximally even scales and a geo-metric 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 ma-terials. Chapters include topics such as “Spatial Relations 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.),55 while Laitz takes the tonal universe as a given, to be explored and appreci-ated, but without asking 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 peda-gogically designed version; and, of course, Laitz’s text integrates every aspect of an undergraduate curriculum, but is more tradi-tional 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 repertory 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# is inter-preted as creating a V7/V chord and effecting a “smooth” modula-tion from the D major to the A major scales (G–G#). It is the scalar connection, more than the chords, that creates the modula-tion and the long-range harmonic progression for Tymoczko. A similar motion is compared from Debussy’s “Le vent dans la

51 Straus (1987).

52 Laitz (2011, xvii). 53 Ibid. (xvii, xix). 54 Johnson ([2003] 2008). 55 Johnson (2008).

example 7. Statistics (in percentages) from Schoenberg’s Op. 11, No. 1




plaine,” where, in an Eb natural minor scale, Bb–Bbb creates a scale equivalent to an F # melodic minor ascending scale, presumably F#–G#–A–B–C#–D#–E#–F # as G b–A b–Bb b–Cb–Db–E b–F–G b. 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 analyses by David Lewin.56 Lewin posits a series of chromatic motions around pentatonic and diatonic collections emphasizing Eb or Bb; 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 col-lections—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 (Bb or Eb as tonic, etc.), Tymoczko’s scale theory is, as stated in another con-text by Pieter van den Toorn, somewhat of an exercise in label-ing.57 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”: he has stated that I10 (F/F, F # /E, G/Eb, Ab/D, A/Db, Bb/C, B/B) maps C diatonic to Gb dia-tonic, B to G acoustic, and C# whole-tone to itself.58 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 struc-tural event such as a scale. Very few pieces before twelve-tone music have this quality; Debussy’s “Voiles” from the first book of Préludes for piano is one example (with only the chromatic mo-tions 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 coin-cides with Y2. This represents the most obvious analysis of the pas-sage, 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 con-trary 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 geometric representation makes the point immediately obvious.

Although this analysis suffers from a similar lack of context and lack of attention to the large-scale motion found through-out the book, Tymoczko’s presentation isolates and focuses at-tention 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.”59 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 consis-tency 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 gener-ate the harmony, creating unusual dissonances, as at bars 6–10.”60 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 invert-ible 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– #4 over 1– #7–1–2; the Y notes, if interpreted in E minor, seem to be a nonsensical (b)6–b7 over 2– #1 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 twelve-bar phrase, which divides into a sequential progression that gradually reduces in number of attacks from eight to two 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 # Locrian) pace Tymoczko. The Locrian implication also allows for the inverse parallel of scale degrees 5–6, 5–#4 in minor with 1– #7, 1–2 in Locrian, and 1– #7, 1–2 in minor and (b)5–6, (b)5–4

56 Tymoczko (2004); Lewin (1987–88). 57 Van den Toorn (2003). 58 Tymoczko (2004, 288, Note 66).

59 MacDonald (1990, 357). 60 Musgrave (1985, 258–59).




example 8. Tymoczko’s Figure 3.5.2




in Locrian (E #–F #–G–A–B–C–D/C–B–A #–G–F#–E–D# or #7–1–2–3–4–5–6 in Locrian/6–5–# 4–3–2–1–# 7 in minor). But this rather forced scalar interpretation raises the question as to what is more relevant to the piece: the parallelism or the mirror-ing 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 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.

We will end with a consideration of Chopin’s F-minor Mazurka, which Tymoczko claims is a “virtual reworking” of the Prelude in E Minor, 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. 4.61 Both passages are unstable tonally, but relatively diatonic, com-pared 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 open-ing phrase (mm. 1–8) without including the opening chord—thus missing the bass arpeggiation of A b–F–C–F which clearly moves through an F minor triad space to the dominant and back

to the tonic (bass notes: A b–G–Gb–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–Fb=E to lead to A). Tymoczko then explores the interesting idea of regarding the piece as a controlled passage of improvisa-tion 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 sev-enth chord by common tones. He then helpfully finds instances of this procedure in other pieces by Chopin, although his inter-pretation 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, how-ever, these readings suffer from the same affliction that ails Tymoczko’s analysis throughout: without a larger-scale under-standing 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 machines and technological advances that allowed them to go beyond human capacities for travel, storing information, and so forth. Of course in music, for hun-dreds 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 sys-tems, favor acoustic consonance, just as we now drive cars and fly planes to get places we could never walk to, or

example 9. Brahms, Opus 116, No. 5, opening phrase reduction with rhythmic normalization, with melodic (M) intervals and harmonic (H) intervals shown in upper staves (t=10, e=11)

61 Kallberg (1985).

Music Theory Spectrum 34.1: Music Examples, p. 23

Headlam, Ex. 9:

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“parallels” between doubling voices




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 imag-ine and create music that challenges our senses and allows us to go beyond our evolutionary boundaries. Despite his prefer-ence for tonality, Tymoczko’s geometric approach actually al-lows 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 for-mation, etc., we can use geometric models for other interval-lic/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.62 In this model there are no fixed endpoints: the magnitude and direc-tion 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 sixty 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.

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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 con-textual/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 conve-nience—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, Sec-tion 1.2.5, 20–21) but this is again, in my opinion, the nature of the Lewin-ian enterprise (i.e., Rings’s “apperceptive multiplicity”). In short, Lewin does provide for multiple paths—in fact, that is an emphasis in his ap-proach.

62 Elsewhere (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 find (I benefit greatly from discussion with Tuukka Ilomäki on Lewin here, although my errors are my own) that Tymoczko’s arguments result from a misreading of Lewin. Briefly, first, Tymoczko as-sumes that his own representations of specific attributes as abstract equiva-lences 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 net-works, 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 spe-cific mappings of specific events—Lewin is almost alone in rejecting equiv-alence 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




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Music Theory Spectrum, Vol. 34, Issue 1, pp. 123–43, ISSN 0195-6167, electronic ISSN 1533-8339. © 2012 by The Society for Music Theory. All rights reserved. Please direct all requests for permission to photocopy or reproduce article content through the University of California Press’s Rights and Permissions website, at http://www.ucpressjournals.com/ reprintinfo.asp. DOI: 10.1525/mts.2012.34.1.123



Mts.2012.34.1.123-143 (Rev) HEADLAM on TYMOCZKO a Geometry of Music

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