Dmitri Tymoczko: A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice (review)

I believe A Geometry of Music (hereinafter AGM) is a ground-breaking book in music theory. According to Dmitri Tymoczko, “While my stated audience consists of composers and music theorists, I have tried to write in a way that is accessible to students and dedicated amateurs,” p. xviii, in the Introduction. If my own experience is any guide, AGM will be especially useful to composers who, like myself, use computers in composition. Indeed, this review is written mainly from the viewpoint of an algorithmic composer. But, as much as my informal education in music theory permits, I will also attempt to give the book some historical and theoretical context.

AGM synthesizes about 15 years of work by the author (and some other theorists) towards developing a geometric understanding of many phenomena in voice leading, chord structure, chord progression, scale theory, and modulation. The starting point is to define each chord as a single point in a continuous Euclidean “chord space” with as many dimensions of pitch as the chord has voices. This simple idea turns out to be tremendously fruitful. Tymoczko convincingly argues that all commonly used measures of voice-leading distance agree with the length of the distance from one chord point to another chord point in chord space.

AGM proposes that music that is tonal in the broadest meaning of the term, from the beginnings of Western polyphony through the extended tonality of today, and across many different classical and popular styles, features conjunct melodic motion, acoustic consonance, harmonic consistency, limited macroharmony (the notion of scale, more or less), and centricity (having a tonal center in the standard sense). I suspect that for academic theorists, the main interest of AGM will be its use of the geometric definition of voice-leading distance to develop a deeper understanding of these phenomena.

According to Tymoczko’s theory, a chord is a point, and voice-leading is movement from one point to another point. Of course, music theorists use many different levels of abstraction in thinking about chords and scales. Theorists almost always abstract from the order of the voices, they usually ignore the particular octave of a pitch, and sometimes even ignore the particular inversion of a chord. And they generally ignore voice doublings. AGM shows that each of these levels of abstraction exactly corresponds to what mathematicians call an “equivalence class” in chord space. For example, the standard definition of a “pitch-class set” corresponds to combining the equivalence classes for octave (“O”), order of voices (“P” for permutation), and number of voices (“C” for cardinality): OPC. Other equivalence classes are “T” (for translational equivalence, i.e., OPTC equivalent chords are the same chord type) and “I” (for inversional equivalence).

When an equivalence class, or combination of classes, is imposed upon a space, then all points that are the same with respect to the equivalence class become “glued together.” The space thus becomes a “quotient space” or “orbifold.” For example, the chord space for trichords under OPC equivalence becomes a tilted prism whose equilateral faces are glued together modulo a one-third twist. (See Figure 1.) The chord points in twelve-tone equal temperament are colored balls. The lines connecting the chords are the semitone voice-leadings. The four augmented triads run up the center of this prism; the twelve major and twelve minor triads surround the augmented triads in six alternating columns of four chords, each glued together to form a twisted torus. The two-pitch trichords are on the sides of the prism, and the one-pitch or unison trichords are on the edges of the prism. Transposition equates to moving a chord in parallel...