Chord Geometry, Part II

In The Geometry of Chords I talked about the mathematics of chords as described by Princeton's Dmitri Tymoczko in his paper "The Geometry of Musical Chords," published in Science magazine. Since that post, I have downloaded his article (not free) and tried to figure out its abstruse mathematics. I don't know how fully successful my attempts have been, but I can say my understanding and appreciation of his ideas have grown.

The articles about Tymoczko's ideas which I cited in the original post seem to have given short shrift to some notions I consider key. One of these is the emphasis Tymoczko places on "voice leading," which is the basis for the contrapuntal music of Bach and others from the 18th century. "Counterpoint (or voice leading) is the technique of connecting the individual notes in a series of chords so as to form simultaneous melodies," writes Tymoczko. Composers of counterpoint aim for a system of felicitous successions of notes: one whose paths are efficient, in that changes in pitch from one note to the next in a given voice are small; one whose paths are independent, such that the separate voices do not move slavishly up and down in parallel; and one whose pitch paths ideally do not cross or intersect .

Counterpoint in (say) four voices produces a progression of four-note chords as a side effect. After the heyday of contrapuntal compositions, composers began thinking in terms of chord-based harmonies per se. There was a new set of constraints as to what sorts of harmonies were permissible. These constraints were not stated with reference to the rules of voice leading common in counterpoint. Yet, Tymoczko finds, post-Bach Western music has generally managed to "satisfy harmonic and contrapuntal constraints at once."

His fundamental purpose is accordingly to show how the two sets of constraints can both be satisfied in the various forms Western music has taken — even the most atonal, avant-garde compositions of the modern era.

Basically — at least as far I can tell — Tymoczko's math reveals that most or all "coherent" chordal harmonies, however consonant, dissonant, or atonal, partake of the same efficient, independent, ideally non-crossing voice leading that true counterpoint does.


In order to see this, you have to first construct various geometrical spaces within which all possible chords have their specific places. These spaces don't adhere to the commonsense axioms of Euclid which form the basis of everyone's (probably forgotten) high school geometry. Tymoczko's geometric spaces, in fact, loop back on themselves like a Möbius strip.

If you take a strip of paper and give it a half-twist, gluing its ends together, you have a Möbius strip. If you trace a line along its center far enough, you wind up on the "other side," exactly opposite the original starting point!

This is how Tymoczko models two-note chords, in fact. For example, take the notes C and D♭. They form two possible two-note chords, one being (C, D♭) and the other being (D♭, C) — see the first example in this movie. Though they are reversed in pitch order, they are identical in sound. They are like a certain point and the point directly opposite it on the Möbius strip.

To get from the first point to the second more expeditiously than following the center line of the strip all the way around, you can "bounce" off the edge or boundary of the strip directly over to the point immediately on the "other side" — though, as the unbroken-center-line journey proves, there is really no "other side" to a Möbius strip, and the two distinct points are the same!

This kind of geometric weirdness, when extended into an arbitrarily large number of dimensions to allow for chords with any conceivable number of notes, makes it possible to describe the various sorts of symmetries and near-symmetries that can occur in the structure of musical chords.


Tymoczko finds that not exact symmetry in chord structure but near-symmetry is the key to understanding why composers pick the chord progressions and voice leadings they do:

In many Western styles, it is desirable to find efficient, independent voice leadings between transpositionally or inversionally related chords .... A chord can participate in such progressions only if it is nearly symmetrical under transposition, permutation, or inversion ....

(C, D♭) and (D♭, C) are nearly symmetrical two-note chords, while (C, C) and (D♭, D♭) are perfectly symmetrical. (C, D♭) can also be represented as (0, 1), since in Tymoczko's math, the note C is equivalent to the "pitch class" 0 and the note D♭is 1. Similarly, (D♭, C) is (1, 0). Both are permutations of the unordered collection {0, 1} — order-independence being indicated by the use of curly braces rather than parentheses.

The idea of a "pitch class," by the way, is simply that tones whose pitches are separated by one or more octaves are essentially identical. Thus, every C in every octave on the piano keyboard is in the same pitch class.

The type of (near) symmetry in the example I just gave is what Tymoczko calls permutational symmetry or P-symmetry, since each pitch class in the first chord corresponds one-for-one with a pitch class in the second. In the most "trivial" case of P-symmetry — for example, the perfect symmetry of (C, C) leading to (C, C) — the voice leading is effectively null, and the result is, says Tymoczko, "musically inert."

Slightly less trivial near-symmetry of the permutational type — for example, (C, D♭) to (D♭, C) — is more interesting, especially when two-note chords become three-note triads such as {B, C, D♭}. Provided the triads as realized are made by three distinct voices that can be told apart by the ear — say, a flute, a clarinet, and an oboe — the resulting progression is dissonant, even atonal, but it is musically valid:

Nearly P-symmetrical chords such as {B, C, D♭} are considered to be extremely dissonant. They are well-suited to static music in which voices move by small distances within an unchanging harmony .... Such practices are characteristic of recent atonal composition, particularly the music of Ligeti and Lutoslawski. From the present perspective, these avant-garde techniques are closely related to those of traditional tonality: They exploit one of three fundamental symmetries permitting efficient, independent voice leading between transpositionally or inversionally related chords.

To get from (B, C, D♭) to, say, (C, D♭, B) involves having the voices or instruments trade off notes. The flute takes over the clarinet's note, the clarinet takes over the oboe's note, and the oboe takes over the flute's note. But the chord remains the same unordered collection, {B, C, D♭}. To capture this idea, Tymoczko has it that a voice-leading progression from (B, C, D♭) to (C, D♭, B) is reflected off the edge or boundary of his geometric space, which is called an "orbifold." As with a Möbius strip for two-note chords, this reflection off the boundary of a triadic, 3-dimensional orbifold results in the chord progression arriving right back where it started.

{C, C, C} is right on a boundary edge of the orbifold, as is {B, B, B}, as is {D♭, D♭, D♭}. Those and other same-note triads are examples of perfectly P-symmetrical chords.

{B, C, D♭}, since it contains three separate notes that are adjacent on a piano keyboard, is a "chromatic cluster," with B as its root. It is close to but not on an orbiform boundary, and so is only nearly P-symmetrical. That proximity to a boundary means the edge-reflected path from (B, C, D♭) to (C, D♭, B) is quite short. It also means that {B, C, D♭}, in any of its ordered permutations, sounds dissonant to the ear.

Taking both characteristics together, we learn why a dissonant, even atonal voice leading from (B, C, D♭) to (C, D♭, B) is said by Tymoczko to be among the "symmetries permitting efficient, independent voice leading between transpositionally or inversionally related chords." (Notice, however, that this particular voice leading is not non-crossing. The third principle of ideal counterpoint is not observed.)

In my next post, I'll take up the subject of how other near-symmetries of chord structure make for voice leadings and harmonies that are less dissonant and more dulcet to Western ears.