The Geometry of Chords

The music of Western culture is quite diverse, including the likes of Chopin and John Coltrane and Deep Purple. Is it all of a piece? How can such disparate uses of melody and harmony all be considered "music"? An answer comes from the world of mathematics — the same math that physicists use to deal with the hidden extra dimensions of the cosmos that string theory reveals.

Princeton University composer Dmitri Tymoczko (tim-OSS-ko), working with professor of mathematics Noam Elkies, a fellow composer, has shown that there are hidden regularities in virtually all pieces of Western music by which we perceive each to be a coherent expression of musical form ... no matter how weird and dissonant its harmonies may sound.

Michael Lemonick's article for TIME, "The Geometry of Music," introduces Tymoczko's work this way:

When you first hear them, a Gregorian chant, a Debussy prelude and a John Coltrane improvisation might seem to have almost nothing in common — except that they all include chord progressions and something you could plausibly call a melody. But music theorists have long known that there's something else that ties these disparate musical forms together. The composers of these and virtually every other style of Western music over the past millennium tend to draw from a tiny fraction of the set of all possible chords. And their chord progressions tend to be efficient, changing as few notes [as possible], by as little as possible, from one chord to the next.

A chord is simply the combination of two or more notes, played at the same time or slightly apart in time. In many forms of music, chords are thought of as such, while in music based on counterpoint, multiple melodies or "voices" form virtual chords as they superimpose their notes on one another. As the music unfolds, we hear a sequence — a "progression" — of chords, though not necessarily identifiable at a conscious level. Even a plain unaccompanied tune that we can whistle easily has implicit chords: the chords that would "go with" the melody if we played it on a piano.

Tymoczko's insight is that chords and their progressions ought somehow to be able to be mapped mathematically. Ordinary musical notation, with its notes on a staff, maps melodies pretty well. By extension, the same notation indicates the harmonies (chord progressions) that are to be played, with the chords shown as as a series of stacked notes on the staff. But complications such as moving notes up or down to the next octave or rearranging a stack of notes in any of numerous possible ways make is hard to see from the notation alone how the chords relate aurally.

To find better ways to map chord progressions, Tymoczko wanted to learn their "geometry." He wanted to be able to draw chords as points on a graph. Here is one such drawing:
It comes from "Mapping Music," an article about Tymoczko in Harvard Magazine. In the graph, a "space" containing all possible three-note chords or triads is represented as a triangular prism. The circles connected by short line segments that form a tinker-toy lattice in the middle of the prism represent the "standard" triads heard in everyday music. For example, the combination of the notes C, E, and G form the ubiquitous C major triad. The C-E-G# triad is called C augmented. It, too, is familiar in Western music. D♭-F-A♭ (or C#-F-G#) is termed C# major. C#-E-G# is C# minor.

All the lattice nodes have names like that, easy to say and familiar to composers and musicians. But there are any number of points, outliers to the central core of the vast prism space, that represent chords that are harder to name. For example, what is C-E-B♭? It can be considered a restacking of B♭-C-E, but even so, it has no common name. Entering it in Tymoczko's ChordGeometries program, available here, gives the unpronounceable name of "B♭ 0_2_6" and a position in the three-dimensional prism space almost squarely in front of the G#G#G# vertex.

The prism-space mapping is fine for three-note chords, but four-note chords are also common, and Tymoczko maps them as tetrachords on a polyhedron with four vertices. The polyhedron represents a four-dimensional "orbifold"; the previous diagram was of a three-dimensional orbifold. Here are four of the vast number of possible tetrachords in this 4D orbifold (note that triads like E minor and A minor can be considered tetrachords missing one note):



These chords are (says Harvard Magazine) among those used in the harmony of "Chopin’s hermetic E minor prelude [Op. 28, No. 4], long baffling to music theorists, [which] traces out a logical pattern of chords (as represented by the moving colored circles) in a four-dimensional orbifold. Each image (from an animation with piano accompaniment that may be viewed at http://music.princeton.edu/~dmitri/chopin3.mov) depicts one chord of the prelude."

As with the central lattice in prism space, the 4D-orbifold mapping of tetrachords has a central cluster of tinker-toy nodes representing a toolkit of common chords. Chopin didn't stick with the chords in the toolkit alone, but his chords flit about in a small space near the central cluster. As the large ball dances and glides in this confined space, no one chord change takes it very far along the intricate path. This is what gives the piece its musical coherence.

The gliding ball also changes color from one chord to the next. Per Harvard Magazine, "Changes in the large ball’s color indicate how evenly the chord divides the octave. Orange represents perfect evenness, blue perfect unevenness." This is a measure of consonance and dissonance. Run-of-the-mill chords — the ones that sound the least dissonant — spread one's fingers out on a keyboard and divide the octave relatively evenly. These are the chords in and around the central cluster in the orbifold that show up as red and orange and (with slightly more dissonance) yellow.

Chords that sound yet more dissonant have notes that cluster closely together on the keyboard, as the clumped notes unevenly divide the octave. The dissonances, shown as green balls, move the harmony out toward the periphery of the orbifold. At the very extreme, where the ball is blue, you find chords that represent multiple instances of the same note (e.g., four C's, possibly in different octaves).

A chord can have more than four notes, and an orbifold more than four dimensions — though showing one of these beasts in the form of a two-dimensional graph presents problems. Here is where the analogy with the extra dimensions of string theory, borrowed from the science of cosmology, comes into the picture. Tymoczko has shown that the same ideas hold for higher-dimensional chord geometries as for triads and tetrachords. Composers generally pick a confined region of the n-dimensional orbiform — even if it's a dissonant one — and stay within it. The path from one chord of the progression to the next, moreover, is generally short. Again, there is musical coherence.

The thing which interests me the most about all this is that the ear can actually hear the geometry of Tymoczko's chord spaces ... even when they of higher dimension. We may not like certain "advanced" or "modern" chord progressions because of their dissonance, but we are hard pressed to call them noise. Coherence is present to us, even if appreciation is not.