The Geometry of Music

By MICHAEL D. LEMONICK

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, by as little as possible, from one chord to the next.

Exactly how one style relates to another, however, has remained a mystery--except over one brief stretch of musical history. That, says Princeton University composer Dmitri Tymoczko, "is why, no matter where you go to school, you learn almost exclusively about classical music from about 1700 to 1900. It's kind of ridiculous."

But Tymoczko may have changed all that. Borrowing some of the mathematics that string theorists invented to plumb the secrets of the physical universe, he has found a way to represent the universe of all possible musical chords in graphic form. "He's not the first to try," says Yale music theorist Richard Cohn. "But he's the first to come up with a compelling answer."

Tymoczko's answer, which led last summer to the first paper on music theory ever published in the journal Science, is that the cosmos of chords consists of weird, multidimensional spaces, known as orbifolds, that turn back on themselves with a twist, like the Moebius strips math teachers love to trot out to prove to students that a two-dimensional figure can have only one side. Indeed, the simplest chords, which consist of just two notes, live on an actual Moebius strip. Three-note chords reside in spaces that look like prisms--except that opposing faces connect to each other. And more complex chords inhabit spaces that are as hard to visualize as the multidimensional universes of string theory.

But if you go to Tymoczko's website music.princeton.edu/~dmitri) you can see exactly what he's getting at by looking at movies he has created to represent tunes by Chopin and, of all things, Deep Purple. In both cases, as the music progresses, one chord after another lights up in patterns that occupy a surprisingly small stretch of musical real estate. According to Tymoczko, most pieces of chord-based music tend to do the same, although they may live in a different part of the orbifold space. Indeed, any conceivable chord lies somewhere in that space, although most of them would sound screechingly harsh to human ears.

The discovery is useful for at least a couple of reasons, says Tymoczko. "One is that composers have been exploring the geometrical structure of these maps since the beginning of Western music without really knowing what they were doing." It's as though you figured out your way around a city like Boston, for example, without realizing that some of your routes intersect. "If someone then showed you a map," he says, "you might say, 'Wow, I didn't realize the Safeway was close to the disco.' We can now go back and look at hundreds of years of this intuitive musical pathmaking and realize that there are some very simple principles that describe the process."

That's likely to help both scholars and teachers, he argues. By showing how compositions of various styles move through his orbifold spaces, says Tymoczko, you can see how different styles of Western music relate to each other and evolve. Tymoczko's maps can also be an aid to composers, says Cohn. Most have a favorite corner in orbifold space, a set of related chord types that they tend to explore over and over in different ways. Venturing into a different part of space can be tough; you have to learn your way around a whole new auditory neighborhood. You can do that intuitively by wandering around and seeing where you get to. But with the maps, you can plot a route that you know in advance will make some sort of sense.

That doesn't mean you can program a computer with Tymoczko's orbifold maps and have it spit out beautiful compositions. "I don't want to sell these maps as the royal road to composition," he warns. "They don't substitute for the hard work of learning how to move notes around." But they can help show when a new idea is promising and when it will probably lead to a dead end. "They might make an O.K. composer good," says Tymoczko, "but they won't make a good composer great."