The Geometry of Music

When I was an undergraduate in college, many of my non-music-major acquaintances and friends used to tease me with jibes about the alleged low academic status of my chosen discipline. "You're getting class credits for singing?? You take classes on how to pronounce words?? Man, I wish I was a music major...that must be easy!"

Of course, I explained that in addition to the music courses, I also had to matriculate and succeed in non-music academic courses, but such protestations fell on deaf ears. Music? Come on, that's just easy!

Oh, yeah?

Exactly how one [musical] 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 Möbius 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 Möbius 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.

Here's the abstract from the paper itself:

A musical chord can be represented as a point in a geometrical space called an orbifold. Line segments represent mappings from the notes of one chord to those of another. Composers in a wide range of styles have exploited the non-Euclidean geometry of these spaces, typically by using short line segments between structurally similar chords. Such line segments exist only when chords are nearly symmetrical under translation, reflection, or permutation. Paradigmatically consonant and dissonant chords possess different near-symmetries and suggest different musical uses.

The entire paper is extremely interesting (to those with a penchant for music theory). You can read it at the above link. Additional materials can be found at Dmitri Tymoczko's Princeton webpage.

Take that academia!