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The geometry of music

— The same mathematical principles that physicists use in string theory can be applied to analyze a string quartet, a music theorist writes in this week's issue of the journal Science. He's devised a new geometrical model that just might serve as a "theory of everything," at least when it comes to Western musical traditions.

The idea of expressing music geometrically goes back centuries. The five-line staffs used in musical notation, for example, can be thought of as grids for plotting the points and curves of a melody. Musicians have looked to the "circle of fifths" as a formula for understanding tonal chord progression since the 1700s. But lots of musical styles lie outside the classical circles, ranging from the chromatic sweep of Wagner to the dissonance of  Schönberg to the fusion of Miles Davis. 

"There have been lots of geometrical representations of music," Princeton music theorist Dmitri Tymoczko told me today. "It's as if we had maps of many small neighborhoods in a city, but we didn't have a sense of how those maps fit together."

To try to fit them together, Tymoczko turned to mathematics that's often applied to the problems of extradimensional physics. He visualized music as a lattice of points in a folded-up, symmetrical space known as an orbifold.

"The mathematical terminiology and technology to do that is only about 50 years old," Tymoczko said. "These spaces, these orbifolds, are familiar to modern string theorists."

For years, string theorists have used music as a metaphor for fundamental particles, and now Tymoczko is usiing the mathematics of string theory to understand the fundamentals of music.

The math makes it easier to understand objectively what great musicians and composers do in their head. "When you sit down to interact with a piano, you're actually interacting with a non-Euclidean space, because there are many different ways you can play a C-major chord on a piano," Tymoczko said.

He said orbifolds capture the multidimensionality of music: how harmony interacts with counterpoint, how chords are connected with each other, even how notes are arranged "to minimize the amount of effort that musicians have to make when moving from chord to chord."

Princeton
Points within a tetrahedron
represent chords from a
Chopin piano prelude.


Tymoczko has done up a QuickTime movie of a particularly tricky Chopin piano prelude in E-minor (Opus 28, No. 4) to illustrate how the orbifold works.

"This prelude is mysterious," he explained in a Princeton news release. "While it uses traditional harmonies, they are connected with nonstandard chord progressions that people have had trouble describing. However, when you plot the chord movement in geometric space, you can see Chopin is moving along very short lines, staying primarily within one region."

On Tymoczko's Web site, you can find additional resources, including his ChordGeometries software, a version of his Science paper and a series of four QuickTime video files that provide further audiovisual explanation. There's even a QuickTime depiction of the famous chords from Deep Purple's "Smoke on the Water."

The scheme works less well for musical styles that don't have the Western notion of chord progression. But even for non-Western styles - say, the rhythms of African music - "you can use my geometric model to think about how you evolve from one rhythmic pattern to another."

Could Tymoczko's geometrical scheme open the way to computer-composed music that might surpass Bach and Beethoven?

"That's not going to happen," he said. "This isn't going to take anyone from being a mediocre composer to a brilliant composer. But it might help you get from being a beginning composer to a pretty good composer. ... We're going to be able to instruct computers to produce musical results. At the very least, we won't ask computers to do impossible things."

Tymoczko said he uses his software as a tool when he's writing his own music. Another composer, Michael Gogins, is to present a paper on the application of the orbifold method at the International Computer Music Conference this November in New Orleans.

Although they may use the same math, the composers have an advantage over the string theorists in at least one respect. String theory has shown that there may be 10500 solutions for the equations that govern the state of the universe, but Tymoczko said "one thing that the spaces show us is that Western music is much more of a unique solution than you might have thought."

That makes it easier to determine which strings of notes make music, even the dissonant kind, and which are just plain noise.

"It's better news than the string theorists are dealing with," Tymoczko said.