The Geometry of Music

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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."