Music's inner map revealed, with some help from geometry

It started with a simple question faced by every composer: From one chord, where can a piece of music go next?
To answer it, Dmitri Tymoczko turned not only to music theory, but to multidimensional geometry.
A lover of mathematics as well as music, Tymoczko (pronounced tim-OSS-ko) was able to construct a theoretical space filled with every possible chord, with similar chords near each other. Any piece of music -- from a Chopin piano prelude to Deep Purple's hard-rock anthem ``Smoke on the Water" -- can be represented as a path through this space.
His musical map could have a variety of applications, specialists said. It could be used to help computers compose music, to teach music theory to students, or perhaps even to develop a musical instrument in which a player drags a pointer through musical space. Since it was published last month in Science -- the first music theory paper to run in the journal's 126-year history -- Tymoczko has even heard from a man interested in building a high-tech lava lamp, with lights that move through space as the music does.
But the work's main importance is much broader. It will give scholars a powerful new tool for understanding how Western music functions, as well as how it has developed over time. And the paper also serves as a striking confirmation that the deep connections between music and mathematics -- in an intuition that goes back at least to Pythagoras -- extends to the many dimensions of modern mathematics.

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Examples - Deep Purple