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.

``It is very, very exciting work," said Richard Cohn , a professor of music theory at Yale University. ``It is an age-old problem, but people have been limited by the limits of the two-dimensional page."

For many centuries, people have devised ways to understand music using visual guides, from simple dots on a staff to more complex constructions with names like the circle of fifths. What Tymoczko has done for the first time is create a kind of ultimate map of musical space that contains all the maps that have come before, and that reveals a deep geometry -- and rich set of connections -- that have never been seen before.

``It was like you had a little map of a couple of blocks of Dorchester, and a few blocks of Brookline, and some of Cambridge," said Tymoczko, who just finished a year of research at Harvard's Radcliffe Institute for Advanced Study . ``Now we have a map of all of Boston."

Much of Western music, from classical to jazz to pop, follows what Austrian composer Arnold Schoenberg once called the law of the shortest way. From one chord, composers generally try to move the individual notes as little as possible to get to the next chord.

This tradition seems to be tied up in how the brain processes music, Tymoczko said. If notes make large leaps, we don't tend to hear it as a single melodic line. But if the steps are small, we can hear many lines of melody at once. Indeed, the Western musical tradition can be thought of as a long exploration of the ways that melodic lines can be combined to make a series of satisfying harmonies, he said.

Although Tymoczko eventually ventured into difficult mathematical territory, he started with the simplest possible chords, made up of two tones. He imagined placing these chords on a flat map, but instead of latitude and longitude, it has two axes representing the tones. At any point on the map, then, would be the chord made up of two tones, just like any point on a traditional map has a latitude and longitude.

Making a musical map this way puts each chord next to the other chords that use nearby notes. However, he had to make two changes to make the map reflect the unique nature of music. The map listed every chord twice, with the notes reversed -- for example the notes C and G in one spot, and the notes G and C in another. So he folded one half of the map onto the other half, so chords were not repeated.

Next, there are many keys on a piano, but it is divided into octaves of twelve notes that cycle around like the numbers on a clock: Moving up twelve notes from C, you end up back at another C that has a very similar sound quality to the first C (though higher in pitch). A chord of C and G sounds similar even if the C is moved up an octave. To handle this problem, he took two edges of the map, gave it a clever twist, and then mathematically ``glued" the edges together. Move twelve notes on the map in one direction, and you end up back where you started.

The resulting space, with a fold and glued edges, is an example of an orbifold, a mathematical structure devised 50 years ago that is now used by physicists to do string theory, an effort to devise a unified explanation for the forces of nature.

On his website is an animation of this two-note chord map, with the opening chords of ``Smoke on the Water." When the dark dot representing the chord being played slides off one end, it appears on the other side.

His paper then extends the same method to higher dimensions -- a three-dimensional space for three-note chords, a four-dimensional space for four-note chords, etc. Another animation on his website shows how the chords in a Chopin prelude move around in space. Seen on Tymoczko's map, it is clear that Chopin is mostly making small moves in one part of space.

This mapping technique helps show the choices that a composer has in moving from one chord to another. One of the more interesting insights, Tymoczko said, is that the familiar chords that sound most pleasant sit like a column, right in the middle of the map. Being in the middle means that the chords are in easy striking distance of many other chords.

It is a fascinating fact, he said, that the very chords that sound nice are also the ones that make it easy to reach other chords in small steps -- the central idea of Western music.

``We have a powerful, systematic language for tying together 1,000 years of musical practice, for expressing what music is about in a very core way," Tymoczko said.

Gareth Cook can be reached at cook@globe.com. Animation and free software can be found at www.music.princeton.edu/~dmitri/