The source: “The Geometry of Musical Chords” by Dmitri Tymoczko, and “Exploring Musical Space” by Julian Hook, in Science, July 7, ­2006.

Discoveries by the ancient Greek philosopher Pythagoras (c. 569–c. 475 bc) forged an unbreakable link between music and mathematics. Pythagoras showed that a string two feet long would vibrate with a cer­tain tone, and that a string half as long would yield a tone an octave higher. Further divisions, into fifths, thirds, and quarters, unlocked the 12 ­tones—­C, D, E, etc., along with intervening sharps and ­flats—­that make up the 12 notes in an octave, the basis for Western music. Given how long this system has been in place, says Julian Hook, a music professor at Indiana University’s Jacobs School of Music, “it is perhaps surprising that our understanding of the mathematical structure of the spaces in which musical phenomena operate remains fragmentary.” Now Dmitri Tymoczko, a music theorist at Princeton University, has developed a way of viewing those spaces that may reveal some of their ­mysteries.

Hook points out that a conven­tional musical score is itself a kind of “graph whose vertical axis represents pitch and whose horizontal axis represents time.” Plotting the positions of those tones, he says, and their internal relations to one another reveals something fundamental about the structure of music. Hugo Riemann (1849–1919) invented one such map, called a Tonnetz, based on the work of mathematician Leonard Euler. The Tonnetz is a ­two-­dimensional model of a musical piece showing the links between individual notes and chords: Perfect fifths get linked diagonally, major thirds vertically, and minor thirds horizontally. A section of a Beethoven string quartet, perhaps not surprisingly, yields a Tonnetz with an elegant geometric structure, like the honeycomb of a ­bee.

What of modern composers, such as Arnold Schönberg (1874–1951) and his successors? Schönberg rejected the notion that any of the 12 familiar tones ought to be more ­dominant—­one might also say pleasing to the ­ear—­than any other, and his work opened the way for experimentation with the spaces between tones, which the Tonnetz cannot ­describe.

Tymoczko’s solution is to create a new kind of musical map, one based on a geometric shape called an ­orbifold.

To mimic the structure of an octave, each half of Tymocz­ko’s map is a reversed mirror of the other, with a ­half-­twist in the middle; this is easiest to visualize in ­two-­note chords, in which the pathway resembles a Möbius strip. Traveling 12 notes in any direction brings one back to the original tone, as the map loops back upon itself. As additional notes are added, and the chords become more complex, the map expands into multiple dimensions, creating a unified framework for all possible chord ­pro­gressions.

Although the relation­ships of perfect fifths and thirds lie within Tymoczko’s orbifold ­map—­and retain their geometric ­structures—­infinite spaces within the 12 tones now emerge, made up of subtones, or fractions, of the intervals between the notes. Notes from music that sounds jarringly dissonant, tellingly, are clustered in very tight spaces in the corners of Tymoczko’s orbifolds. Major chords, on the other hand, lie toward the center, allowing them efficient linking with minor keys and inverted chords. Many composers exploit such connections to inject counterpoint into their ­compositions.

Using the orbifold map, says Tymoczko, it is possible to track common chord progressions in classical music and see that they lie along a predictable trajectory. He can discern, for instance, how certain ­chords—­
C, ­D-­flat, ­E-­flat—­and chords closely related to them define the music of Schubert, Wagner, and Debussy. “My geometric models show us that there are important strands of commonality running through the last thousand years of music,” Tymoczko says, that previously went unrecognized. Tymoc­zko also believes that his sys­tem is invaluable for studying the music of ­non-­Western cultures, which frequently employ tones and pitches off the 12-tone scale. The orbifold map might even open up new tonal possibilities for contemporary com­posers to explore, though with no guarantee that they will inspire listenable ­music.