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Published in the February 9, 2011, issue

**A few years ago, Princeton University music theorist and composer Dmitri Tymoczko was sitting in the living room of his home playing with a piece of paper. Printed on the sheet were rows and columns of dots representing all the two-note chords that can be played on a piano — AA, AB** b **, AB, and so on for the rows; AA, B** b **A, BA, and so on for the columns. It was a simple drawing, something a child could make, yet Tymoczko felt that the piece of paper was trying to show him something that no one ever had seen before.**

Suddenly Tymoczko (pronounced tim-OSS-ko) realized that if he cut two triangles from the piece of paper, turned one of the triangles upside down, and reconnected the two triangles where the chords overlapped, the two-note chords on one edge of the resulting strip of paper would be the reversed versions of those on the opposite edge. If he then twisted the paper and attached the two edges, the chords would line up. “That’s when I got a tingly feeling in my fingers,” he says.

Tymoczko had discovered the fundamental geometric shape of two-note chords. They occupy the space of a Möbius strip, a two-dimensional surface embedded in a three-dimensional space. Music is not just something that can be heard, he realized. It has a shape.

He soon saw that he could transform more complex chords the same way. Three-note chords occupy a twisted three-dimensional space, and four-note chords live in a corresponding but impossible-to-visualize four-dimensional space. In fact, it worked for any number of notes — each chord inhabit ed a multidimensional space that twisted back on itself in unusual ways — a non-Euclidean space that does not adhere to the classical rules of geometry. A physicist friend told him that these odd multidimensional spaces were called orbifolds — a name chosen by the graduate students of Princeton mathematician William Thurston, who first described them in the 1970s. In the 1980s, physicists found a few applications for orbifolds in arcane areas of string theory. Now Tymoczko had discovered that music exists in a universe of orbifolds.

Tymoczko’s insight, made possible through a research collaboration with Clifton Callender from Florida State University and Ian Quinn from Yale University, has created “quite a buzz in Anglo-American music-theory circles,” says Scott Burnham, the Scheide Professor of Music History at Princeton. His work has “physicalized” music. It provides a way to convert melodies and harmonies into movements in higher dimensional spaces. It has given composers new tools to write music, has revealed new ways to teach music students, and has revealed surprising musical connections between composers as distant as Palestrina — the Italian Renaissance composer — and Paul McCartney.

In a book to be published in March by Oxford University Press, *A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice*, Tymoczko uses the connection between music and geometry to analyze the music of the last millennium and position modern composers in a new landscape. He rejects the idea that music can be divided into distinct genres. As Tymoczko sees it, medieval polyphony, the high classical music of Beethoven and Mozart, the chromatic romanticism of Wagner and Debussy, the jazz improvisations of Bill Evans, and the Beatles’ *Sgt. Pepper’s Lonely Hearts Club Band* all are built on the same handful of principles. Tymoczko writes in the preface: “It would make me happy to think that these ideas will be helpful to some young musician, brimming with excitement over the world of musical possibilities, eager to understand how classical music, jazz, and rock all fit together — and raring to make some new contribution to musical culture.”

**The link between** geometry and music has deep roots. Sometime between 530 and 500 B.C., in the town of Kroton on the rocky southern coast of Italy, Pythagoras and his followers made one of the most consequential discoveries in the history of science. If the string of a harp is shortened by half, it creates a tone one octave above that of the unshortened string. If the original string is shortened by two-thirds, the resulting tone is separated from the octave tone by a euphonious interval we know today as a fifth. Further experimentation showed that dividing the string into four parts produces intervals now known as fourths, with fur ther divisions of the string producing the familiar 12-note chromatic scale that the Greeks bequeathed to history.

Pythagoras and his followers thought big. The rational division of the musical scale was not just beautiful or pleasing — it was a sign that the universe was constructed on a rational basis and could be understood. “It was the first consistent realization that there is a mathematical rationality in the universe and that the human mind can make sense of that rationality,” says Kitty Ferguson, the author of *The Music of Pythagoras: How an Ancient Brotherhood Cracked the Code of the Universe and Lit the Path from Antiquity to Outer Space*.

Two-and-a-half millennia later, fifths and fourths are still the basis not just of three-chord rock-and-roll, but of much basic music theory. Students learn how to recognize intervals and relate those intervals to different kinds of scales, like major and minor scales. They practice transposing, inverting, and modulating melodies and chords. They absorb, perhaps without fully realizing it, the mathematician Gottfried Leibniz’s injunction that music is the “unknowing exercise of our mathematical faculties.”

Tymoczko falls squarely into the mathematical tradition in music. His father, Thomas, was a well-known philosopher of mathematics at Smith College who was fascinated by the use of computers in mathematics. His sister, Julianna, is a mathematician specializing in algebraic geometry at the University of Iowa.

But Tymoczko, growing up in the 1980s in Northhampton, Mass., spent more time listening to the Talking Heads, John Coltrane, and Brian Eno than solving equations. He swapped his piano lessons for guitar lessons and began playing in bands. He entered Harvard intending to study music, but the abstract and atonal music his professors preferred left him cold, and he switched from composition to philosophy. After studying philosophy at Oxford on a Rhodes scholarship, he kicked around Harvard for a few years as a teaching assistant, composing on the side and dabbling in journalism. Finally he decided to become serious about music again and enrolled in music graduate school at the University of California, Berkeley.

Möbius strip representation of two-note chords: The black lines in the illustration above outline the rows and columns of Tymoczko’s original diagram, in which all the two-note chords that can be played on a piano were arranged vertically and horizontally. To see what he found, copy this page and cut out the diagram. Twist it, and attach the 1s to each other and the 2s to each other. The chords match up in the resulting Möbius strip.

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CURRENT ISSUE: Feb. 9, 2011

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Video: Geometry of music

Watch a Chopin prelude as it travels through four-dimensional space, and Deep Purple's "Smoke on the Water" on a Mobius strip.

Watch a Chopin prelude as it travels through four-dimensional space, and Deep Purple's "Smoke on the Water" on a Mobius strip.