September/October 2008

The Geometry of Sound

Elaine Chew, SM '98, PhD '00, returns to campus to explore the intersection of music and math

By Larry Hardesty

Software codeveloped by Elaine Chew yielded this visualization of a tonal analysis of P. D. Q. Bach’s Fugue No. 6 from The Short-Tempered Clavier. Credit: Elaine Chew

Pythagoras is the father of harmonic theory, and during World War II, musicians proved exceptional code breakers. In May, Elaine Chew, SM '98, PhD '00, who directs the Music Computation and Cognition Laboratory at the University of Southern California, came to Killian Hall for a concert and lecture exploring the long-established connection between music and mathematics.

The concert opened with Ivan ­Tcherepnin's Fêtes, a fantasia on the melody of "Happy Birthday." Chew used the piece to demonstrate techniques of "melodic transformation" such as stretto, in which one voice repeats a tune before another has finished it, and inversion, or turning a melody upside down. A projected display of the real-time output of software developed by Chew and USC's Alexandre François accompanied the performance. As Chew played, the software mapped the piece's chord progression as a series of geometric shapes moving around a three-dimensional helix.

The final three pieces were written specifically for Chew. Or perhaps "written" is the wrong word, since the next piece, A Simple Gift for Elaine, was generated in part by computer. Rodney Waschka, a professor of arts studies at North Carolina State University, wrote an arrangement of the Shaker melody "Simple Gifts" and fed it through a set of what are called "genetic" algorithms, which randomly deleted notes and recombined bars in new ways. The resulting melodies lurched and feinted surprisingly, but the piece also had passages of great rhythmic vigor.

Next was Sudoku Variations, by Tamar Diesendruck, a visiting professor at USC. For each of the numbers 1 through 9, ­Diesendruck wrote a melodic fragment of as many beats. The first nine variations present the fragments in an order determined by the rows of a Sudoku puzzle; the next nine combine row sequences with column sequences; and a 19th variation wraps things up. Around the hall, heads bobbed to the piece's strange metrical shifts as people read off the rows of the Sudoku puzzle printed in the program.

Chew closed with a set of excerpts from Doubles III, by MIT music professor Peter Child. The piece relies on a fairly common 20th-century practice called polytonality, in which strands of melody in different keys are played simultaneously. But its beauty more than made up for its tenuous connection to the concert's theme. Chew grew up in multiethnic Singapore, and Doubles III borrows themes from songs of her youth. Before performing the piece, she sang those songs for the audience in an untrained but clear voice. It was a moving reminder that while music may be a uniquely mathematical art form, its power derives from a different, deeper source.