July 8, 2006
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Princeton music professor Dmitri Tymoczko has an intriguing article in the journal Science this week explaining how he uses  topology and non-Euclidean geometry to explore elusive musical structures.

 "Western music theory has developed impressive tools for thinking about traditional harmonies, but it doesn’t have the same sophisticated tools for thinking about these newer chords," Tymoczko says. "This led me to want to develop a general geometrical model in which every conceivable chord is represented by a point in space. That way, if you hear any sequence of chords, no matter how unfamiliar, you can still represent it as a series of points in the space. To understand the melodic relationship between these chords, you connect the points with lines that represent how you have to change their notes to get from one chord to the next."

Clear as the azure sky of deepest summer? Check out the short movie he made of Chopin's  E minor piano prelude (Opus 28, No. 4) -- it traces the harmonic movement in a sort of triangular prism, in which points representing traditionally familiar harmonies such as major chords gather near the center of the triangle, forming neat geometric shapes with other common chords that relate to them closely.

More of these movies (and a pdf of the article, "The Geometry of Musical Chords") can be found here.