Gödel, Escher, Chopin

Inuitive links between musical chords and geometries

Familiar relationships between sets of musical notes, such as transposition between chords, directly translate into geometrical structures such as this Möbius strip — where each dot represents a whole class of equivalent two-note chords — or into more complex structures with many dimensions.

SHAPE OF CHORDS

Composers have an understanding of these geometries without realizing it, says music theorist Dmitri Tymoczko of PrincetonUniversity. “Musicians like Chopin had a very direct, intuitive understanding of these spaces at a time when mathematicians still didn’t know much about high-dimensional geometry,” he says.

Wandering around these spaces, Tymoczko and his collaborators have found subtle relationships between progressions of chords that traditional musical theory would classify as unrelated — for example, between progressions in Mozart’s Fantasy in C-minor and in Beethoven’s Ninth Symphony, the team reported in the April 18 Science.

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