Familiar relationships between setsrnof musical notes, such as transposition between chords, directly translate into geometrical structures such as this Mobiusrnstrip-where each dot represents a whole class of equivalent two-note chords - or into structures with many dimensions.rnComposers understand these geometries without realizing it, says music theorist Dmitri Tymoczko of Princeton University. "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.rnWandering 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 Beethoven's Ninth Symphony, the team reported in the April 18 Science.
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