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
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.