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.