” Coooool site. I never thought of music this way…” wrote Brian, a gifted, highly trained musician, when he sent me links that sparked this learn node from the work at Princeton by Dmitri Tymoczko. The illustration to the right is 2 frames from a Tymoczko animation of chords in four-dimensional space.
An article in Science about The Geometry of Musical Chords begins with this abstract:
A musical chord can be represented as a point in a geometrical space called an orbifold. Line segments represent mappings from the notes of one chord to those of another. Composers in a wide range of styles have exploited the non-Euclidean geometry of these spaces, typically by using short line segments between structurally similar chords. Such line segments exist only when chords are nearly symmetrical under translation, reflection, or permutation. Paradigmatically consonant and dissonant chords possess different near-symmetries and suggest different musical uses.
The article is by Dmitri Tymoczko who illustrates ChordGeometries on animations with sounds on his Web pages at Princeton University. He invites you to: “Watch as Chopin moves around in a circle, a Mobius strip, and in four-dimensional space! Or try Deep Purple on a Mobius strip!”
For some background to the topic, an introduction to Form in Music by Catherine Schmidt-Jones is available in Rice University’s Connexions. Anthony Brandt, also in Connexions, gives an overview of Musical Form, with examples from Schumann, Bach, Boulez, and Beethoven.
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