Posts tagged with orbifold

Dmitri Tymoczko — author of The Geometry of Music

• how to make visual representations of music
• (in paintings, video games, sculpture)
• 5 constraints on a composition that are necessary (but not sufficient) for it to sound good
• global statistical properties of songs
• why 20th century classical music had little audience
• a random painting is much less offensive to the eye than random notes are to the ear
• "I came up with these 5 principles using my brain, which is a kind of crude statistical device”
• the piano is essentially a line
• [NB: linear ⊃ monotonic ⊃ totally ordered]
• violin/voice musicians know that notes ⊂ continuous space, but the piano does us a favour by constraining us to a subset of those notes
• line `mod 13` = circle
• (equivalence classes of octaves — `A1=A2=A9` and `E4=E7=E12` etc.)
• directed segments, unordered tuples
• musical translation = mathematical transposition, musical inversion = mathematical rotation
• The fact that most people don’t have most perfect pitch (things sound the same in different keys) may be so that we can understand that, despite pitch differences in male/female adults’ speech and children’s speech, they are saying the same words.
• "It’s as if we couldn’t tell the difference between red and blue, but we were highly sensitive to the-difference-between-red-and-orange and the-difference-between-blue-and-green.
• [Also: this.]
• Minor vs major is the other isometry of the circle (besides rotation): reflection.
• "Harmonic progression is like zone defence"
• Minute 26: Awesome. Watch how to move around in 2-chord space — seen on a circle and on Tymoczko’s grid

## The Geometry of Musical Chords

by Dmitri Tymoczko

A musical chord can be represented as a point in an orbifold. (An orbifold is a quotient manifold.)

Line segments join 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.

(via Artemy Kolchinsky)

book