Music Structure Transformations

Rotation in Lag space




The trajectory below is a three dimensional embedding of the first 6 measures of Bach's Courante from Cello Suit no.4. There are 48 notes in the rhythmic sequence, but because multiple states (points) are visited multiple times, only 18 point are visible in the embedded trajectory.

There are a total of 8*8*8=512 45 degree rotations, but only 26 of the projections are different. Below are the 26 variations of the Bach rhythmic sequence resulting from 45 degrees rotations of the embedded trajectory.


3D viewer from www.javaview.de
 The original rhythmic sequence found in Bach's Courante.
rotation z: 0, y: 2π(1/8), x: 0
rotation z: 0, y: 2π(2/8), x: 0
rotation z: 0, y: 2π(3/8), x: 0
rotation z: 0, y: 2π(4/8), x: 0
rotation z: 0, y: 2π(5/8), x: 0
rotation z: 0, y: 2π(6/8), x: 0
rotation z: 0, y: 2π(7/8), x: 0
rotation z: 2π(1/8), y: 0, x: 0
rotation z: 2π(1/8), y: 2π(1/8), x: 0
rotation z: 2π(1/8), y: 2π(3/8), x: 0
rotation z: 2π(1/8), y: 2π(4/8), x: 0
rotation z: 2π(1/8), y: 2π(5/8), x: 0
rotation z: 2π(1/8), y: 2π(7/8), x: 0
rotation z: 2π(2/8), y: 0, x: 0
rotation z: 2π(2/8), y: 2π(1/8), x: 0
rotation z: 2π(2/8), y: 2π(3/8), x: 0
rotation z: 2π(2/8), y: 2π(4/8), x: 0
rotation z: 2π(2/8), y: 2π(5/8), x: 0
rotation z: 2π(2/8), y: 2π(7/8), x: 0
rotation z: 2π(3/8), y: 0, x: 0
rotation z: 2π(3/8), y: 2π(1/8), x: 0
rotation z: 2π(3/8), y: 2π(3/8), x: 0
rotation z: 2π(3/8), y: 2π(4/8), x: 0
rotation z: 2π(3/8), y: 2π(5/8), x: 0
rotation z: 2π(3/8), y: 2π(7/8), x: 0