Music Structure Transformations

Pitch Rotations in Lag Space

The following are three-dimensional rotations of the pitch sequence from the popular tune "Frere Jaques" embedded in lag-space. 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 tune resulting from 45 degrees rotations of the embedded trajectory. Notice that rotations z: 0, y: 2π(4/8), x: 0, is the classical inversion of the sequence.




original sequence


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