15-limit tonality diamond

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The 15-limit tonality diamond has the following notes:

1/1 9/8 5/4 11/8 3/2 13/8 7/4 15/8
16/9 1/1 10/9 11/9 4/3 13/9 14/9 5/3
8/5 9/5 1/1 11/10 6/5 13/10 7/5 3/2
16/11 18/11 20/11 1/1 12/11 13/11 14/11 15/11
4/3 3/2 5/3 11/6 1/1 13/12 7/6 5/4
16/13 18/13 20/13 22/13 24/13 1/1 14/13 15/13
8/7 9/7 10/7 11/7 12/7 13/7 1/1 15/14
16/15 6/5 4/3 22/15 8/5 26/15 28/15 1/1

Symmetry group

The symmetry group of the 15-limit tonality diamond has 24 elements. The following are its generators:

  • Transformation R: 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 5:3, 14:9, 11:9, 13:9
  • Transformation S: 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 11:8, 7:4, 13:8
  • Transformation S': 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 7:4, 13:8, 11:8
  • Transformation T: 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 8:5, 8:7, 16:11, 16:13

These generators have the relations R² = S² = T² = S² = I, (SS)³ = I, RS = SR, RS = SR, and T commutes with the three other generators. Thus the symmetry group is isomorphic to S₃ × C₂².

Orbits and Invariant Subsets

The Hendrix diamond is invariant under action by R, S' and T, and the images of the action of S and S² on the Hendrix diamond are the 11-Hendrix diamond and 13-Hendrix diamond respectively.

Two other interesting invariant subsets are the 5-limit tonality diamond and the tonality diamond constructed from the harmonics 1, 3, 5, 9 and 15.