User:AthiTrydhen/15-limit tonality diamond

This page may be difficult to understand to those unfamiliar with the mathematical concepts involved. A more accessible version will be worked on; in the meantime, feel free to ask questions in the Xenharmonic Alliance Discord server or Facebook group.

Todo: Incorporate Hendrix diamond ideas

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.