# 15-limit tonality diamond

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 *R*² = *S*² = *T*² = *S ² = I, (*

**SS**)³ = I,

*RS*=

*SR*,

*RS*

**=****S**

*R*, and

*T*commutes with the three other generators. Thus the symmetry group is isomorphic to

*S*₃ ×

*C*₂².

### 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 *S*² 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.