Heptacircle chords

Heptacircle chords are essentially tempered chords tempered by the heptacircle comma, 2431/2430.

Heptacircle chords are of pattern 2 in the 2.3.5.11.13.17 subgroup 17-odd-limit, meaning that there are 6 triads, 15 tetrads and 6 pentads, for a total of 27 distinct chord structures.

For triads, there are three pairs of chords in inverse relationship:

• 1-11/9-18/13 with steps of 11/9-17/15-13/9, and its inverse
• 1-13/9-18/11 with steps of 13/9-17/15-11/9;
• 1-15/11-13/9 with steps of 15/11-18/17-18/13, and its inverse
• 1-18/17-13/9 with steps of 18/17-15/11-18/13;
• 1-15/13-11/9 with steps of 15/13-18/17-18/11, and its inverse
• 1-18/17-11/9 with steps of 18/17-15/13-18/11.

For tetrads, there are three palindromic chords and six pairs of chords in inverse relationship. The palindromic chords are

• 1-11/9-18/13-22/13 with steps of 11/9-17/15-11/9-13/11;
• 1-15/11-13/9-17/9 with steps of 15/11-18/17-17/13-18/17;
• 1-15/13-11/9-17/9 with steps of 15/13-18/17-17/11-18/17.

The inversely related pairs of chords are

• 1-11/9-18/13-17/9 with steps of 11/9-17/15-15/11-18/17, and its inverse
• 1-15/11-17/11-17/9 with steps of 15/11-17/15-11/9-18/17;
• 1-15/11-5/3-17/9 with steps of 15/11-11/9-17/15-18/17, and its inverse
• 1-17/15-18/13-17/9 with steps of 17/15-11/9-15/11-18/17;
• 1-18/13-18/11-17/9 with steps of 18/13-13/11-15/13-18/17, and its inverse
• 1-15/13-15/11-17/9 with steps of 15/13-13/11-18/13-18/17;
• 1-15/13-18/13-17/9 with steps of 15/13-6/5-15/11-18/17, and its inverse
• 1-15/11-18/11-17/9 with steps of 15/11-6/5-15/13-18/17;
• 1-17/15-18/11-17/9 with steps of 17/15-13/9-15/13-18/17, and its inverse
• 1-15/13-5/3-17/9 with steps of 15/13-13/9-17/15-18/17;
• 1-13/9-18/11-17/9 with steps of 13/9-17/15-15/13-18/17, and its inverse
• 1-15/13-17/13-17/9 with steps of 15/13-17/15-13/9-18/17.

For pentads, there are three pairs of chords in inverse relationship:

• 1-15/13-15/11-5/3-17/9 with steps of 15/13-13/11-11/9-17/15-18/17, and its inverse
• 1-17/15-18/13-18/11-17/9 with steps of 17/15-11/9-13/11-15/13-18/17;
• 1-15/11-13/9-18/13-17/9 with steps of 15/11-18/17-17/15-15/13-18/17, and its inverse
• 1-15/13-17/13-18/13-17/9 with steps of 15/13-17/15-18/17-15/11-18/17;
• 1-15/13-11/9-18/13-17/9 with steps of 15/13-18/17-17/15-15/11-18/17, and its inverse
• 1-15/11-17/11-18/11-17/9 with steps of 15/11-17/15-18/17-15/13-18/17.

Equal temperaments with heptacircle chords include 24, 34, 53, 58, 72, 87, 94, 111, 159, 183, 217, 270, 277, 311, 494, 581, 605, 677 and 764.