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.