Kestrel chords

Kestrel chords are essentially tempered chords tempered by the kestrel comma, 1188/1183.

Kestrel chords are of pattern 2 in the 2.3.7.11.13 subgroup 13-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:

• 1–7/6–18/13 with steps of 7/6, 13/11, 13/9, and its inverse
• 1–13/11–18/13 with steps of 13/11, 7/6, 13/9;
• 1–14/11–18/13 with steps of 14/11, 13/12, 13/9, and its inverse
• 1–13/12–18/13 with steps of 13/12, 14/11, 13/9;
• 1–13/11–9/7 with steps of 13/11, 13/12, 14/9, and its inverse
• 1–13/12–9/7 with steps of 13/12, 13/11, 14/9.

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

• 1–13/11–18/13–18/11 with steps of 13/11, 7/6, 13/11, 11/9;
• 1–13/12–18/13–3/2 with steps of 13/12, 14/11, 13/12, 4/3;
• 1–13/12–13/11–9/7 with steps of 13/12, 12/11, 13/12, 14/9.

The inversely related pairs of chords are

• 1–11/9–13/9–11/7 with steps of 11/9, 13/11, 13/12, 14/11, and its inverse
• 1–13/12–9/7–11/7 with steps of 13/12, 13/11, 11/9, 14/11;
• 1–7/6–18/13–3/2 with steps of 7/6, 13/11, 13/12, 4/3, and its inverse
• 1–13/12–9/7–3/2 with steps of 13/12, 13/11, 7/6, 4/3;
• 1–13/11–9/7–18/13 with steps of 13/11, 13/12, 14/13, 13/9, and its inverse
• 1–14/13–7/6–18/13 with steps of 14/13, 13/12, 13/11, 13/9;
• 1–13/11–14/11–18/13 with steps of 13/11, 14/13, 13/12, 13/9, and its inverse
• 1–13/12–7/6–18/13 with steps of 13/12, 14/13, 13/11, 13/9;
• 1–7/6–14/11–18/13 with steps of 7/6, 12/11, 13/12, 13/9, and its inverse
• 1–13/12–13/11–18/13 with steps of 13/12, 12/11, 7/6, 13/9;
• 1–13/12–9/7–18/13 with steps of 13/12, 13/11, 14/13, 13/9, and its inverse
• 1–14/13–14/11–18/13 with steps of 14/13, 13/11, 13/12, 13/9.

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

• 1–13/11–9/7–18/13–18/11 with steps of 13/11, 13/12, 14/13, 13/11, 11/9, and its inverse
• 1–13/11–14/11–18/13–18/11 with steps of 13/11, 14/13, 13/12, 13/11, 11/9;
• 1–13/12–7/6–18/13–3/2 with steps of 13/12, 14/13, 13/11, 13/12, 4/3, and its inverse
• 1–13/12–9/7–18/13–3/2 with steps of 13/12, 13/11, 14/13, 13/12, 4/3;
• 1–13/12–13/11–9/7–18/13 with steps of 13/12, 12/11, 13/12, 14/13, 13/9, and its inverse
• 1–14/13–7/6–14/11–18/13 with steps of 14/13, 13/12, 12/11, 13/12, 13/9.

Equal temperaments with kestrel chords include 17, 24, 26, 41, 53, 58, 77, 94, 103, 111, 161, 205, 214 and 255, with 255edo giving the optimal patent val.