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.