Horizmic chords

Horizmic chords are essentially tempered chords tempered by the horizma, 833/832.

Horizmic chords are of pattern 1a in the 2.7.13.17 subgroup 17-odd-limit, meaning that there are 3 triads, 6 tetrads and 2 pentads, for a total of 11 distinct chord structures.

For triads, there are one palindromic chord and a pair of chords in inverse relationship. The palindromic chord is

• 1–8/7–17/13 with steps of 8/7, 8/7, 26/17.

The inversely related pair of chords is

• 1–14/13–8/7 with steps of 14/13, 17/16, 7/4, and its inverse
• 1–17/16–8/7 with steps of 17/16, 14/13, 7/4.

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

• 1–14/13–8/7–16/13 with steps of 14/13, 17/16, 14/13, 13/8;
• 1–17/16–8/7–17/14 with steps of 17/16, 14/13, 17/16, 28/17.

The inversely related pairs of chords are

• 1–8/7–16/13–17/13 with steps of 8/7, 14/13, 17/16, 26/17, and its inverse
• 1–17/16–8/7–17/13 with steps of 17/16, 14/13, 8/7, 26/17;
• 1–8/7–17/14–17/13 with steps of 8/7, 17/16, 14/13, 26/17, and its inverse
• 1–14/13–8/7–17/13 with steps of 14/13, 17/16, 8/7, 26/17.

They can be extended to pentads:

• 1–14/13–8/7–16/13–17/13 with steps of 14/13, 17/16, 14/13, 17/16, 26/17, and its inverse
• 1–17/16–8/7–17/14–17/13 with steps of 17/16, 14/13, 17/16, 14/13, 26/17.

Equal temperaments with horizmic chords include 10, 11, 21, 26, 31, 36, 46, 47, 57, 67, 83, 93, 103, 104 and 150.