Twosquare chords

Twosquare chords are essentially tempered chords tempered by the twosquare comma, 1089/1088.

Twosquare chords are of pattern 1b in the 2.3.11.17 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

• 1–11/8–17/9 with steps of 11/8, 11/8, 18/17.

The inversely related pair of chords is

• 1–11/8–17/11 with steps of 11/8, 9/8, 22/17, and its inverse
• 1–9/8–17/11 with steps of 9/8, 11/8, 22/17.

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

• 1–9/8–11/8–17/11 with steps of 9/8, 11/9, 9/8, 22/17;
• 1–9/8–16/11–17/11 with steps of 9/8, 22/17, 17/16, 22/17.

The inversely related pairs of chords are

• 1–11/8–17/11–17/9 with steps of 11/8, 9/8, 11/9, 18/17, and its inverse
• 1–11/9–11/8–17/9 with steps of 11/9, 9/8, 11/8, 18/17;
• 1–11/8–16/9–17/9 with steps of 11/8, 22/17, 17/16, 18/17, and its inverse
• 1–17/16–11/8–17/9 with steps of 17/16, 22/17, 11/8, 18/17.

Then there is an inversely related pair of pentads:

• 1–11/9–11/8–16/9–17/9 with steps of 11/9, 9/8, 22/17, 17/16, 18/17, and its inverse
• 1–17/16–11/8–17/11–17/9 with steps of 17/16, 22/17, 9/8, 11/9, 18/17.

Equal temperaments with twosquare chords include 22, 24, 41, 46, 65, 70, 89, 94, 111, 135, 159, 183, 207 and 231.