Pentacircle chords

A pentacircle chord is an essentially tempered dyadic chord in the 2.9.7.11 subgroup in the 11-odd-limit, tempering out the pentacircle comma, 896/891.

Pentacircle chords are of pattern 1b in the 2.9.7.11 subgroup 11-odd-limit, meaning that there are 3 triads, 6 tetrads and 2 pentads, for a total of 11 distinct chord structures.

The pentacircle triads are

• 1-9/7-16/9 with steps 9/7-11/8-9/8;
• 1-9/7-16/11 with steps 9/7-9/8-11/8;
• 1-11/7-16/9 with steps 11/7-9/8-9/8.

The tetrads include the palindromic

• 1-9/8-14/9-7/4 with steps 9/8-11/8-9/8-8/7;
• 1-9/8-11/8-14/9 with steps 9/8-11/9-9/8-9/7.

And the inversely related pairs

• 1-11/8-11/7-16/9 with steps 11/8-8/7-9/8-9/8, and its inverse
• 1-11/8-14/9-7/4 with steps 11/8-9/8-9/8-8/7;
• 1-11/9-11/7-16/9 with steps 11/9-9/7-9/8-9/8, and its inverse
• 1-11/9-11/8-14/9 with steps 11/9-9/8-9/8-9/7.

Finally, there are two pentacircle pentads, inversely related:

• 1-11/9-11/8-11/7-16/9 with steps 11/9-9/8-8/7-9/8-9/8, and its inverse
• 1-11/9-11/8-14/9-16/9 with steps 11/9-9/8-9/8-8/7-9/8.

Equal divisions of the octave with pentacircle chords include 22, 27, 41, 46, 58, 68, 80, 87, 121, 145, 167, and 208, with 208edo giving the optimal patent val.