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.

The pentacircle triads are three in number:

• 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.

There are six pentacircle tetrads, including 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.

The count of chords is triads: 3, tetrads: 6, pentads: 2, for a total of 11.

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.