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 three pentacircle triads include a palindrome since it identifies 14/11 by a stack of two 9/8's:

• 1–9/8–14/11 with steps 9/8, 9/8, 11/7.

And an inversely related pair:

• 1–9/8–16/11 with steps 9/8, 9/7, 11/8;
• 1–9/8–14/9 with steps 9/8, 11/8, 9/7;

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