Werckismic chords

A werckismic chord is an essentially tempered dyadic chord tempered by the werckisma, 441/440.

Werckismic chords are of pattern 2 in the 11-odd-limit, meaning that there are 6 triads, 15 tetrads and 6 pentads, for a total of 27 distinct chord structures.

The werckismic triads consist of three pairs of inversely related chords:

• 1-11/9-7/5 with steps 11/9-8/7-10/7, and its inverse
• 1-11/9-7/4 with steps 11/9-10/7-8/7;
• 1-11/7-7/4 with steps 11/7-10/9-8/7, and its inverse
• 1-11/7-9/5 with steps 11/7-8/7-10/9;
• 1-7/5-11/7 with steps 7/5-9/8-14/11, and its inverse
• 1-14/11-10/7 with steps 14/11-9/8-7/5.

The werckismic tetrads consist of three palindromic (self-inversive) chords and five pairs of chords in an inverse relationship. The palindromic chords are

• 1-5/4-10/7-7/4 with steps 5/4-8/7-11/9-8/7;
• 1-9/8-10/7-11/7 with steps 9/8-14/11-11/10-14/11;
• 1-14/11-7/5-16/9 with steps 14/11-11/10-14/11-9/8.

The pairs of chords are:

• 1-11/9-11/7-7/4 with steps 11/9-9/7-10/9-8/7, and its inverse
• 1-9/7-11/7-9/5 with steps 9/7-11/9-8/7-10/9;
• 1-10/7-11/7-7/4 with steps 10/7-11/10-10/9-8/7, and its inverse
• 1-8/7-14/11-7/5 with steps 8/7-10/9-11/10-10/7;
• 1-9/8-10/7-7/4 with steps 9/8-14/11-11/9-8/7, and its inverse
• 1-11/9-14/9-7/4 with steps 11/9-14/11-9/8-8/7;
• 1-11/9-11/8-7/4 with steps 11/9-9/8-14/11-8/7, and its inverse
• 1-14/11-10/7-7/4 with steps 14/11-9/8-11/9-8/7;
• 1-9/8-5/4-10/7 with steps 9/8-10/9-8/7-7/5, and its inverse
• 1-10/9-5/4-7/4 with steps 10/9-9/8-7/5-8/7;
• 1-9/8-9/7-10/7 with steps 9/8-8/7-10/9-7/5, and its inverse
• 1-9/8-11/7-7/4 with steps 9/8-7/5-10/9-8/7.

The werckismic pentads consist of three pairs of chords in an inverse relationship:

• 1-9/8-5/4-10/7-7/4 with steps 9/8-10/9-8/7-11/9-8/7, and its inverse
• 1-11/9-7/5-14/9-7/4 with steps 11/9-8/7-10/9-9/8-8/7;
• 1-11/9-11/8-11/7-7/4 with steps 11/9-9/8-8/7-10/9-8/7, and its inverse
• 1-9/8-11/8-11/7-7/4 with steps 9/8-11/9-8/7-10/9-8/7;
• 1-9/8-9/7-10/7-11/7 with steps 9/8-8/7-10/9-11/10-14/11, and its inverse
• 1-9/8-10/7-11/7-7/4 with steps 9/8-14/11-11/10-10/9-8/7.

Equal temperaments with werckismic chords include 31, 41, 43, 46, 58, 72, 77, 84, 103, 118, 130, 159, 171, 190, 248, 289, and 320, with 320edo giving the optimal patent val.