Mynucumic chords

Mynucumic chords are essentially tempered chords tempered by the mynucuma, 196/195.

There are 10 triads, 33 tetrads, 26 pentads and 6 hexads as 2.3.5.7.13 subgroup 13-odd-limit essentially tempered chords.

For triads, there are five pairs of chords in inverse relationship.

The mynucumic triads are temperings of

• 1-16/13-10/7 with steps of 16/13-7/6-7/5, and its inverse
• 1-7/6-10/7 with steps of 7/6-16/13-7/5.

The rest four inversely related pairs of triads contain semitones, such as 13/12 or 14/13:

• 1-7/5-3/2 with steps of 7/5-14/13-4/3, and its inverse
• 1-14/13-3/2 with steps of 14/13-7/5-4/3;
• 1-7/6-5/4 with steps of 7/6-14/13-8/5, and its inverse
• 1-14/13-5/4 with steps of 14/13-7/6-8/5;
• 1-6/5-9/7 with steps of 6/5-14/13-14/9, and its inverse
• 1-14/13-9/7 with steps of 14/13-6/5-14/9;
• 1-9/7-7/5 with steps of 9/7-13/12-10/7, and its inverse
• 1-13/12-7/5 with steps of 13/12-9/7-10/7.

For tetrads, there are five palindromic chords and fourteen pairs of chords in inverse relationship.

The mynucumic tetrad is palindromic, a tempering of

• 1-7/6-7/5-13/8 chord with steps of 7/6-6/5-7/6-16/13.

The rest four palindromic tetrads contain semitones:

• 1-14/13-7/5-3/2 chord with steps of 14/13-13/10-14/13-4/3;
• 1-14/13-7/6-5/4 chord with steps of 14/13-13/12-14/13-8/5;
• 1-6/5-9/7-13/7 chord with steps of 6/5-14/13-13/9-14/13;
• 1-9/7-7/5-9/5 chord with steps of 9/7-13/12-9/7-10/9.

The inversely related pairs of tetrads also contain semitones:

• 1-9/7-7/5-3/2 with steps of 9/7-13/12-14/13-4/3, and its inverse
• 1-14/13-7/6-3/2 with steps of 14/13-13/12-9/7-4/3;
• 1-6/5-9/7-3/2 with steps of 6/5-14/13-7/6-4/3, and its inverse
• 1-7/6-5/4-3/2 with steps of 7/6-14/13-6/5-4/3;
• 1-6/5-7/5-3/2 with steps of 6/5-7/6-14/13-4/3, and its inverse
• 1-14/13-5/4-3/2 with steps of 14/13-7/6-6/5-4/3;
• 1-7/6-7/5-3/2 with steps of 7/6-6/5-14/13-4/3, and its inverse
• 1-14/13-9/7-3/2 with steps of 14/13-6/5-7/6-4/3;
• 1-13/12-7/5-3/2 with steps of 13/12-9/7-14/13-4/3, and its inverse
• 1-14/13-18/13-3/2 with steps of 14/13-9/7-13/12-4/3;
• 1-7/5-3/2-9/5 with steps of 7/5-14/13-6/5-10/9, and its inverse
• 1-14/13-3/2-5/3 with steps of 14/13-7/5-10/9-6/5;
• 1-7/5-3/2-7/4 with steps of 7/5-14/13-7/6-8/7, and its inverse
• 1-14/13-3/2-12/7 with steps of 14/13-7/5-8/7-7/6;
• 1-7/5-3/2-12/7 with steps of 7/5-14/13-8/7-7/6, and its inverse
• 1-14/13-3/2-7/4 with steps of 14/13-7/5-7/6-8/7;
• 1-9/7-3/2-24/13 with steps of 9/7-7/6-16/13-13/12, and its inverse
• 1-7/6-3/2-13/8 with steps of 7/6-9/7-13/12-16/13;
• 1-7/5-3/2-13/8 with steps of 7/5-14/13-13/12-16/13, and its inverse
• 1-14/13-3/2-24/13 with steps of 14/13-7/5-16/13-13/12;
• 1-7/6-5/4-13/8 with steps of 7/6-14/13-13/10-16/13, and its inverse
• 1-14/13-5/4-20/13 with steps of 14/13-7/6-16/13-13/10;
• 1-7/6-5/4-10/7 with steps of 7/6-14/13-8/7-7/5, and its inverse
• 1-14/13-5/4-7/4 with steps of 14/13-7/6-7/5-8/7;
• 1-6/5-13/10-13/7 with steps of 6/5-13/12-10/7-14/13, and its inverse
• 1-6/5-9/7-24/13 with steps of 6/5-14/13-10/7-13/12;
• 1-6/5-12/7-13/7 with steps of 6/5-10/7-13/12-14/13, and its inverse
• 1-6/5-9/7-7/5 with steps of 6/5-14/13-13/12-10/7.

For pentads, there are thirteen pairs of chords in inverse relationship, all of them involve semitones and the perfect fifth:

• 1-6/5-9/7-7/5-3/2 with steps of 6/5-14/13-13/12-14/13-4/3, and its inverse
• 1-14/13-7/6-5/4-3/2 with steps of 14/13-13/12-14/13-6/5-4/3;
• 1-13/12-7/6-7/5-3/2 with steps of 13/12-14/13-6/5-14/13-4/3, and its inverse
• 1-14/13-9/7-18/13-3/2 with steps of 14/13-6/5-14/13-13/12-4/3;
• 1-14/13-9/7-7/5-3/2 with steps of 14/13-6/5-13/12-14/13-4/3, and its inverse
• 1-14/13-7/6-7/5-3/2 with steps of 14/13-13/12-6/5-14/13-4/3;
• 1-9/7-7/5-3/2-9/5 with steps of 9/7-13/12-14/13-6/5-10/9, and its inverse
• 1-14/13-7/6-3/2-5/3 with steps of 14/13-13/12-9/7-10/9-6/5;
• 1-6/5-9/7-3/2-9/5 with steps of 6/5-14/13-7/6-6/5-10/9, and its inverse
• 1-7/6-5/4-3/2-5/3 with steps of 7/6-14/13-6/5-10/9-6/5;
• 1-6/5-9/7-3/2-12/7 with steps of 6/5-14/13-7/6-8/7-7/6, and its inverse
• 1-7/6-5/4-3/2-7/4 with steps of 7/6-14/13-6/5-7/6-8/7;
• 1-6/5-7/5-3/2-12/7 with steps of 6/5-7/6-14/13-8/7-7/6, and its inverse
• 1-14/13-5/4-3/2-7/4 with steps of 14/13-7/6-6/5-7/6-8/7;
• 1-14/13-7/5-3/2-7/4 with steps of 14/13-13/10-14/13-7/6-8/7, and its inverse
• 1-14/13-7/5-3/2-12/7 with steps of 14/13-13/10-14/13-8/7-7/6;
• 1-9/7-18/13-3/2-24/13 with steps of 9/7-14/13-13/12-16/13-13/12, and its inverse
• 1-13/12-7/6-3/2-13/12 with steps of 13/12-14/13-9/7-13/12-16/13;
• 1-6/5-9/7-3/2-24/13 with steps of 6/5-14/13-7/6-16/13-13/12, and its inverse
• 1-7/6-5/4-3/2-13/8 with steps of 7/6-14/13-6/5-13/12-16/13;
• 1-7/6-7/5-3/2-13/8 with steps of 7/6-6/5-14/13-13/12-16/13, and its inverse
• 1-14/13-9/7-3/2-24/13 with steps of 14/13-6/5-7/6-16/13-13/12;
• 1-9/7-3/2-12/7-24/13 with steps of 9/7-7/6-8/7-14/13-13/12, and its inverse
• 1-7/6-3/2-13/8-7/4 with steps of 7/6-9/7-13/12-14/13-8/7;
• 1-7/5-3/2-13/8-7/4 with steps of 7/5-14/13-13/12-14/13-8/7, and its inverse
• 1-14/13-3/2-12/7-24/13 with steps of 14/13-7/5-8/7-14/13-13/12.

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

• 1-14/13-7/6-5/4-3/2-5/3 chord with steps of 14/13-13/12-14/13-6/5-10/9-6/5;
• 1-13/12-7/6-7/5-3/2-13/8 chord with steps of 13/12-14/13-6/5-14/13-13/12-16/13.

The inversely related pairs of chords are

• 1-6/5-9/7-7/5-3/2-12/7 with steps of 6/5-14/13-13/12-14/13-8/7-7/6, and its inverse
• 1-14/13-7/6-5/4-3/2-7/4 with steps of 14/13-13/12-14/13-6/5-7/6-8/7;
• 1-14/13-9/7-7/5-3/2-12/7 with steps of 14/13-6/5-13/12-14/13-8/7-7/6, and its inverse
• 1-14/13-7/6-7/5-3/2-7/4 with steps of 14/13-13/12-6/5-14/13-7/6-8/7.

Equal temperaments with mynucumic chords include 19, 27, 31, 41, 46, 58, 68, 77, 87, 118, 145 and 232.