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.