Major minthmic chords

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Major minthmic chords are essentially tempered dyadic chords in the 13-limit rank-5 temperament, and in addition the rank-3 2.3.11.13 subgroup temperament, tempered by the major minthma, 352/351.

There are 8 triads, 27 tetrads, 28 pentads, 12 hexads and 2 heptads as 2.3.11.13 subgroup 13-odd-limit essentially tempered chords.

For triads, there are four pairs of chords in inverse relationship:

  • 1–16/13–3/2 with steps of 16/13, 11/9, 4/3, and its inverse
  • 1–11/9–3/2 with steps of 11/9, 16/13, 4/3;
  • 1–13/11–4/3 with steps of 13/11, 9/8, 3/2, and its inverse
  • 1–9/8–4/3 with steps of 9/8, 13/11, 3/2;
  • 1–9/8–16/13 with steps of 9/8, 12/11, 13/8, and its inverse
  • 1–12/11–16/13 with steps of 12/11, 9/8, 13/8;
  • 1–9/8–11/9 with steps of 9/8, 13/12, 18/11, and its inverse
  • 1–13/12–11/9 with steps of 13/12, 9/8, 18/11.

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

  • 1–11/9–3/2–13/8 with steps of 11/9, 16/13, 13/12, 16/13;
  • 1–11/9–4/3–13/8 with steps of 11/9, 12/11, 11/9, 16/13;
  • 1–9/8–4/3–3/2 with steps of 9/8, 13/11, 9/8, 4/3;
  • 1–9/8–16/13–18/13 with steps of 9/8, 12/11, 9/8, 13/9;
  • 1–9/8–11/9–11/8 with steps of 9/8, 13/12, 9/8, 16/11.

The inversely related pairs of chords are

  • 1–11/9–11/8–13/8 with steps of 11/9, 9/8, 13/11, 16/13, and its inverse
  • 1–13/11–4/3–13/8 with steps of 13/11, 9/8, 11/9, 16/13;
  • 1–11/9–13/9–13/8 with steps of 11/9, 13/11, 9/8, 16/13, and its inverse
  • 1–9/8–4/3–13/8 with steps of 9/8, 13/11, 11/9, 16/13;
  • 1–16/13–18/13–3/2 with steps of 16/13, 9/8, 13/12, 4/3, and its inverse
  • 1–13/12–11/9–3/2 with steps of 13/12, 9/8, 16/13, 4/3;
  • 1–16/13–4/3–3/2 with steps of 16/13, 13/12, 9/8, 4/3, and its inverse
  • 1–9/8–11/9–3/2 with steps of 9/8, 13/12, 16/13, 4/3;
  • 1–11/9–11/8–3/2 with steps of 11/9, 9/8, 12/11, 4/3, and its inverse
  • 1–12/11–16/13–3/2 with steps of 12/11, 9/8, 11/9, 4/3;
  • 1–11/9–4/3–3/2 with steps of 11/9, 12/11, 9/8, 4/3, and its inverse
  • 1–9/8–16/13–3/2 with steps of 9/8, 12/11, 11/9, 4/3;
  • 1–13/11–4/3–16/11 with steps of 13/11, 9/8, 12/11, 11/8, and its inverse
  • 1–12/11–16/13–16/11 with steps of 12/11, 9/8, 13/11, 11/8;
  • 1–13/11–4/3–13/9 with steps of 13/11, 9/8, 13/12, 18/13, and its inverse
  • 1–13/12–11/9–13/9 with steps of 13/12, 9/8, 13/11, 18/13;
  • 1–9/8–16/13–4/3 with steps of 9/8, 12/11, 13/12, 3/2, and its inverse
  • 1–13/12–13/11–4/3 with steps of 13/12, 12/11, 9/8, 3/2;
  • 1–9/8–11/9–4/3 with steps of 9/8, 13/12, 12/11, 3/2, and its inverse
  • 1–12/11–13/11–4/3 with steps of 12/11, 13/12, 9/8, 3/2;
  • 1–12/11–16/13–4/3 with steps of 12/11, 9/8, 13/12, 3/2, and its inverse
  • 1–13/12–11/9–4/3 with steps of 13/12, 9/8, 12/11, 3/2.

For pentads, there are fourteen pairs of chords in inverse relationship:

  • 1–13/11–4/3–16/11–18/11 with steps of 13/11, 9/8, 12/11, 9/8, 11/9, and its inverse
  • 1–9/8–16/13–18/13–18/11 with steps of 9/8, 12/11, 9/8, 13/11, 11/9;
  • 1–9/8–4/3–3/2–18/11 with steps of 9/8, 13/11, 9/8, 12/11, 11/9, and its inverse
  • 1–12/11–16/13–16/11–18/11 with steps of 12/11, 9/8, 13/11, 9/8, 11/9;
  • 1–9/8–16/13–3/2–18/11 with steps of 9/8, 12/11, 11/9, 12/11, 11/9, and its inverse
  • 1–12/11–4/3–16/11–18/11 with steps of 12/11, 11/9, 12/11, 9/8, 11/9;
  • 1–11/9–11/8–3/2–13/8 with steps of 11/9, 9/8, 12/11, 13/12, 16/13, and its inverse
  • 1–13/12–13/11–4/3–13/8 with steps of 13/12, 12/11, 9/8, 11/9, 16/13;
  • 1–11/9–4/3–3/2–13/8 with steps of 11/9, 12/11, 9/8, 13/12, 16/13 and, its inverse
  • 1–13/12–11/9–4/3–13/8 with steps of 13/12, 9/8, 12/11, 11/9, 16/13;
  • 1–11/9–4/3–13/9–13/8 with steps of 11/9, 12/11, 13/12, 9/8, 16/13, and its inverse
  • 1–9/8–11/9–4/3–13/8 with steps of 9/8, 13/12, 12/11, 11/9, 16/13;
  • 1–13/11–4/3–13/9–13/8 with steps of 13/11, 9/8, 13/12, 9/8, 16/13, and its inverse
  • 1–9/8–11/9–11/8–13/8 with steps of 9/8, 13/12, 9/8, 13/11, 16/13;
  • 1–9/8–4/3–3/2–13/8 with steps of 9/8, 13/11, 9/8, 13/12, 16/13, and its inverse
  • 1–13/12–11/9–13/9–13/8 with steps of 13/12, 9/8, 13/11, 9/8, 16/13;
  • 1–9/8–11/9–3/2–13/8 with steps of 9/8, 13/12, 16/13, 13/12, 16/13, and its inverse
  • 1–13/12–4/3–13/9–13/8 with steps of 13/12, 16/13, 13/12, 9/8, 16/13;
  • 1–9/8–16/13–4/3–3/2 with steps of 9/8, 12/11, 13/12, 9/8, 4/3, and its inverse
  • 1–9/8–11/9–4/3–3/2 with steps of 9/8, 13/12, 12/11, 9/8, 4/3;
  • 1–9/8–16/13–18/13–3/2 with steps of 9/8, 12/11, 9/8, 13/12, 4/3, and its inverse
  • 1–13/12–11/9–4/3–3/2 with steps of 13/12, 9/8, 12/11, 9/8, 4/3;
  • 1–9/8–11/9–11/8–3/2 with steps of 9/8, 13/12, 9/8, 12/11, 4/3, and its inverse
  • 1–12/11–16/13–4/3–3/2 with steps of 12/11, 9/8, 13/12, 9/8, 4/3;
  • 1–12/11–16/13–4/3–16/11 with steps of 12/11, 9/8, 13/12, 12/11, 11/8, and its inverse
  • 1–12/11–13/11–4/3–16/11 with steps of 12/11, 13/12, 9/8, 12/11, 11/8;
  • 1–13/12–11/9–4/3–13/9 with steps of 13/12, 9/8, 12/11, 13/12, 18/13, and its inverse
  • 1–13/12–13/11–4/3–13/9 with steps of 13/12, 12/11, 9/8, 13/12, 18/13.

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

  • 1–12/11–16/13–4/3–3/2–18/11 with steps of 12/11, 9/8, 13/12, 9/8, 12/11, 11/9;
  • 1–13/12–11/9–4/3–3/2–13/8 with steps of 13/12, 9/8, 12/11, 9/8, 13/12, 16/13.

The inversely related pairs of chords are

  • 1–9/8–16/13–4/3–3/2–22/13 with steps of 9/8, 12/11, 9/8, 13/12, 9/8, 13/11, and its inverse
  • 1–9/8–11/9–11/8–3/2–22/13 with steps of 9/8, 13/12, 9/8, 12/11, 9/8, 13/11;
  • 1–9/8–16/13–18/13–3/2–18/11 with steps of 9/8, 12/11, 9/8, 13/12, 12/11, 11/9, and its inverse
  • 1–12/11–13/11–4/3–16/11–18/11 with steps of 12/11, 13/12, 9/8, 12/11, 9/8, 11/9;
  • 1–9/8–16/13–4/3–3/2–18/11 with steps of 9/8, 12/11, 13/12, 9/8, 12/11, 11/9, and its inverse
  • 1–12/11–16/13–4/3–16/11–18/11 with steps of 12/11, 9/8, 13/12, 12/11, 9/8, 11/9;
  • 1–9/8–11/9–11/8–3/2–13/8 with steps of 9/8, 13/12, 9/8, 12/11, 13/12, 16/13, and its inverse
  • 1–13/12–13/11–4/3–13/9–13/8 with steps of 13/12, 12/11, 9/8, 13/12, 9/8, 16/13;
  • 1–9/8–11/9–4/3–3/2–13/8 with steps of 9/8, 13/12, 12/11, 9/8, 13/12, 16/13, and its inverse
  • 1–13/12–11/9–4/3–13/9–13/8 with steps of 13/12, 9/8, 12/11, 13/12, 9/8, 16/13.

Finally, there is a pair of heptads in inverse relationship:

  • 1–9/8–16/13–4/3–3/2–18/11–24/13 with steps of 9/8, 12/11, 13/12, 9/8, 12/11, 9/8, 13/12 (→ minthmic7a), and its inverse
  • 1–9/8–11/9–4/3–3/2–13/8–11/6 with steps of 9/8, 13/12, 12/11, 9/8, 13/12, 9/8, 12/11 (→ minthmic7b).

Equal temperaments with major minthmic chords include 22, 29, 46, 53, 80, 87, 111, 121, 140, 198 and 205.

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