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).

Mintha was a nymph turned into a mint plant by a goddess whom she got in the way of [citation needed ], and major minthmic tempering has the slightly sharp, minty-fresh fifths some people appreciate (Margo Schulter has expressed great fondness for this comma, for example). Equal temperaments with major minthmic chords include 22, 29, 46, 53, 80, 87, 111, 121, 140, 198 and 205.

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