Minor minthmic chords

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Minor minthmic chords are essentially tempered chords tempered by the minor minthma, 364/363.

There are 10 triads, 33 tetrads, 26 pentads and 6 hexads as 2.3.7.11.13 subgroup 13-odd-limit essentially tempered chords.

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

The supermajor gentle triad (or gentle major triad) is a tempering of

  • 1-14/11-3/2 with steps of 14/11-13/11-4/3;

and its inversion the subminor gentle triad (or gentle minor triad) is a tempering of

  • 1-13/11-3/2 with steps of 13/11-14/11-4/3.

The gothic gentle triads are temperings of

  • 1-13/11-11/8 with steps of 13/11-7/6-16/11,

and its inversion,

  • 1-7/6-11/8 with steps of 7/6-13/11-16/11.

The names refer to Margo Schulter's Neo-gothic theory of harmony, which features a gentle region with a slightly sharpened fifth in which gentle triads and neogothic triads flourish.

The rest three inversely related pairs of triads contain semitones, such as 12/11 or 13/12:

  • 1-14/11-18/13 with steps of 14/11-12/11-13/9, and its inverse
  • 1-12/11-18/13 with steps of 12/11-14/11-13/9;
  • 1-14/11-11/8 with steps of 14/11-13/12-16/11, and its inverse
  • 1-13/12-11/8 with steps of 13/12-14/11-16/11;
  • 1-13/11-9/7 with steps of 13/11-12/11-14/9, and its inverse
  • 1-12/11-9/7 with steps of 12/11-13/11-14/9.

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

The gentle major tetrad is a tempering of

  • 1-14/11-3/2-7/4 with steps of 14/11-13/11-7/6-8/7;

and its inversion the gentle minor tetrad is a tempering of

  • 1-13/11-3/2-12/7 with steps of 13/11-14/11-8/7-7/6.

The gothic gentle tetrad is palindromic, a tempering of

  • 1-13/11-11/8-13/8 with steps of 13/11-7/6-13/11-16/13.

The rest four palindromic tetrads contain semitones, such as 12/11, 13/12 or 14/13:

  • 1-13/11-14/11-3/2 with steps of 13/11-14/13-13/11-4/3;
  • 1-14/11-11/8-7/4 with steps of 14/11-13/12-14/11-8/7;
  • 1-12/11-14/11-18/13 with steps of 12/11-7/6-12/11-13/9;
  • 1-12/11-13/11-9/7 with steps of 12/11-13/12-12/11-14/9;

as well as the rest thirteen inversely related pairs of tetrads:

  • 1-14/11-3/2-24/13 with steps of 14/11-13/11-16/13-13/12, and its inverse
  • 1-13/11-3/2-13/8 with steps of 13/11-14/11-13/12-16/13;
  • 1-14/11-3/2-11/6 with steps of 14/11-13/11-11/9-12/11, and its inverse
  • 1-13/11-3/2-18/11 with steps of 13/11-14/11-12/11-11/9;
  • 1-14/11-3/2-18/11 with steps of 14/11-13/11-12/11-11/9, and its inverse
  • 1-13/11-3/2-11/6 with steps of 13/11-14/11-11/9-12/11;
  • 1-7/6-14/11-3/2 with steps of 7/6-12/11-13/11-4/3, and its inverse
  • 1-13/11-9/7-3/2 with steps of 13/11-12/11-7/6-4/3;
  • 1-7/6-11/8-3/2 with steps of 7/6-13/11-12/11-4/3, and its inverse
  • 1-12/11-9/7-3/2 with steps of 12/11-13/11-7/6-4/3;
  • 1-14/11-11/8-3/2 with steps of 14/11-13/12-12/11-4/3, and its inverse
  • 1-12/11-13/11-3/2 with steps of 12/11-13/12-14/11-4/3;
  • 1-14/11-18/13-3/2 with steps of 14/11-12/11-13/12-4/3, and its inverse
  • 1-13/12-13/11-3/2 with steps of 13/12-12/11-14/11-4/3;
  • 1-13/11-11/8-3/2 with steps of 13/11-7/6-12/11-4/3, and its inverse
  • 1-12/11-14/11-3/2 with steps of 12/11-7/6-13/11-4/3;
  • 1-12/11-18/13-3/2 with steps of 12/11-14/11-13/12-4/3, and its inverse
  • 1-13/12-11/8-3/2 with steps of 13/12-14/11-12/11-4/3;
  • 1-11/9-13/9-11/7 with steps of 11/9-13/11-12/11-14/11, and its inverse
  • 1-12/11-9/7-11/7 with steps of 12/11-13/11-11/9-14/11;
  • 1-12/11-9/7-18/13 with steps of 12/11-13/11-14/13-13/9, and its inverse
  • 1-14/13-14/11-18/13 with steps of 14/13-13/11-12/11-13/9;
  • 1-13/11-14/11-11/8 with steps of 13/11-14/13-13/12-16/11, and its inverse
  • 1-13/12-7/6-11/8 with steps of 13/12-14/13-13/11-16/11;
  • 1-7/6-14/11-11/8 with steps of 7/6-12/11-13/12-16/11, and its inverse
  • 1-13/12-13/11-11/8 with steps of 13/12-12/11-7/6-16/11.

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

  • 1-14/11-11/8-3/2-7/4 with steps of 14/11-13/12-12/11-7/6-8/7, and its inverse
  • 1-12/11-13/11-3/2-12/7 with steps of 12/11-13/12-14/11-8/7-7/6;
  • 1-13/11-9/7-3/2-12/7 with steps of 13/11-12/11-7/6-8/7-7/6, and its inverse
  • 1-7/6-14/11-3/2-7/4 with steps of 7/6-12/11-13/11-7/6-8/7;
  • 1-14/11-11/8-3/2-11/6 with steps of 14/11-13/12-12/11-11/9-12/11, and its inverse
  • 1-12/11-13/11-3/2-18/11 with steps of 12/11-13/12-14/11-12/11-11/9;
  • 1-14/11-18/13-3/2-18/11 with steps of 14/11-12/11-13/12-12/11-11/9, and its inverse
  • 1-13/12-13/11-3/2-11/6 with steps of 13/12-12/11-14/11-11/9-12/11;
  • 1-13/11-14/11-3/2-11/6 with steps of 13/11-14/13-13/11-11/9-12/11, and its inverse
  • 1-13/11-14/11-3/2-18/11 with steps of 13/11-14/13-13/11-12/11-11/9;
  • 1-13/11-9/7-3/2-18/11 with steps of 13/11-12/11-7/6-12/11-11/9, and its inverse
  • 1-7/6-14/11-3/2-11/6 with steps of 7/6-12/11-13/11-11/9-12/11;
  • 1-13/11-11/8-3/2-11/6 with steps of 13/11-7/6-12/11-11/9-12/11, and its inverse
  • 1-12/11-14/11-3/2-18/11 with steps of 12/11-7/6-13/11-12/11-11/9;
  • 1-14/11-18/13-3/2-24/13 with steps of 14/11-12/11-13/12-16/13-13/12, and its inverse
  • 1-13/12-13/11-3/2-13/8 with steps of 13/12-12/11-14/11-13/12-16/13;
  • 1-13/11-11/8-3/2-13/8 with steps of 13/11-7/6-12/11-13/12-16/13, and its inverse
  • 1-12/11-14/11-3/2-24/13 with steps of 12/11-7/6-13/11-16/13-13/12;
  • 1-13/11-14/11-11/8-3/2 with steps of 13/11-14/13-13/12-12/11-4/3, and its inverse
  • 1-12/11-13/11-14/11-3/2 with steps of 12/11-13/12-14/13-13/11-4/3;
  • 1-7/6-14/11-11/8-3/2 with steps of 7/6-12/11-13/12-12/11-4/3, and its inverse
  • 1-12/11-13/11-9/7-3/2 with steps of 12/11-13/12-12/11-7/6-4/3;
  • 1-12/11-9/7-18/13-3/2 with steps of 12/11-13/11-14/13-13/12-4/3, and its inverse
  • 1-13/12-7/6-11/8-3/2 with steps of 13/12-14/13-13/11-12/11-4/3;
  • 1-12/11-14/11-18/13-3/2 with steps of 12/11-7/6-12/11-13/12-4/3, and its inverse
  • 1-13/12-13/11-11/8-3/2 with steps of 13/12-12/11-7/6-12/11-4/3.

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

  • 1-7/6-14/11-11/8-3/2-7/4 with steps of 7/6-12/11-13/12-12/11-7/6-8/7;
  • 1-12/11-14/11-18/13-3/2-24/13 with steps of 12/11-7/6-12/11-13/12-16/13-13/12.

The inversely related pairs of chords are

  • 1-7/6-14/11-11/8-3/2-11/6 with steps of 7/6-12/11-13/12-12/11-11/9-12/11, and its inverse
  • 1-12/11-13/11-9/7-3/2-18/11 with steps of 12/11-13/12-12/11-7/6-12/11-11/9;
  • 1-13/11-14/11-11/8-3/2-11/6 with steps of 13/11-14/13-13/12-12/11-11/9-12/11, and its inverse
  • 1-12/11-13/11-14/11-3/2-18/11 with steps of 12/11-13/12-14/13-13/11-12/11-11/9.

Equal temperaments with minor minthmic chords include 17, 22, 29, 41, 46, 58, 72, 87, 104, 121, 130, 217, 232, 234, 289 and 456.

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