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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