Minor minthmic chords: Difference between revisions

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A gentle tetrad is any of three 13-limit [[Dyadic_chord|essentially tempered tetrads]] in the gentle temperament, tempering out 364/363. In close position, these have steps of size 14/11-13/11-7/6-8/7, 13/11-14/11-8/7-7/6 and 13/11-7/6-13/11-16/13, leading to the temperings of 1-14/11-3/2-7/4, 1-13/11-3/2-12/7 and 1-13/11-11/8-13/8. A gentle triad is any of four 13-limit essentially tempered triads in the gentle temperament. The supermajor gentle triad is a tempering of 1-14/11-3/2, and its inversion the subminor gentle triad is a tempering of 1-13/11-3/2. The gothic gentle triads are the temperings of 1-7/6-11/8 and its inversion 1-13/11-11/8. The names refer to [[Margo_Schulter|Margo Schulter]]'s Neo-gothic theory of harmony, which features a [[Gentle_region|gentle region]] with a slightly sharpened fifth in which gentle triads and neogothic triads flourish. Equal divisions with gentle triads include 17, 22, 29, 41, 46, 58, 72, 87, 104, 121, 130, 217, 232, 234, 289 and 456.
'''Minor minthmic chords''' are [[Dyadic chord|essentially tempered chords]] tempered by the minor minthma, [[364/363]].
[[Category:13-limit]]
 
[[Category:chord]]
There are 10 triads, 33 tetrads, 26 pentads and 6 hexads as 2.3.7.11.13 [[subgroup]] [[13-odd-limit]] essentially tempered chords.
[[Category:dyadic]]
 
[[Category:gentle]]
For triads, there are five pairs of chords in inverse relationship.
[[Category:neo-gothic]]
 
[[Category:todo:add_links]]
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 {{Optimal ET sequence| 17, 22, 29, 41, 46, 58, 72, 87, 104, 121, 130, 217, 232, 234, 289 and 456 }}.
 
[[Category:13-odd-limit chords]]
[[Category:Essentially tempered chords]]
[[Category:Triads]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Hexads]]
[[Category:Minor minthmic]]
[[Category:Neo-gothic]]

Latest revision as of 14:29, 19 March 2025

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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