Minor minthmic chords: Difference between revisions

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A '''gentle chord''' is a [[13-odd-limit]] [[Dyadic chord|essentially tempered chord]] in the gentle temperament, tempering out the gentle comma, [[364/363]].  
'''Minor minthmic chords''' are [[Dyadic chord|essentially tempered chords]] tempered by the minor minthma, [[364/363]].


A gentle triad is any of four 13-odd-limit essentially tempered triads in the gentle temperament.  
There are 10 triads, 33 tetrads, 26 pentads and 6 hexads as 2.3.7.11.13 [[subgroup]] [[13-odd-limit]] essentially tempered chords.


The ''supermajor gentle triad'' is a tempering of  
For triads, there are five pairs of chords in inverse relationship.
* 1-14/11-3/2


and its inversion the ''subminor gentle triad'' is a tempering of  
The ''supermajor gentle triad'' (or ''gentle major triad'') is a tempering of  
* 1-13/11-3/2
* 1–14/11–3/2 with steps of 14/11, 13/11, 4/3;


The ''gothic gentle triads'' are the temperings of  
and its inversion the ''subminor gentle triad'' (or ''gentle minor triad'') is a tempering of  
* 1-7/6-11/8
* 1–13/11–3/2 with steps of 13/11, 14/11, 4/3.


and its inversion  
The ''gothic gentle triads'' are temperings of
* 1-13/11-11/8.  
* 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 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.  


A gentle tetrad is any of three 13-limit essentially tempered tetrads in the gentle temperament.  
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.


In close position, these are temperings of
The inversely related pairs of chords are  
* 1-14/11-3/2-7/4, with steps of 14/11-13/11-7/6-8/7
* 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
* 1-13/11-3/2-12/7, with steps of 13/11-14/11-8/7-7/6
* 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;
* 1-13/11-11/8-13/8, with steps of 13/11-7/6-13/11-16/13
* 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.


Edo with gentle chords include {{EDOs| 17, 22, 29, 41, 46, 58, 72, 87, 104, 121, 130, 217, 232, 234, 289 and 456 }}.
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:Essentially tempered chords]]
[[Category:Gentle]]
[[Category:Triads]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Hexads]]
[[Category:Minor minthmic]]
[[Category:Neo-gothic]]
[[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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