Major minthmic chords: Difference between revisions

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A ''minthmic chord'' is an [[Dyadic chord|essentially tempered dyadic chord]] which is defined in the 13-limit rank five temperament, and in addition the rank three 2.3.11.13 temperament, tempering out the minthma, 352/351. We have the four minthmic tetrads, with steps the minthmic tempering of 11/9-16/13-7/6-8/7, 11/9-16/13-8/7-7/6, 16/13-11/9-7/6-8/7, and 16/13-11/9-8/7-7/6, leading to temperings of 1-11/9-3/2-7/4, 1-11/9-3/2-12/7, 1-16/13-3/2-7/4 and 1-16/13-3/2-12/7. These in turn have subchords such as the minthmic temperings of 1-11/9-3/2 and 1-16/13-3/2. Note that tempering 1-11/9-3/2-7/4 and 1-11/9-3/2-12/7 in [[jove tetrads|jove]] is also possible, leading to a similar but not identical chord.
'''Major minthmic chords''' are [[Dyadic chord|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]].  


Mintha was a nymph turned into a mint plant by a goddess whom she got in the way of, and 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 minthmic chords include 22, 29, 46, 53, 80, 87, 111, 121, 140, 198 and 205.
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.
[[Category:13-limit]]
 
[[Category:Chords]]
For triads, there are four pairs of chords in inverse relationship:
[[Category:Minthmic]]
* 1–16/13–3/2 with steps of 16/13, 11/9, 4/3, and its inverse
[[Category:Todo:add links]]
* 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]]).  
 
Equal temperaments with major minthmic chords include {{Optimal ET sequence| 22, 29, 46, 53, 80, 87, 111, 121, 140, 198 and 205 }}.
 
[[Category:13-odd-limit chords]]
[[Category:Essentially tempered chords]]
[[Category:Triads]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Hexads]]
[[Category:Heptads]]
[[Category:Major minthmic]]

Latest revision as of 14:35, 20 March 2025

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).

Equal temperaments with major minthmic chords include 22, 29, 46, 53, 80, 87, 111, 121, 140, 198 and 205.

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