Major minthmic chords: Difference between revisions

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'''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, tempering out the minthma, [[352/351]].  
'''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 minthma, [[352/351]].  


We have two pairs of inversely related minthmic triads, all involving the [[3/2|perfect fifth]]:
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.
* 1-[[11/9]]-[[3/2]] with steps 11/9-[[16/13]]-[[4/3]], and its inverse
* 1-16/13-3/2 with steps 16/13-11/9-4/3;
* 1-3/2-[[16/9]] with steps 3/2-[[13/11]]-[[9/8]], and its inverse
* 1-3/2-[[22/13]] with steps 3/2-9/8-13/11.  


For tetrads, first there is the palindrome, a sus2sus4 chord with the m3 serving a nice ~13/11:
For triads, there are four pairs of chords in inverse relationship:
* 1-9/8-4/3-3/2 with steps 9/8-13/11-9/8-4/3.  
* 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.


Then there are inversely related pairs that are concatenations of the triads:
For tetrads, there are five palindromic chords and eleven pairs of chords in inverse relationship. The palindromic chords are
* 1-11/9-3/2-16/9 with steps 11/9-16/13-13/11-9/8, and its inverse
* 1-11/9-3/2-13/8 with steps of 11/9-16/13-13/12-16/13;
* 1-16/13-3/2-22/13 with steps 16/13-11/9-9/8-13/11;  
* 1-11/9-4/3-13/8 with steps of 11/9-12/11-11/9-16/13;
* 1-11/9-3/2-22/13 with steps 11/9-16/13-9/8-13/11, and its inverse
* 1-9/8-4/3-3/2 with steps of 9/8-13/11-9/8-4/3;
* 1-16/13-3/2-16/9 with steps 16/13-11/9-13/11-9/8.  
* 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.


As well as
The inversely related pairs of chords are
* 1-3/2-13/8-11/6 with steps 3/2-13/12-9/8-12/11, and its inverse
* 1-11/9-11/8-13/8 with steps of 11/9-9/8-13/11-16/13, and its inverse
* 1-3/2-18/11-16/9 with steps 3/2-12/11-13/12-9/8.  
* 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.


If we add prime 7 to the mix, we additionally get
For pentads, there are fourteen pairs of chords in inverse relationship:
* 1-11/9-3/2-[[7/4]] with steps 11/9-16/13-[[7/6]]-[[8/7]], and its inverse
* 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
* 1-16/13-3/2-[[12/7]] with steps 16/13-11/9-8/7-7/6;
* 1-9/8-16/13-18/13-18/11 with steps of 9/8-12/11-9/8-13/11-11/9;
* 1-11/9-3/2-12/7, withs steps 11/9-16/13-8/7-7/6, and its inverse
* 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
* 1-16/13-3/2-7/4 with steps 16/13-11/9-7/6-8/7.  
* 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.


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.
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.


Finally, there is a pair of minthmic heptads:
The inversely related pairs of chords are
* 1-9/8-11/9-4/3-3/2-13/8-11/6 with steps 9/8-13/12-12/11-9/8-13/12-9/8-12/11, and its inverse
* 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
* 1-9/8-16/13-4/3-3/2-18/11-24/13 with steps 9/8-12/11-13/12-9/8-12/11-9/8-13/12.  
* 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.


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 {{EDOs| 22, 29, 46, 53, 80, 87, 111, 121, 140, 198 and 205 }}.
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]]).
 
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 {{Optimal ET sequence| 22, 29, 46, 53, 80, 87, 111, 121, 140, 198 and 205 }}.


[[Category:13-odd-limit]]
[[Category:13-odd-limit]]
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[[Category:Triads]]
[[Category:Triads]]
[[Category:Tetrads]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Hexads]]
[[Category:Heptads]]
[[Category:Heptads]]
[[Category:Minthmic]]
[[Category:Minthmic]]

Revision as of 05:33, 21 October 2023

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 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).

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.

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