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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]].  
'''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]].  


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]]).  


[[Category:13-odd-limit]]
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:Essentially tempered chords]]
[[Category:Triad]]
[[Category:Triads]]
[[Category:Tetrad]]
[[Category:Tetrads]]
[[Category:Heptad]]
[[Category:Pentads]]
[[Category:Minthmic]]
[[Category:Hexads]]
[[Category:Heptads]]
[[Category:Major minthmic]]
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