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-5 temperament, and in addition the rank-3 2.3.11.13 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 the four minthmic~~ tetrads, ~~with steps the minthmic tempering of~~

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.

* [[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

For triads, there are four pairs of chords in inverse relationship:

* 1-11/9~~-[[~~3/2~~]]-[[7~~/4]],

* 1–16/13–3/2 with steps of 16/13, 11/9, 4/3, and its inverse

* ~~1-11~~/9-3/2~~-[[~~12/~~7]]~~,

* 1–11/9–3/2 with steps of 11/9, 16/13, 4/3;

* ~~1-16~~/13-3/~~2-7~~/4 and

* 1–13/11–4/3 with steps of 13/11, 9/8, 3/2, and its inverse

* ~~1-16~~/13-3/2-12/7.

* 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.

~~These~~ in ~~turn have subchords such as the minthmic temperings~~ of

For tetrads, there are five palindromic chords and eleven pairs of chords in inverse relationship. The palindromic chords are

* 1-11/9-3/~~2 and~~

* 1–11/9–3/2–13/8 with steps of 11/9, 16/13, 13/12, 16/13;

* ~~1-16~~/13-3/2.

* 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.

~~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~~.

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.

~~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 }}~~.

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:~~Minthmic~~]]

[[Category:Triads]]

[[Category:Tetrads]]

[[Category:Pentads]]

[[Category:Hexads]]

[[Category:Heptads]]

[[Category:Major minthmic]]

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-A '''minthmic chord''' is an [[Dyadic chord|essentially tempered dyadic chord]] which is defined in the [[13-limit]] rank-5 temperament, and in addition the rank-3 2.3.11.13 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 the four minthmic tetrads, with steps the minthmic tempering of
+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.
-* [[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
+For triads, there are four pairs of chords in inverse relationship:
-* 1-11/9-[[3/2]]-[[7/4]],
+* 1–16/13–3/2 with steps of 16/13, 11/9, 4/3, and its inverse
-* 1-11/9-3/2-[[12/7]],
+* 1–11/9–3/2 with steps of 11/9, 16/13, 4/3;
-* 1-16/13-3/2-7/4 and
+* 1–13/11–4/3 with steps of 13/11, 9/8, 3/2, and its inverse
-* 1-16/13-3/2-12/7.
+* 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.
-These in turn have subchords such as the minthmic temperings of
+For tetrads, there are five palindromic chords and eleven pairs of chords in inverse relationship. The palindromic chords are
-* 1-11/9-3/2 and
+* 1–11/9–3/2–13/8 with steps of 11/9, 16/13, 13/12, 16/13;
-* 1-16/13-3/2.
+* 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.
-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.
+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.
-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 }}.
+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:Minthmic]]
+[[Category:Triads]]
+[[Category:Tetrads]]
+[[Category:Pentads]]
+[[Category:Hexads]]
+[[Category:Heptads]]
+[[Category:Major minthmic]]