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]].

'''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]].

We have ~~the four~~ minthmic ~~tetrads~~, ~~with steps~~ the ~~minthmic tempering of~~

We have two pairs of inversely related minthmic triads, all involving the [[3/2|perfect fifth]]:

* [[11/9]]-[[16/13]]-[[7/6]]-[[8/7]],

* 1-[[11/9]]-[[3/2]] with steps 11/9-[[16/13]]-[[4/3]], and its inverse

* 11/9-16/13-8/7-7/6,

* 1-16/13-3/2 with steps 16/13-11/9-4/3;

* 16/13-11/9-7/6-8/7, and

* 1-3/2-[[16/9]] with steps 3/2-[[13/11]]-[[9/8]], and its inverse

* 16/13-11/9-8/7-7/6,

* 1-3/2-[[22/13]] with steps 3/2-9/8-13/11.

~~leading to temperings of~~

For tetrads, first there is the palindrome, a sus2sus4 chord with the m3 serving a nice ~13/11:

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

* 1-9/8-4/3-3/2 with steps 9/8-13/11-9/8-4/3.

* 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~~

Then there are inversely related pairs that are concatenations of the triads:

* 1-11/9-3/2 and

* 1-11/9-3/2-16/9 with steps 11/9-16/13-13/11-9/8, and its inverse

* 1-16/13-3/2.

* 1-16/13-3/2-22/13 with steps 16/13-11/9-9/8-13/11;

* 1-11/9-3/2-22/13 with steps 11/9-16/13-9/8-13/11, and its inverse

* 1-16/13-3/2-16/9 with steps 16/13-11/9-13/11-9/8.

As well as

* 1-3/2-13/8-11/6 with steps 3/2-13/12-9/8-12/11, and its inverse

* 1-3/2-18/11-16/9 with steps 3/2-12/11-13/12-9/8.

If we add prime 7 to the mix, we additionally get

* 1-11/9-3/2-[[7/4]] with steps 11/9-16/13-[[7/6]]-[[8/7]], and its inverse

* 1-16/13-3/2-[[12/7]] with steps 16/13-11/9-8/7-7/6;

* 1-11/9-3/2-12/7, withs steps 11/9-16/13-8/7-7/6, and its inverse

* 1-16/13-3/2-7/4 with steps 16/13-11/9-7/6-8/7.

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.

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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 }}.

[[Category:13-odd-limit]]

[[Category:Essentially tempered chords]]

[[Category:Triads]]

[[Category:Tetrads]]

[[Category: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]].
+'''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]].
-We have the four minthmic tetrads, with steps the minthmic tempering of
+We have two pairs of inversely related minthmic triads, all involving the [[3/2|perfect fifth]]:
-* [[11/9]]-[[16/13]]-[[7/6]]-[[8/7]],
+* 1-[[11/9]]-[[3/2]] with steps 11/9-[[16/13]]-[[4/3]], and its inverse
-* 11/9-16/13-8/7-7/6,
+* 1-16/13-3/2 with steps 16/13-11/9-4/3;
-* 16/13-11/9-7/6-8/7, and
+* 1-3/2-[[16/9]] with steps 3/2-[[13/11]]-[[9/8]], and its inverse
-* 16/13-11/9-8/7-7/6,
+* 1-3/2-[[22/13]] with steps 3/2-9/8-13/11.
-leading to temperings of
+For tetrads, first there is the palindrome, a sus2sus4 chord with the m3 serving a nice ~13/11:
-* 1-11/9-[[3/2]]-[[7/4]],
+* 1-9/8-4/3-3/2 with steps 9/8-13/11-9/8-4/3.
-* 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
+Then there are inversely related pairs that are concatenations of the triads:
-* 1-11/9-3/2 and
+* 1-11/9-3/2-16/9 with steps 11/9-16/13-13/11-9/8, and its inverse
-* 1-16/13-3/2.
+* 1-16/13-3/2-22/13 with steps 16/13-11/9-9/8-13/11;
+* 1-11/9-3/2-22/13 with steps 11/9-16/13-9/8-13/11, and its inverse
+* 1-16/13-3/2-16/9 with steps 16/13-11/9-13/11-9/8.
+As well as
+* 1-3/2-13/8-11/6 with steps 3/2-13/12-9/8-12/11, and its inverse
+* 1-3/2-18/11-16/9 with steps 3/2-12/11-13/12-9/8.
+If we add prime 7 to the mix, we additionally get
+* 1-11/9-3/2-[[7/4]] with steps 11/9-16/13-[[7/6]]-[[8/7]], and its inverse
+* 1-16/13-3/2-[[12/7]] with steps 16/13-11/9-8/7-7/6;
+* 1-11/9-3/2-12/7, withs steps 11/9-16/13-8/7-7/6, and its inverse
+* 1-16/13-3/2-7/4 with steps 16/13-11/9-7/6-8/7.
 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.
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 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 }}.
+[[Category:13-odd-limit]]
 [[Category:Essentially tempered chords]]
+[[Category:Triads]]
+[[Category:Tetrads]]
 [[Category:Minthmic]]