2.3.7 subgroup: Difference between revisions

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* Ratios without a 7 are pythagorean and sound much like 12edo intervals

* Ratios with a 7 in the numerator (7-over or '''zo''' in color notation) sound [[Supermajor and subminor|subminor]]

* Ratios with a 7 in the denominator (7-under or '''ru''' in color notation) sound supermajor

* Ratios with a 7 in the denominator (7-under or '''ru''' in color notation) sound [[Supermajor and subminor|supermajor]]

This subgroup is notably well-represented by [[5edo]] for its size, and therefore many of its simple intervals tend to cluster around the notes of 5edo: [[9/8]]~[[8/7]]~[[7/6]] representing a pentatonic "second", [[9/7]]~[[21/16]]~[[4/3]] representing a pentatonic "third", and so on. Therefore, one way to approach the 2.3.7 subgroup is to think of a pentatonic framework for composition as natural to it, rather than the diatonic framework associated with the [[5-limit]], and a few of the scales below reflect that nature.

=== Scales ===

==== Minor ====

* zo pentatonic: 1/1 7/6 4/3 3/2 7/4 2/1

* zo [[wikipedia:In_scale|in]]: 1/1 9/8 7/6 3/2 14/9 2/1 (the in scale is a minor scale with no 4th or 7th)

* zo: 1/1 9/8 7/6 4/3 3/2 14/9 7/4 2/1

* za harmonic minor: 1/1 9/8 7/6 4/3 3/2 14/9 27/14 2/1 (zo scale with a ru 7th)

~~If one wishes to use color notation, these qualities may be remembered by thinking of ru as "red" (sharp, alarmed, bright) and zo as "azure" (moody, dark).~~

==== Major ====

* ru pentatonic: 1/1 9/8 9/7 3/2 12/7 2/1

* ru: 1/1 9/8 9/7 4/3 3/2 12/7 27/14 2/1

This subgroup is notably well-represented by [[5edo]] for its size, and therefore many of its simple intervals tend to cluster around the notes of 5edo: [[9/8]]~[[8/7]]~[[7/6]] representing a pentatonic "second", [[9/7]]~[[21/16]]~[[4/3]] representing a pentatonic "third", and so on. Therefore, one way to approach the 2.3.7 subgroup is to think of a pentatonic framework for composition as natural to it, rather than the diatonic framework associated with the [[5-limit]], and a few of the scales below reflect that nature.

==== Misc ====

~~=== Scales ===~~

* subminor pentatonic: 1/1 7/6 4/3 3/2 7/4 2/1

* supermajor pentatonic: 1/1 9/8 9/7 3/2 12/7 2/1

* subminor [[wikipedia:In_scale|in]]: 1/1 9/8 7/6 3/2 14/9 2/1 (the in scale is a minor scale with no 4th or 7th)

* subminor: 1/1 9/8 7/6 4/3 3/2 14/9 7/4 2/1

* supermajor: 1/1 9/8 9/7 4/3 3/2 12/7 27/14 2/1

* 2.3.7 harmonic minor: 1/1 9/8 7/6 4/3 3/2 14/9 27/14 2/1 (subminor scale with a supermajor 7th)

* [[diasem]]/Tas[9] ([[Chiral|left-handed]]): 1/1 9/8 7/6 21/16 4/3 3/2 14/9 7/4 16/9 2/1

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<nowiki />* In 2.3.7-targeted DKW tuning

==== ~~Gamelic~~ ====

==== Slendric ====

'''~~Gamelic~~''' temperament, ~~better~~ known as [[slendric]], tempers out the comma [[1029/1024]] = S7/S8 in the 2.3.7 subgroup, which splits the perfect fifth into three intervals of [[8/7]]. It is one of the most accurate temperaments of its simplicity. While semaphore and archy equate each middle interval of each triplet with either the major or the minor, gamelic makes it a true "neutral" intermediate between them.

'''Slendric''' temperament, also known as [[slendric|gamelic]], tempers out the comma [[1029/1024]] = S7/S8 in the 2.3.7 subgroup, which splits the perfect fifth into three intervals of [[8/7]]. It is one of the most accurate temperaments of its simplicity. While semaphore and archy equate each middle interval of each triplet with either the major or the minor, gamelic makes it a true "neutral" intermediate between them.

The [[DKW theory|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 699.126c, and therefore ~8/7 to 233.042c; a chart of mistunings of simple intervals is below.

@@ Line 14: / Line 14: @@
 * Ratios without a 7 are pythagorean and sound much like 12edo intervals
 * Ratios with a 7 in the numerator (7-over or '''zo''' in color notation) sound [[Supermajor and subminor|subminor]]
-* Ratios with a 7 in the denominator (7-under or '''ru''' in color notation) sound supermajor
+* Ratios with a 7 in the denominator (7-under or '''ru''' in color notation) sound [[Supermajor and subminor|supermajor]]
+This subgroup is notably well-represented by [[5edo]] for its size, and therefore many of its simple intervals tend to cluster around the notes of 5edo: [[9/8]]~[[8/7]]~[[7/6]] representing a pentatonic "second", [[9/7]]~[[21/16]]~[[4/3]] representing a pentatonic "third", and so on. Therefore, one way to approach the 2.3.7 subgroup is to think of a pentatonic framework for composition as natural to it, rather than the diatonic framework associated with the [[5-limit]], and a few of the scales below reflect that nature.
+=== Scales ===
+==== Minor ====
+* zo pentatonic: 1/1 7/6 4/3 3/2 7/4 2/1
+* zo [[wikipedia:In_scale|in]]: 1/1 9/8 7/6 3/2 14/9 2/1 (the in scale is a minor scale with no 4th or 7th)
+* zo: 1/1 9/8 7/6 4/3 3/2 14/9 7/4 2/1
+* za harmonic minor: 1/1 9/8 7/6 4/3 3/2 14/9 27/14 2/1 (zo scale with a ru 7th)
-If one wishes to use color notation, these qualities may be remembered by thinking of ru as "red" (sharp, alarmed, bright) and zo as "azure" (moody, dark).
+==== Major ====
+* ru pentatonic: 1/1 9/8 9/7 3/2 12/7 2/1
+* ru: 1/1 9/8 9/7 4/3 3/2 12/7 27/14 2/1
-This subgroup is notably well-represented by [[5edo]] for its size, and therefore many of its simple intervals tend to cluster around the notes of 5edo: [[9/8]]~[[8/7]]~[[7/6]] representing a pentatonic "second", [[9/7]]~[[21/16]]~[[4/3]] representing a pentatonic "third", and so on. Therefore, one way to approach the 2.3.7 subgroup is to think of a pentatonic framework for composition as natural to it, rather than the diatonic framework associated with the [[5-limit]], and a few of the scales below reflect that nature.
+==== Misc ====
-=== Scales ===
-* subminor pentatonic: 1/1 7/6 4/3 3/2 7/4 2/1
-* supermajor pentatonic: 1/1 9/8 9/7 3/2 12/7 2/1
-* subminor [[wikipedia:In_scale|in]]: 1/1 9/8 7/6 3/2 14/9 2/1 (the in scale is a minor scale with no 4th or 7th)
-* subminor: 1/1 9/8 7/6 4/3 3/2 14/9 7/4 2/1
-* supermajor: 1/1 9/8 9/7 4/3 3/2 12/7 27/14 2/1
-* 2.3.7 harmonic minor: 1/1 9/8 7/6 4/3 3/2 14/9 27/14 2/1 (subminor scale with a supermajor 7th)
 * [[diasem]]/Tas[9] ([[Chiral|left-handed]]): 1/1 9/8 7/6 21/16 4/3 3/2 14/9 7/4 16/9 2/1
@@ Line 117: / Line 121: @@
 <nowiki />* In 2.3.7-targeted DKW tuning
-==== Gamelic ====
+==== Slendric ====
-'''Gamelic''' temperament, better known as [[slendric]], tempers out the comma [[1029/1024]] = S7/S8 in the 2.3.7 subgroup, which splits the perfect fifth into three intervals of [[8/7]]. It is one of the most accurate temperaments of its simplicity. While semaphore and archy equate each middle interval of each triplet with either the major or the minor, gamelic makes it a true "neutral" intermediate between them.
+'''Slendric''' temperament, also known as [[slendric|gamelic]], tempers out the comma [[1029/1024]] = S7/S8 in the 2.3.7 subgroup, which splits the perfect fifth into three intervals of [[8/7]]. It is one of the most accurate temperaments of its simplicity. While semaphore and archy equate each middle interval of each triplet with either the major or the minor, gamelic makes it a true "neutral" intermediate between them.
 The [[DKW theory|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 699.126c, and therefore ~8/7 to 233.042c; a chart of mistunings of simple intervals is below.