2.3.7 subgroup: Difference between revisions

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The '''2.3.7 subgroup'''<ref>Sometimes incorrectly named '''2.3.7-limit''' or '''2.3.7-prime limit'''; a [[prime limit]] is a subgroup spanned by all primes up to a given prime, and "limit" used alone usually implies prime limit.</ref> ('''za''' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 3, and 7 are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 3, and 7. This is an infinite set ~~and still infinite~~ even ~~if we restrict consideration~~ to a single octave. Some examples within the [[octave]] include [[3/2]], [[7/4]], [[7/6]], [[9/7]], [[9/8]], [[21/16]], and so on.

The '''2.3.7 subgroup'''<ref group="note">Sometimes incorrectly named '''2.3.7-limit''' or '''2.3.7-prime limit'''; a [[prime limit]] is a subgroup spanned by all primes up to a given prime, and "limit" used alone usually implies prime limit.</ref> sometimes called '''septal''' or, in [[color notation]], '''za''' is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 3, and 7 are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 3, and 7. This is an infinite set, even when restricted to a single octave. Some examples within the [[octave]] include [[3/2]], [[7/4]], [[7/6]], [[9/7]], [[9/8]], [[21/16]], and so on.

The 2.3.7 subgroup is a retraction of the [[7-limit]], obtained by removing prime 5. Its simplest expansion is the [[2.3.7.11 subgroup]], which adds prime 11.

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== Properties ==

The simpler ratios fall into 3 categories:

* 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 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 ===

* ~~Zo minor~~ pentatonic: 1/1 7/6 4/3 3/2 7/4 2/1

* ~~Ru pentatonic: 1/1 9/8 9/7 3/2 12/7 2/1~~

==== Minor ====

* Zo [[wikipedia:In_scale|in]]: 1/1 9/8 7/6 3/2 14/9 2/1

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

* ~~Zo minor~~: 1/1 9/8 7/6 4/3 3/2 14/9 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)

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

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

* Za [[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

* 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)

==== 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

==== Misc ====

* [[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

== Regular temperaments ==

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{{Main|Tour of regular temperaments#Clans defined by a 2.3.7 (za) comma}}

In the below tables, the generator of the temperament is highlighted in bold. Intervals in the tables reflect the 1.3.7.9.21 tonality diamond. It is structurally notable that ~~they~~ come in four triplets, each centered around one note of 5edo, consisting of a "minor", "middle", and "major" interval of each set.

In the below tables, the generator of the temperament is highlighted in bold. Intervals in the tables reflect the 1.3.7.9.21 tonality diamond. It is structurally notable that the intervals come in four triplets, each centered around one note of 5edo, consisting of a "minor", "middle", and "major" interval of each set.

==== Semaphore ====

'''Semaphore''' temperament tempers out the comma [[49/48]] = S7 in the 2.3.7 subgroup, which equates [[8/7]] with [[7/6]], creating a single neutral semifourth which serves as the generator. Similarly to [[dicot]], semaphore can be regarded as an exotemperament that elides fundamental distinctions within the subgroup ~~(from~~ the perspective of a pentatonic framework, this is equivalent to erasing the major-minor distinction as dicot does), though the comma involved is half the size of dicot's [[25/24]].

'''[[Semaphore]]''' temperament tempers out the comma [[49/48]] = S7 in the 2.3.7 subgroup, which equates [[8/7]] with [[7/6]], creating a single neutral semifourth which serves as the generator. Similarly to [[dicot]], semaphore can be regarded as an exotemperament that elides fundamental distinctions within the subgroup. From the perspective of a pentatonic framework, this is equivalent to erasing the major-minor distinction as dicot does, though the comma involved is half the size of dicot's [[25/24]].

The [[DKW theory|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 696.230c; a chart of mistunings of simple intervals is below.

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==== Archy ====

'''Archy''' temperament tempers out the comma [[64/63]] = S8 in the 2.3.7 subgroup, which equates [[9/8]] with [[8/7]], and [[4/3]] with [[21/16]]. It serves as a septimal analogue of [[meantone]], favoring fifths sharp of just rather than flat.

'''[[Archy]]''' temperament tempers out the comma [[64/63]] = S8 in the 2.3.7 subgroup, which equates [[9/8]] with [[8/7]], and [[4/3]] with [[21/16]]. It serves as a septimal analogue of [[meantone]], favoring fifths sharp of just rather than flat.

The [[DKW theory|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 712.585c (though most other optimizations tune this a few cents flatter); a chart of mistunings of simple intervals is below.

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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.

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== Notes ==

[[Category:Subgroup]]

@@ Line 1: / Line 1: @@
-The '''2.3.7 subgroup'''<ref>Sometimes incorrectly named '''2.3.7-limit''' or '''2.3.7-prime limit'''; a [[prime limit]] is a subgroup spanned by all primes up to a given prime, and "limit" used alone usually implies prime limit.</ref> ('''za''' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 3, and 7 are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 3, and 7. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the [[octave]] include [[3/2]], [[7/4]], [[7/6]], [[9/7]], [[9/8]], [[21/16]], and so on.
+The '''2.3.7 subgroup'''<ref group="note">Sometimes incorrectly named '''2.3.7-limit''' or '''2.3.7-prime limit'''; a [[prime limit]] is a subgroup spanned by all primes up to a given prime, and "limit" used alone usually implies prime limit.</ref> sometimes called '''septal''' or, in [[color notation]], '''za''' is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 3, and 7 are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 3, and 7. This is an infinite set, even when restricted to a single octave. Some examples within the [[octave]] include [[3/2]], [[7/4]], [[7/6]], [[9/7]], [[9/8]], [[21/16]], and so on.
 The 2.3.7 subgroup is a retraction of the [[7-limit]], obtained by removing prime 5. Its simplest expansion is the [[2.3.7.11 subgroup]], which adds prime 11.
@@ Line 10: / Line 10: @@
 == Properties ==
+The simpler ratios fall into 3 categories:
+* 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 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 ===
-* Zo minor pentatonic: 1/1 7/6 4/3 3/2 7/4 2/1
-* Ru pentatonic: 1/1 9/8 9/7 3/2 12/7 2/1
+==== Minor ====
-* Zo [[wikipedia:In_scale|in]]: 1/1 9/8 7/6 3/2 14/9 2/1
+* zo pentatonic: 1/1 7/6 4/3 3/2 7/4 2/1
-* Zo minor: 1/1 9/8 7/6 4/3 3/2 14/9 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)
-* Za harmonic minor:  1/1 9/8 7/6 4/3 3/2 14/9 27/14 2/1 (zo minor with a ru 7th)
+* zo: 1/1 9/8 7/6 4/3 3/2 14/9 7/4 2/1
-* Za [[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
+* 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)
+==== 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
+==== Misc ====
+* [[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
 == Regular temperaments ==
@@ Line 29: / Line 43: @@
 {{Main|Tour of regular temperaments#Clans defined by a 2.3.7 (za) comma}}
-In the below tables, the generator of the temperament is highlighted in bold. Intervals in the tables reflect the 1.3.7.9.21 tonality diamond. It is structurally notable that they come in four triplets, each centered around one note of 5edo, consisting of a "minor", "middle", and "major" interval of each set.
+In the below tables, the generator of the temperament is highlighted in bold. Intervals in the tables reflect the 1.3.7.9.21 tonality diamond. It is structurally notable that the intervals come in four triplets, each centered around one note of 5edo, consisting of a "minor", "middle", and "major" interval of each set.
 ==== Semaphore ====
-'''Semaphore''' temperament tempers out the comma [[49/48]] = S7 in the 2.3.7 subgroup, which equates [[8/7]] with [[7/6]], creating a single neutral semifourth which serves as the generator. Similarly to [[dicot]], semaphore can be regarded as an exotemperament that elides fundamental distinctions within the subgroup (from the perspective of a pentatonic framework, this is equivalent to erasing the major-minor distinction as dicot does), though the comma involved is half the size of dicot's [[25/24]].
+'''[[Semaphore]]''' temperament tempers out the comma [[49/48]] = S7 in the 2.3.7 subgroup, which equates [[8/7]] with [[7/6]], creating a single neutral semifourth which serves as the generator. Similarly to [[dicot]], semaphore can be regarded as an exotemperament that elides fundamental distinctions within the subgroup. From the perspective of a pentatonic framework, this is equivalent to erasing the major-minor distinction as dicot does, though the comma involved is half the size of dicot's [[25/24]].
 The [[DKW theory|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 696.230c; a chart of mistunings of simple intervals is below.
@@ Line 70: / Line 84: @@
 ==== Archy ====
-'''Archy''' temperament tempers out the comma [[64/63]] = S8 in the 2.3.7 subgroup, which equates [[9/8]] with [[8/7]], and [[4/3]] with [[21/16]]. It serves as a septimal analogue of [[meantone]], favoring fifths sharp of just rather than flat.
+'''[[Archy]]''' temperament tempers out the comma [[64/63]] = S8 in the 2.3.7 subgroup, which equates [[9/8]] with [[8/7]], and [[4/3]] with [[21/16]]. It serves as a septimal analogue of [[meantone]], favoring fifths sharp of just rather than flat.
 The [[DKW theory|DKW]] (2.3.7) optimum tuning states ~3/2 is tuned to 712.585c (though most other optimizations tune this a few cents flatter); a chart of mistunings of simple intervals is below.
@@ Line 107: / 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.
@@ Line 153: / Line 167: @@
 == Notes ==
-<references />
+<references group="note"/>
 [[Category:Subgroup]]