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" alone ~~is ambiguous~~.</ref> 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>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> 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.

A notable subset of the 2.3.7 subgroup is the 1.3.7 [[tonality diamond]], comprised of all intervals in which 1, 3 and 7 are the only allowable odd numbers, once all powers of 2 are removed, either for the intervals of the scale or the ratios between successive or simultaneously sounding notes of the composition. The complete list of intervals in the 1.3.7 tonality diamond within the octave is [[1/1]], [[8/7]], [[7/6]], [[4/3]], [[3/2]], [[12/7]], [[7/4]], and [[2/1]].

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== 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

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* 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

== ~~Music~~ ==

== Regular temperaments ==

=== Rank-1 temperaments (edos) ===

* Michael Harrison, From Ancient Worlds (~~For harmonic Piano~~) ~~1992~~

* Michael Harrison, Revelation: Music In Pure Intonation, 2007

== ~~Edos~~ ==

A list of edos with progressively better tunings for the 2.3.7 subgroup: {{EDOs| 5, 12, 14, 17, 22, 31, 36, 77, 94, 130, 135, 171, 265, 306, 400, 571, 706, 1277 }} and so on.

Another list of edos which provides relatively good tunings for the 2.3.7 subgroup (relative error < 2.5%): {{EDOs| 36, 41, 77, 94, 99, 130, 135, 171, 207, 229, 265, 301, 306, 364, 400, 436, 441, 477, 494, 535, 571, 576, 607, 648, 665, 670, 701, 706, 742, 747, 783, 836, 841, 877, 913, 935, 971, 976, 1007, 1012, 1048, 1106, 1147, 1178, 1183, 1236, 1241, 1277 }} and so on.

== Rank-2 temperaments ==

=== Rank-2 temperaments ===

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

== Music ==

; [[Michael Harrison]]

* From Ancient Worlds (for harmonic piano), 1992

* Revelation: Music in Pure Intonation, 2007

== Notes ==

[[Category:Subgroup]]

[[Category:7-limit]]

[[Category:Rank 3]]

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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" alone is ambiguous.</ref> 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>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> 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.
 A notable subset of the 2.3.7 subgroup is the 1.3.7 [[tonality diamond]], comprised of all intervals in which 1, 3 and 7 are the only allowable odd numbers, once all powers of 2 are removed, either for the intervals of the scale or the ratios between successive or simultaneously sounding notes of the composition. The complete list of intervals in the 1.3.7 tonality diamond within the octave is [[1/1]], [[8/7]], [[7/6]], [[4/3]], [[3/2]], [[12/7]], [[7/4]], and [[2/1]].
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 == 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
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 * 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
-== Music ==
+== Regular temperaments ==
+=== Rank-1 temperaments (edos) ===
-* Michael Harrison, From Ancient Worlds (For harmonic Piano) 1992
-* Michael Harrison, Revelation: Music In Pure Intonation, 2007
-== Edos ==
 A list of edos with progressively better tunings for the 2.3.7 subgroup: {{EDOs| 5, 12, 14, 17, 22, 31, 36, 77, 94, 130, 135, 171, 265, 306, 400, 571, 706, 1277 }} and so on.
 Another list of edos which provides relatively good tunings for the 2.3.7 subgroup (relative error < 2.5%): {{EDOs| 36, 41, 77, 94, 99, 130, 135, 171, 207, 229, 265, 301, 306, 364, 400, 436, 441, 477, 494, 535, 571, 576, 607, 648, 665, 670, 701, 706, 742, 747, 783, 836, 841, 877, 913, 935, 971, 976, 1007, 1012, 1048, 1106, 1147, 1178, 1183, 1236, 1241, 1277 }} and so on.
-== Rank-2 temperaments ==
+=== Rank-2 temperaments ===
 {{Main|Tour of regular temperaments#Clans defined by a 2.3.7 (za) comma}}
+== Music ==
+; [[Michael Harrison]]
+* From Ancient Worlds (for harmonic piano), 1992
+* Revelation: Music in Pure Intonation, 2007
 == Notes ==
 <references />
+[[Category:Subgroup]]
 [[Category:7-limit]]
 [[Category:Rank 3]]