2.3.7 subgroup: Difference between revisions

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The 2.3.7-limit or 2.3.7-prime-limit is a [[~~Just~~ intonation subgroup]] consisting of rational ~~intervals~~ where 2, 3, and 7 are the only allowable prime ~~factors~~, 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~~-odd-limit refers to~~ a ~~constraint on~~ the ~~selection of~~ [[Just intonation|just]] [[Interval class|intervals]] for a scale or composition such that 3 and 7 are the only allowable odd numbers, either for the intervals of the scale, or the ratios between successive or simultaneously sounding notes of the composition. The complete list of 2.3.7~~-odd~~-limit ~~intervals within the octave is [[1/1~~]], [[~~8/7]], [[7/6]], [[4/~~3~~]], [[3/2]], [[12/~~7~~]], [[7/4]], and [[2/1~~]], which ~~is known as the 2.3.7-limit tonality diamond~~.

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.

~~The phrase "~~2.3.7~~-limit just intonation" usually refers to~~ the 2.3.7~~-prime-limit~~ and ~~includes primes~~ 2, 3, and 7. When octave equivalence is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3 and 7, which can be represented in 2-dimensional [[~~Harmonic~~ lattice diagram~~|lattice diagrams~~]], each prime represented by a different dimension.

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]].

Another such subset is the 1.3.7.9 tonality diamond, which adds the following intervals to the previous list: [[9/8]], [[9/7]], [[14/9]], and [[16/9]].

When [[octave equivalence]] is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3 and 7, which can be represented in a 2-dimensional [[lattice diagram]], each prime represented by a different dimension, such that each point on the lattice represents a different [[interval class]].

== 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 ===

==== 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)

==== 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 ==

=== Rank-1 temperaments (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.

=== Commas and rank-2 temperaments ===

{{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 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]].

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.

{| class="wikitable center-1 center-2 center-3 center-4"

|+ style="font-size: 105%;" | Semaphore (49/48)

|-

! rowspan="2" | Interval !! rowspan="2" | Just tuning !! colspan="2" | Tunings*

|-

! Optimal tuning !! Deviation

|-

| 9/8 || 203.910 || 192.460 || -11.450

|-

| '''8/7''' || 231.174 || '''251.885''' || +20.711

|-

| 7/6 || 266.871 || '''251.885''' || -14.986

|-

| 9/7 || 435.084 || 444.345 || +9.261

|-

| 21/16 || 470.781 || 444.345 || -26.436

|-

| 4/3 || 498.045 || 503.770 || +5.725

|-

| 3/2 || 701.955 || 696.230 || -5.725

|-

| 32/21 || 729.219 || 755.655 || +26.436

|-

| 14/9 || 764.916 || 755.655 || -9.261

|-

| 12/7 || 933.129 || 948.115 || +14.986

|-

| 7/4 || 968.826 || 948.115 || -20.711

|-

| 16/9 || 996.090 || 1007.540 || +11.450

|}

<nowiki />* In 2.3.7-targeted DKW tuning

==== 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.

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.

{| class="wikitable center-1 center-2 center-3 center-4"

|+ style="font-size: 105%;" | Archy (64/63)

|-

! rowspan="2" | Interval !! rowspan="2" | Just tuning !! colspan="2" | Tunings*

|-

! Optimal tuning !! Deviation

|-

| 9/8 || 203.910 || 225.171 || +21.261

|-

| 8/7 || 231.174 || 225.171 || -6.003

|-

| 7/6 || 266.871 || 262.244 || -4.627

|-

| 9/7 || 435.084 || 450.341 || +15.257

|-

| 21/16 || 470.781 || 487.415 || +16.634

|-

| 4/3 || 498.045 || 487.415 || -10.630

|-

| '''3/2''' || 701.955 || '''712.585''' || +10.630

|-

| 32/21 || 729.219 || '''712.585''' || -16.634

|-

| 14/9 || 764.916 || 749.659 || -15.257

|-

| 12/7 || 933.129 || 937.756 || +4.627

|-

| 7/4 || 968.826 || 974.829 || +6.003

|-

| 16/9 || 996.090 || 974.829 || -21.261

|}

<nowiki />* In 2.3.7-targeted DKW tuning

==== Slendric ====

'''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.

{| class="wikitable center-1 center-2 center-3 center-4"

|+ style="font-size: 105%;" | Gamelic (1029/1024)

|-

! rowspan="2" | Interval !! rowspan="2" | Just tuning !! colspan="2" | Tunings*

|-

! Optimal tuning !! Deviation

|-

| 9/8 || 203.910 || 198.253 || -5.657

|-

| '''8/7''' || 231.174 || '''233.042''' || +1.868

|-

| 7/6 || 266.871 || 267.831 || +0.960

|-

| 9/7 || 435.084 || 431.295 || -3.789

|-

| 21/16 || 470.781 || 466.084 || -4.697

|-

| 4/3 || 498.045 || 500.874 || +2.829

|-

| 3/2 || 701.955 || 699.126 || -2.829

|-

| 32/21 || 729.219 || 733.916 || +4.697

|-

| 14/9 || 764.916 || 768.705 || +3.789

|-

| 12/7 || 933.129 || 932.169 || -0.960

|-

| 7/4 || 968.826 || 966.958 || -1.868

|-

| 16/9 || 996.090 || 1001.747 || +5.657

|}

<nowiki />* In 2.3.7-targeted DKW tuning

== Music ==

; [[Michael Harrison]]

* From Ancient Worlds (for harmonic piano), 1992

* Revelation: Music in Pure Intonation, 2007

; [[La Monte Young]]

* The Well-Tuned Piano, 1974

== Notes ==

[[Category:Subgroup]]

[[Category:7-limit]]

[[Category:Rank 3]]

[[Category:~~Stub~~]]

[[Category:Lists of scales]]

@@ Line 1: / Line 1: @@
-The 2.3.7-limit or 2.3.7-prime-limit is a [[Just intonation subgroup]] consisting of rational intervals where 2, 3, and 7 are the only allowable prime factors, 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-odd-limit refers to a constraint on the selection of [[Just intonation|just]] [[Interval class|intervals]] for a scale or composition such that 3 and 7 are the only allowable odd numbers, either for the intervals of the scale, or the ratios between successive or simultaneously sounding notes of the composition. The complete list of 2.3.7-odd-limit intervals within the octave is [[1/1]], [[8/7]], [[7/6]], [[4/3]], [[3/2]], [[12/7]], [[7/4]], and [[2/1]], which is known as the 2.3.7-limit tonality diamond.
+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.
-The phrase "2.3.7-limit just intonation" usually refers to the 2.3.7-prime-limit and includes primes 2, 3, and 7. When octave equivalence is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3 and 7, which can be represented in 2-dimensional [[Harmonic lattice diagram|lattice diagrams]], each prime represented by a different dimension.
+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]].
+Another such subset is the 1.3.7.9 tonality diamond, which adds the following intervals to the previous list: [[9/8]], [[9/7]], [[14/9]], and [[16/9]].
+When [[octave equivalence]] is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3 and 7, which can be represented in a 2-dimensional [[lattice diagram]], each prime represented by a different dimension, such that each point on the lattice represents a different [[interval class]].
+== 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 ===
+==== 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)
+==== 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 ==
+=== Rank-1 temperaments (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.
+=== Commas and rank-2 temperaments ===
+{{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 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]].
+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.
+{| class="wikitable center-1 center-2 center-3 center-4"
+|+ style="font-size: 105%;" | Semaphore (49/48)
+|-
+! rowspan="2" | Interval !! rowspan="2" | Just tuning !! colspan="2" | Tunings*
+|-
+! Optimal tuning !! Deviation
+|-
+| 9/8 || 203.910 || 192.460 || -11.450
+|-
+| '''8/7''' || 231.174 || '''251.885''' || +20.711
+|-
+| 7/6 || 266.871 || '''251.885''' || -14.986
+|-
+| 9/7 || 435.084 || 444.345 || +9.261
+|-
+| 21/16 || 470.781 || 444.345 || -26.436
+|-
+| 4/3 || 498.045 || 503.770 || +5.725
+|-
+| 3/2 || 701.955 || 696.230 || -5.725
+|-
+| 32/21 || 729.219 || 755.655 || +26.436
+|-
+| 14/9 || 764.916 || 755.655 || -9.261
+|-
+| 12/7 || 933.129 || 948.115 || +14.986
+|-
+| 7/4 || 968.826 || 948.115 || -20.711
+|-
+| 16/9 || 996.090 || 1007.540 || +11.450
+|}
+<nowiki />* In 2.3.7-targeted DKW tuning
+==== 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.
+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.
+{| class="wikitable center-1 center-2 center-3 center-4"
+|+ style="font-size: 105%;" | Archy (64/63)
+|-
+! rowspan="2" | Interval !! rowspan="2" | Just tuning !! colspan="2" | Tunings*
+|-
+! Optimal tuning !! Deviation
+|-
+| 9/8 || 203.910 || 225.171 || +21.261
+|-
+| 8/7 || 231.174 || 225.171 || -6.003
+|-
+| 7/6 || 266.871 || 262.244 || -4.627
+|-
+| 9/7 || 435.084 || 450.341 || +15.257
+|-
+| 21/16 || 470.781 || 487.415 || +16.634
+|-
+| 4/3 || 498.045 || 487.415 || -10.630
+|-
+| '''3/2''' || 701.955 || '''712.585''' || +10.630
+|-
+| 32/21 || 729.219 || '''712.585''' || -16.634
+|-
+| 14/9 || 764.916 || 749.659 || -15.257
+|-
+| 12/7 || 933.129 || 937.756 || +4.627
+|-
+| 7/4 || 968.826 || 974.829 || +6.003
+|-
+| 16/9 || 996.090 || 974.829 || -21.261
+|}
+<nowiki />* In 2.3.7-targeted DKW tuning
+==== Slendric ====
+'''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.
+{| class="wikitable center-1 center-2 center-3 center-4"
+|+ style="font-size: 105%;" | Gamelic (1029/1024)
+|-
+! rowspan="2" | Interval !! rowspan="2" | Just tuning !! colspan="2" | Tunings*
+|-
+! Optimal tuning !! Deviation
+|-
+| 9/8 || 203.910 || 198.253 || -5.657
+|-
+| '''8/7''' || 231.174 || '''233.042''' || +1.868
+|-
+| 7/6 || 266.871 || 267.831 || +0.960
+|-
+| 9/7 || 435.084 || 431.295 || -3.789
+|-
+| 21/16 || 470.781 || 466.084 || -4.697
+|-
+| 4/3 || 498.045 || 500.874 || +2.829
+|-
+| 3/2 || 701.955 || 699.126 || -2.829
+|-
+| 32/21 || 729.219 || 733.916 || +4.697
+|-
+| 14/9 || 764.916 || 768.705 || +3.789
+|-
+| 12/7 || 933.129 || 932.169 || -0.960
+|-
+| 7/4 || 968.826 || 966.958 || -1.868
+|-
+| 16/9 || 996.090 || 1001.747 || +5.657
+|}
+<nowiki />* In 2.3.7-targeted DKW tuning
+== Music ==
+; [[Michael Harrison]]
+* From Ancient Worlds (for harmonic piano), 1992
+* Revelation: Music in Pure Intonation, 2007
+; [[La Monte Young]]
+* The Well-Tuned Piano, 1974
+== Notes ==
+<references group="note"/>
+[[Category:Subgroup]]
 [[Category:7-limit]]
 [[Category:Rank 3]]
-[[Category:Stub]]
+[[Category:Lists of scales]]