2.3.7.11 subgroup - Revision history

Eufalesio: Colors go from simplest to complexest in limits

2026-04-19T11:32:46Z

Colors go from simplest to complexest in limits

← Older revision		Revision as of 11:32, 19 April 2026
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	The '''2.3.7.11 subgroup''' ('''~~laza~~''' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 3, 7, and 11 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, 7, and 11. 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]], [[9/7]], [[21/16]], [[11/9]], [[22/21]], and so on.		The '''2.3.7.11 subgroup''' ('''zala''' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 3, 7, and 11 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, 7, and 11. 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]], [[9/7]], [[21/16]], [[11/9]], [[22/21]], and so on.

	The 2.3.7.11 subgroup is a retraction of the [[11-limit]], obtained by removing prime 5. Its simplest expansion is the 2.3.7.11.13 subgroup, which adds prime 13. It can also be retracted to the [[2.3.7 subgroup]] by removing prime 11.		The 2.3.7.11 subgroup is a retraction of the [[11-limit]], obtained by removing prime 5. Its simplest expansion is the 2.3.7.11.13 subgroup, which adds prime 13. It can also be retracted to the [[2.3.7 subgroup]] by removing prime 11.

FloraC: Notation & grammar

2026-03-10T05:12:44Z

Notation & grammar

← Older revision		Revision as of 05:12, 10 March 2026
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	The 2.3.7.11 subgroup is a retraction of the [[11-limit]], obtained by removing prime 5. Its simplest expansion is the 2.3.7.11.13 subgroup, which adds prime 13. It can also be retracted to the [[2.3.7 subgroup]] by removing prime 11.		The 2.3.7.11 subgroup is a retraction of the [[11-limit]], obtained by removing prime 5. Its simplest expansion is the 2.3.7.11.13 subgroup, which adds prime 13. It can also be retracted to the [[2.3.7 subgroup]] by removing prime 11.

	A notable subset of the 2.3.7.11 subgroup is the 1.3.7.9.11 [[tonality diamond]], ~~comprised of~~ all intervals in which 1, 3, 7, 9, and 11 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 1.3.7.9.11~~ tonality diamond within the octave is [[1/1]], [[12/11]], [[9/8]], [[8/7]], [[7/6]], [[11/9]], [[14/11]], [[9/7]], [[4/3]], [[11/8]], [[16/11]], [[3/2]], [[14/9]], [[11/7]], [[18/11]], [[12/7]], [[7/4]], [[16/9]], [[11/6]], and [[2/1]].		A notable subset of the 2.3.7.11 subgroup is the {1, 3, 7, 9, 11} [[tonality diamond]], comprising all intervals in which 1, 3, 7, 9, and 11 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 this tonality diamond within the octave is [[1/1]], [[12/11]], [[9/8]], [[8/7]], [[7/6]], [[11/9]], [[14/11]], [[9/7]], [[4/3]], [[11/8]], [[16/11]], [[3/2]], [[14/9]], [[11/7]], [[18/11]], [[12/7]], [[7/4]], [[16/9]], [[11/6]], and [[2/1]].

	When [[octave equivalence]] is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3, 7, and 11, which can be represented in a 3-dimensional [[lattice diagram]], each prime represented by a different dimension, such that each point on the lattice represents a different [[interval class]].		When [[octave equivalence]] is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3, 7, and 11, which can be represented in a 3-dimensional [[lattice diagram]], each prime represented by a different dimension, such that each point on the lattice represents a different [[interval class]].

FloraC: /* Regular temperaments */ - copypasting error

2026-03-10T05:09:43Z

Regular temperaments: - copypasting error

← Older revision		Revision as of 05:09, 10 March 2026
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	== Regular temperaments ==		== Regular temperaments ==
	=== Rank-1 temperaments (edos) ===		=== Rank-1 temperaments (edos) ===
	The 2.3.7.11 subgroup is relatively well approximated by the following edos (decreasing [[TE error]], bold ones do particularly well in this subgroup): {{EDOs\| '''5''', 9, 10, 12, 14, '''17''', 31~~, 34~~, '''41''', 58, 63, '''72''', 94, 118, 130, '''135''', 342, … }}		The 2.3.7.11 subgroup is relatively well approximated by the following edos (decreasing [[TE error]], bold ones do particularly well in this subgroup): {{EDOs\| '''5''', 9, 10, 12, 14, '''17''', 31, '''41''', 58, 63, '''72''', 94, 118, 130, '''135''', 342, … }}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===

FloraC: /* Regular temperaments */ + edo list

2026-03-10T05:07:09Z

Regular temperaments: + edo list

← Older revision		Revision as of 05:07, 10 March 2026
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	== Regular temperaments ==		== Regular temperaments ==
	=== Rank-1 temperaments (edos) ===		=== Rank-1 temperaments (edos) ===
	{{~~Todo~~\|~~inline=1\|expand~~}}		The 2.3.7.11 subgroup is relatively well approximated by the following edos (decreasing [[TE error]], bold ones do particularly well in this subgroup): {{EDOs\| '''5''', 9, 10, 12, 14, '''17''', 31, 34, '''41''', 58, 63, '''72''', 94, 118, 130, '''135''', 342, … }}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===

FloraC: Recategorize

2026-02-11T10:25:23Z

Recategorize

← Older revision		Revision as of 10:25, 11 February 2026
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	{{Todo\|inline=1\|expand}}		{{Todo\|inline=1\|expand}}

	[[Category:~~Subgroup~~]]		[[Category:Just intonation subgroups\|#]]
	[[Category:Rank-4 temperaments]]		[[Category:Rank-4 temperaments\|#]]
	[[Category:11-limit]]		[[Category:11-limit\|#]]
	[[Category:Lists of scales]]		[[Category:Lists of scales\|#]]

FloraC: Recategorize. Mark todo

2026-02-06T08:25:07Z

Recategorize. Mark todo

← Older revision		Revision as of 08:25, 6 February 2026
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	== Regular temperaments ==		== Regular temperaments ==
	=== Rank-1 temperaments (edos) ===		=== Rank-1 temperaments (edos) ===
			{{Todo\|inline=1\|expand}}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{{Main\|Tour of regular temperaments#Temperaments defined by an 11-limit comma}}		{{Main\|Tour of regular temperaments#Temperaments defined by an 11-limit comma}}
			{{Todo\|inline=1\|expand}}
	~~== Music =~~=

	[[Category:Subgroup]]		[[Category:Subgroup]]
			[[Category:Rank-4 temperaments]]
	[[Category:11-limit]]		[[Category:11-limit]]
	~~[[Category:Rank 4]]~~
	[[Category:Lists of scales]]		[[Category:Lists of scales]]

FloraC: Recategorize

2024-12-18T10:12:45Z

Recategorize

← Older revision		Revision as of 10:12, 18 December 2024
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	[[Category:Subgroup]]		[[Category:Subgroup]]
	[[Category:11-limit]]		[[Category:11-limit]]
	[[Category:Rank 3]]		[[Category:Rank 4]]
	[[Category:Lists of scales]]		[[Category:Lists of scales]]

Fredg999: Add lead section and link to temperaments, cleanup

2023-05-29T04:04:37Z

Add lead section and link to temperaments, cleanup

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-<ul><li>Genus(3*7*11) 12/11, 14/11, 21/16, 16/11, 3/2, 7/4, 21/11, 2</li><li>Genus(3*7^2*11) 49/48, 49/44, 7/6, 14/11, 4/3, 16/11, 49/33, 49/32, 56/33, 7/4, 64/33, 2</li><li>Ptolemy's Intense Chromatic at 1/1 22/21 8/7 4/3 3/2 11/7 12/7 2/1 (disjunct form)</li><li>a conjunct Rast: 1/1 9/8 27/22 4/3 3/2 11/8 16/9 2/1 (note: actually 2.3.11 subgroup)</li><li>Qutb al-Din al-Shirazi's version of Hijaz: 1/1-12/11-14/11-4/3 (12:11-7:6-22:21), itself a permutation of Ptolemy's Intense Chromatic<ul><li>[[Margo_Schulter|Margo Schulter]] notes: "A modern form of Maqam Hijaz based on this tuning might be 1/1-12/11-14/11-4/3-32-18/11-16/9-2/1 ascending, and 1/1-12/11-14/11-4/3-3/2-128/81-16/9-2/1 descending (with the minor sixth maybe a bit smaller, say 11/7 or the like)."</li></ul></li><li>a septimal flavor of Sazkar, which has a minor third above the 1/1 ascending but a tone descending: thus ascending 1/1-7/6-11/9-4/3-3/2-27/16-11/6-2/1, and descending 1/1-9/8-11/9-4/3-3/2-27/16-11/6-2/1<ul><li>Margo Schulter adds: "I should caution that an Arab Rast, of which Sazkar is an offshoot, might usually have a Zalzalian third more like 27/22 or 16/13 rather than 11/9, so this may be more of a new tuning than a traditional Arab Sazkar."</li></ul></li><li>The 1-3-7-9-11 [[Combination_product_sets|2(5 Dekany]] within a 12-tone [[Periodic_scale#Constant Structure|constant structure]] (see [http://anaphoria.com/dekanyconstantstructures.pdf link]): 1*3, 9*11, 3*9, 1*7, 3*7*11, 7*9, 3*11, 1*9, 3*9*11, 7*11, 3*7, 1*11.</li><li>A 12f, {352/351, 364/363} 2.3.7.11.13 elf transversal: 28/27-9/8-13/11-9/7-4/3-11/8-3/2-14/9-22/13-16/9-27/14-2</li></ul>      [[Category:11-limit]]
+The '''2.3.7.11 subgroup''' ('''laza''' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 3, 7, and 11 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, 7, and 11. 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]], [[9/7]], [[21/16]], [[11/9]], [[22/21]], and so on.
-[[Category:rank_3]]
 [[Category:Lists of scales]]

Fredg999: Fredg999 moved page Gallery of 2.3.7.11 Subgroup Scales to 2.3.7.11 subgroup: Merge list of scales into a page about the subgroup more generally

2023-05-29T03:29:07Z

Fredg999 moved page Gallery of 2.3.7.11 Subgroup Scales to 2.3.7.11 subgroup: Merge list of scales into a page about the subgroup more generally

← Older revision	Revision as of 03:29, 29 May 2023
(No difference)

Fredg999 category edits: Removing from Category:Gallery using Cat-a-lot

2023-04-13T21:43:31Z

Removing from Category:Gallery using Cat-a-lot

← Older revision		Revision as of 21:43, 13 April 2023
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	<ul><li>Genus(3711) 12/11, 14/11, 21/16, 16/11, 3/2, 7/4, 21/11, 2</li><li>Genus(37^211) 49/48, 49/44, 7/6, 14/11, 4/3, 16/11, 49/33, 49/32, 56/33, 7/4, 64/33, 2</li><li>Ptolemy's Intense Chromatic at 1/1 22/21 8/7 4/3 3/2 11/7 12/7 2/1 (disjunct form)</li><li>a conjunct Rast: 1/1 9/8 27/22 4/3 3/2 11/8 16/9 2/1 (note: actually 2.3.11 subgroup)</li><li>Qutb al-Din al-Shirazi's version of Hijaz: 1/1-12/11-14/11-4/3 (12:11-7:6-22:21), itself a permutation of Ptolemy's Intense Chromatic<ul><li>[[Margo_Schulter\|Margo Schulter]] notes: "A modern form of Maqam Hijaz based on this tuning might be 1/1-12/11-14/11-4/3-32-18/11-16/9-2/1 ascending, and 1/1-12/11-14/11-4/3-3/2-128/81-16/9-2/1 descending (with the minor sixth maybe a bit smaller, say 11/7 or the like)."</li></ul></li><li>a septimal flavor of Sazkar, which has a minor third above the 1/1 ascending but a tone descending: thus ascending 1/1-7/6-11/9-4/3-3/2-27/16-11/6-2/1, and descending 1/1-9/8-11/9-4/3-3/2-27/16-11/6-2/1<ul><li>Margo Schulter adds: "I should caution that an Arab Rast, of which Sazkar is an offshoot, might usually have a Zalzalian third more like 27/22 or 16/13 rather than 11/9, so this may be more of a new tuning than a traditional Arab Sazkar."</li></ul></li><li>The 1-3-7-9-11 [[Combination_product_sets\|2(5 Dekany]] within a 12-tone [[Periodic_scale#Constant Structure\|constant structure]] (see [http://anaphoria.com/dekanyconstantstructures.pdf link]): 13, 911, 39, 17, 3711, 79, 311, 19, 3911, 711, 37, 111.</li><li>A 12f, {352/351, 364/363} 2.3.7.11.13 elf transversal: 28/27-9/8-13/11-9/7-4/3-11/8-3/2-14/9-22/13-16/9-27/14-2</li></ul> [[Category:11-limit]]		<ul><li>Genus(3711) 12/11, 14/11, 21/16, 16/11, 3/2, 7/4, 21/11, 2</li><li>Genus(37^211) 49/48, 49/44, 7/6, 14/11, 4/3, 16/11, 49/33, 49/32, 56/33, 7/4, 64/33, 2</li><li>Ptolemy's Intense Chromatic at 1/1 22/21 8/7 4/3 3/2 11/7 12/7 2/1 (disjunct form)</li><li>a conjunct Rast: 1/1 9/8 27/22 4/3 3/2 11/8 16/9 2/1 (note: actually 2.3.11 subgroup)</li><li>Qutb al-Din al-Shirazi's version of Hijaz: 1/1-12/11-14/11-4/3 (12:11-7:6-22:21), itself a permutation of Ptolemy's Intense Chromatic<ul><li>[[Margo_Schulter\|Margo Schulter]] notes: "A modern form of Maqam Hijaz based on this tuning might be 1/1-12/11-14/11-4/3-32-18/11-16/9-2/1 ascending, and 1/1-12/11-14/11-4/3-3/2-128/81-16/9-2/1 descending (with the minor sixth maybe a bit smaller, say 11/7 or the like)."</li></ul></li><li>a septimal flavor of Sazkar, which has a minor third above the 1/1 ascending but a tone descending: thus ascending 1/1-7/6-11/9-4/3-3/2-27/16-11/6-2/1, and descending 1/1-9/8-11/9-4/3-3/2-27/16-11/6-2/1<ul><li>Margo Schulter adds: "I should caution that an Arab Rast, of which Sazkar is an offshoot, might usually have a Zalzalian third more like 27/22 or 16/13 rather than 11/9, so this may be more of a new tuning than a traditional Arab Sazkar."</li></ul></li><li>The 1-3-7-9-11 [[Combination_product_sets\|2(5 Dekany]] within a 12-tone [[Periodic_scale#Constant Structure\|constant structure]] (see [http://anaphoria.com/dekanyconstantstructures.pdf link]): 13, 911, 39, 17, 3711, 79, 311, 19, 3911, 711, 37, 111.</li><li>A 12f, {352/351, 364/363} 2.3.7.11.13 elf transversal: 28/27-9/8-13/11-9/7-4/3-11/8-3/2-14/9-22/13-16/9-27/14-2</li></ul> [[Category:11-limit]]
	~~[[Category:gallery]]~~
	[[Category:rank_3]]		[[Category:rank_3]]
	[[Category:Lists of scales]]		[[Category:Lists of scales]]
	[[Category:todo:add_definition]]		[[Category:todo:add_definition]]