2.3.7.11 subgroup - Revision history

Domin at 09:47, 14 August 2026

2026-08-14T09:47:37Z

← Older revision		Revision as of 09:47, 14 August 2026
Line 38:		Line 38:

	[[Olympian]] equates 33/32 with a stack of two 64/63's. It is very accurate yet still easy to notate on the staff, using the septimal comma as spacer added to the chain of fifths to finds intervals of 7 and 11.		[[Olympian]] equates 33/32 with a stack of two 64/63's. It is very accurate yet still easy to notate on the staff, using the septimal comma as spacer added to the chain of fifths to finds intervals of 7 and 11.

			== Music ==
			; [[Domin]]
			* [https://www.youtube.com/watch?v=Hr4WtKdvs_4 ''Zutsz (2.3.7.11 JI)''] (2024)
			* [https://www.youtube.com/watch?v=jjD9IGhCVFQ ''Tzakh (2.3.7.11 JI)''] (2024)


	[[Category:Just intonation subgroups\|#]]		[[Category:Just intonation subgroups\|#]]

FloraC: /* Regular temperaments */ expand

2026-08-03T03:43:08Z

Regular temperaments: expand

← Older revision		Revision as of 03:43, 3 August 2026
Line 24:		Line 24:

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	~~{{Main\|Tour~~ of ~~regular temperaments#Temperaments defined~~ by an 11-~~limit~~ comma}}		[[Supra]], which extends [[archy]], provides a simple yet high-damage approximation to the subgroup. It is generated by a perfect fifth, tuned a little sharp so that two make [[8/7]][[~]][[9/8]] and six make [[16/11]], tempering out [[64/63]] and [[99/98]]. Alternatively, [[suhajira]] can be considered an extension of archy that adds neutral intervals, with the perfect fifth split into two neutral third generators, each representing [[11/9]]~[[27/22]], thus tempering out 64/63 and [[243/242]].
	~~{{Todo\|inline=1\|expand}}~~
			[[Skwares]] takes the 11/9~27/22 neutral third, adds an octave to it and splits it in halves for ~[[11/7]], tempering out 99/98 and 243/242.

			[[Radon]], which adds prime 11 to [[slendric]] by tempering out [[896/891]], provides a more complex entry, well represented by 41edo and 46edo. A similar temperament at this level is [[hemif]], which tempers out 243/242 and 896/891 and can be tuned to 58edo. In both cases, the diatonic major third represents [[14/11]].

			On the high-accuracy side, [[gary]] is an important temperament that finds 7 and 11 far into the [[chain of fifths]].

			=== Rank-3 temperaments ===
			[[Parapyth]] equates [[28/27]] with [[33/32]] and uses this interval as a spacer added to the chain of fifths to finds intervals of 7 and 11.

			[[Symbiotian]], which makes 33/32 and 28/27 sum to the [[Pythagorean apotome]], is an efficient high-accuracy counterpart of parapyth.

			[[Olympian]] equates 33/32 with a stack of two 64/63's. It is very accurate yet still easy to notate on the staff, using the septimal comma as spacer added to the chain of fifths to finds intervals of 7 and 11.

	[[Category:Just intonation subgroups\|#]]		[[Category:Just intonation subgroups\|#]]

Fredg999: Text replacement - "[[Combination_product_sets|" to "[[Combination product set|"

2026-07-27T20:36:03Z

Text replacement - "[[Combination_product_sets|" to "[[Combination product set|"

← Older revision		Revision as of 20:36, 27 July 2026
Line 16:		Line 16:
	* 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		* 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
	** 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."		** 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."
	* 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.		* The 1-3-7-9-11 [[Combination product set\|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.
	* (with prime 13 added) 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		* (with prime 13 added) 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

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
Line 1:		Line 1:
	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
Line 3:		Line 3:
	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
Line 21:		Line 21:
	== 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
Line 21:		Line 21:
	== 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
Line 27:		Line 27:
	{{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
Line 21:		Line 21:
	== 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
Line 29:		Line 29:
	[[Category:Subgroup]]		[[Category:Subgroup]]
	[[Category:11-limit]]		[[Category:11-limit]]
	[[Category:Rank 3]]		[[Category:Rank 4]]
	[[Category:Lists of scales]]		[[Category:Lists of scales]]