4L 2s - Revision history

FloraC: /* Scale tree */ review

2026-02-25T05:36:46Z

Scale tree: review

← Older revision		Revision as of 05:36, 25 February 2026
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	{{MOS tuning spectrum		{{MOS tuning spectrum
	\| 5/4 = [[Antikythera]]		\| 5/4 = [[Antikythera]]
	\| 13/8 = Golden [[~~Jubilismic_clan#Lemba\|~~lemba]]		\| 9/7 = [[Wizard]]
	\| 7/4 = Lemba ~~is around here~~		\| 13/8 = Golden [[lemba]]
	\| 2/1 = ~~Optimum rank range~~		\| 5/3 = Lemba
	\| 6/1 = [[Doublewide]] ~~is around here~~		\| 12/5 = [[Sruti]]
			\| 6/1 = [[Doublewide]]
	}}		}}

	[[Category:6-tone scales]]		[[Category:6-tone scales]]

Lucius Chiaraviglio: /* Low harmonic entropy scales */ Fix link to doublewide

2025-11-24T19:38:15Z

Low harmonic entropy scales: Fix link to doublewide

← Older revision		Revision as of 19:38, 24 November 2025
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	There are three scales with this [[MOS]] pattern that are significant minima of harmonic entropy. The first is [[antikythera]], or no-3's [[Diaschismic_family\|srutal/pajara]], which is srutal/pajara without any intervals containing 3 in their prime factorization, so it becomes a 2.5.9 or 2.5.7.9 subgroup temperament. This means that the generator is twice that of srutal/pajara (210–220{{c}} rather than 105–110), since odd numbers of generators are only needed for intervals with 3. So this is a basically "whole tone" scale, but made uneven so some 2-step intervals are 5/4 and others are 9/7.		There are three scales with this [[MOS]] pattern that are significant minima of harmonic entropy. The first is [[antikythera]], or no-3's [[Diaschismic_family\|srutal/pajara]], which is srutal/pajara without any intervals containing 3 in their prime factorization, so it becomes a 2.5.9 or 2.5.7.9 subgroup temperament. This means that the generator is twice that of srutal/pajara (210–220{{c}} rather than 105–110), since odd numbers of generators are only needed for intervals with 3. So this is a basically "whole tone" scale, but made uneven so some 2-step intervals are 5/4 and others are 9/7.

	The second is [[Dicot_family#Decimal\|decimal]], in which two generators make a 4/3, and the third is [[Jubilismic_clan\|~~Doublewide~~]], in which the generator is 7/6 so the period minus the generator is 6/5.		The second is [[Dicot_family#Decimal\|decimal]], in which two generators make a 4/3, and the third is [[Jubilismic_clan#Doublewide\|doublewide]], in which the generator is 7/6 so the period minus the generator is 6/5.

	== Scale properties ==		== Scale properties ==

Lucius Chiaraviglio: /* Low harmonic entropy scales */ Fix link to decimal

2025-11-24T19:36:16Z

Low harmonic entropy scales: Fix link to decimal

← Older revision		Revision as of 19:36, 24 November 2025
Line 21:		Line 21:
	There are three scales with this [[MOS]] pattern that are significant minima of harmonic entropy. The first is [[antikythera]], or no-3's [[Diaschismic_family\|srutal/pajara]], which is srutal/pajara without any intervals containing 3 in their prime factorization, so it becomes a 2.5.9 or 2.5.7.9 subgroup temperament. This means that the generator is twice that of srutal/pajara (210–220{{c}} rather than 105–110), since odd numbers of generators are only needed for intervals with 3. So this is a basically "whole tone" scale, but made uneven so some 2-step intervals are 5/4 and others are 9/7.		There are three scales with this [[MOS]] pattern that are significant minima of harmonic entropy. The first is [[antikythera]], or no-3's [[Diaschismic_family\|srutal/pajara]], which is srutal/pajara without any intervals containing 3 in their prime factorization, so it becomes a 2.5.9 or 2.5.7.9 subgroup temperament. This means that the generator is twice that of srutal/pajara (210–220{{c}} rather than 105–110), since odd numbers of generators are only needed for intervals with 3. So this is a basically "whole tone" scale, but made uneven so some 2-step intervals are 5/4 and others are 9/7.

	The second is [[Dicot_family\|decimal]], in which two generators make a 4/3, and the third is [[Jubilismic_clan\|Doublewide]], in which the generator is 7/6 so the period minus the generator is 6/5.		The second is [[Dicot_family#Decimal\|decimal]], in which two generators make a 4/3, and the third is [[Jubilismic_clan\|Doublewide]], in which the generator is 7/6 so the period minus the generator is 6/5.

	== Scale properties ==		== Scale properties ==

Lucius Chiaraviglio: /* Scale tree */ Add links to temperaments

2025-11-22T23:48:35Z

Scale tree: Add links to temperaments

← Older revision		Revision as of 23:48, 22 November 2025
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	== Scale tree ==		== Scale tree ==
	{{MOS tuning spectrum		{{MOS tuning spectrum
	\| 5/4 = Antikythera		\| 5/4 = [[Antikythera]]
	\| 13/8 = Golden lemba		\| 13/8 = Golden [[Jubilismic_clan#Lemba\|lemba]]
	\| 7/4 = Lemba is around here		\| 7/4 = Lemba is around here
	\| 2/1 = Optimum rank range		\| 2/1 = Optimum rank range
	\| 6/1 = Doublewide is around here		\| 6/1 = [[Doublewide]] is around here
	}}		}}

	[[Category:6-tone scales]]		[[Category:6-tone scales]]

ArrowHead294 at 19:00, 3 March 2025

2025-03-03T19:00:24Z

← Older revision		Revision as of 19:00, 3 March 2025
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	== Scale properties ==		== Scale properties ==
	{{TAMNAMS use}}		{{TAMNAMS use}}
	{{MOS ~~data~~}}
			=== Intervals ===
			{{MOS intervals}}

			=== Generator chain ===
			{{MOS genchain}}

			=== Modes ===
			{{MOS mode degrees}}

	== Scale tree ==		== Scale tree ==

ArrowHead294 at 18:01, 28 February 2025

2025-02-28T18:01:52Z

← Older revision		Revision as of 18:01, 28 February 2025
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	{{MOS intro}}		{{MOS intro}}

	4L 2s can be seen as a [[Warped diatonic\|warped diatonic scale]], where one large step of diatonic (5L 2s) is removed, or as the equal-tempered whole-tone scale ([[6edo]]), but with two "whole tones" that are smaller than the others.		4L 2s can be seen as a [[Warped diatonic\|warped diatonic scale]], where one large step of diatonic (5L 2s) is removed, or as the equal-tempered whole-tone scale ([[6edo]]), but with two "whole tones" that are smaller than the others.

	Scales with the true MOS pattern are always [[Rothenberg propriety\|proper]], because there is only one small step per period. In addition, there are near-MOS patterns, such as LLLsLs, in which the period is the only generic interval that has more than two specific representatives. The near-MOS is only proper if the generator is smaller than 2\10 of an octave (240 ~~cents~~).		Scales with the true MOS pattern are always [[Rothenberg propriety\|proper]], because there is only one small step per period. In addition, there are near-MOS patterns, such as LLLsLs, in which the period is the only generic interval that has more than two specific representatives. The near-MOS is only proper if the generator is smaller than 2\10 of an octave (240{{c}}).

	== Name==		== Name ==
	[[TAMNAMS]] suggests the temperament-agnostic name '''citric''' for this scale.		[[TAMNAMS]] suggests the temperament-agnostic name '''citric''' for this scale.

	==Theory==		== Theory ==
			=== Low harmonic entropy scales ===
	===Low harmonic entropy scales===		There are three scales with this [[MOS]] pattern that are significant minima of harmonic entropy. The first is [[antikythera]], or no-3's [[Diaschismic_family\|srutal/pajara]], which is srutal/pajara without any intervals containing 3 in their prime factorization, so it becomes a 2.5.9 or 2.5.7.9 subgroup temperament. This means that the generator is twice that of srutal/pajara (210–220{{c}} rather than 105–110), since odd numbers of generators are only needed for intervals with 3. So this is a basically "whole tone" scale, but made uneven so some 2-step intervals are 5/4 and others are 9/7.
	There are three scales with this [[~~MOSScales\|~~MOS]] pattern that are significant minima of harmonic entropy. The first is [[antikythera]], or no-3's [[Diaschismic_family\|srutal/pajara]], which is srutal/pajara without any intervals containing 3 in their prime factorization, so it becomes a 2.5.9 or 2.5.7.9 subgroup temperament. This means that the generator is twice that of srutal/pajara (~~210-220 cents~~ rather than ~~105-110~~), since odd numbers of generators are only needed for intervals with 3. So this is a basically "whole tone" scale, but made uneven so some 2-step intervals are 5/4 and others are 9/7.

	The second is [[Dicot_family\|decimal]], in which two generators make a 4/3, and the third is [[Jubilismic_clan\|Doublewide]], in which the generator is 7/6 so the period minus the generator is 6/5.		The second is [[Dicot_family\|decimal]], in which two generators make a 4/3, and the third is [[Jubilismic_clan\|Doublewide]], in which the generator is 7/6 so the period minus the generator is 6/5.

	==~~Modes~~==		== Scale properties ==
	{{MOS ~~mode degrees~~}}		{{TAMNAMS use}}
			{{MOS data}}

	== ~~Intervals~~ ==		== Scale tree ==
	{{MOS ~~intervals~~}}		{{MOS tuning spectrum
			\| 5/4 = Antikythera
			\| 13/8 = Golden lemba
			\| 7/4 = Lemba is around here
			\| 2/1 = Optimum rank range
			\| 6/1 = Doublewide is around here
			}}

	~~==Scale tree==~~
	~~{{Scale tree\|Comments=5/4: Antikythera;~~
	~~13/8: Golden lemba;~~
	~~7/4: Lemba is around here;~~
	~~2/1: Optimum rank range;~~
	~~6/1: Doublewide is around here}}~~
	[[Category:6-tone scales]]		[[Category:6-tone scales]]

BudjarnLambeth: == Intervals == {{MOS intervals}}

2024-12-16T00:14:43Z

== Intervals == {{MOS intervals}}

← Older revision		Revision as of 00:14, 16 December 2024
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	\| Name = citric		\| Name = citric
	}}		}}
			{{MOS intro}}

	~~{{MOS intro}}~~
	4L 2s can be seen as a [[Warped diatonic\|warped diatonic scale]], where one large step of diatonic (5L 2s) is removed, or as the equal-tempered whole-tone scale ([[6edo]]), but with two "whole tones" that are smaller than the others.		4L 2s can be seen as a [[Warped diatonic\|warped diatonic scale]], where one large step of diatonic (5L 2s) is removed, or as the equal-tempered whole-tone scale ([[6edo]]), but with two "whole tones" that are smaller than the others.

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	==Modes==		==Modes==
	{{MOS mode degrees}}		{{MOS mode degrees}}

			== Intervals ==
			{{MOS intervals}}

	==Scale tree==		==Scale tree==

Nick Vuci: /* Modes */ added MOS mode degrees template

2024-07-24T23:18:13Z

Modes: added MOS mode degrees template

← Older revision		Revision as of 23:18, 24 July 2024
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	==Modes==		==Modes==
	{{MOS ~~modes~~}}		{{MOS mode degrees}}

	==Scale tree==		==Scale tree==
	{{Scale tree\|Comments=5/4: Antikythera;		{{Scale tree\|Comments=5/4: Antikythera;

Ganaram inukshuk: /* Scale tree */ Added scale tree template, transferred all entries (except for e and pi tunings)

2024-04-24T19:19:17Z

Scale tree: Added scale tree template, transferred all entries (except for e and pi tunings)

Show changes

Ganaram inukshuk: /* Temperament interpretations */ This section name was "Low harmonic entropy scales" on other similarly formatted pages

2023-10-07T09:19:18Z

Temperament interpretations: This section name was "Low harmonic entropy scales" on other similarly formatted pages

← Older revision		Revision as of 09:19, 7 October 2023
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	==Theory==		==Theory==

	===~~Temperament interpretations~~===		===Low harmonic entropy scales===
	There are three scales with this [[MOSScales\|MOS]] pattern that are significant minima of harmonic entropy. The first is [[antikythera]], or no-3's [[Diaschismic_family\|srutal/pajara]], which is srutal/pajara without any intervals containing 3 in their prime factorization, so it becomes a 2.5.9 or 2.5.7.9 subgroup temperament. This means that the generator is twice that of srutal/pajara (210-220 cents rather than 105-110), since odd numbers of generators are only needed for intervals with 3. So this is a basically "whole tone" scale, but made uneven so some 2-step intervals are 5/4 and others are 9/7.		There are three scales with this [[MOSScales\|MOS]] pattern that are significant minima of harmonic entropy. The first is [[antikythera]], or no-3's [[Diaschismic_family\|srutal/pajara]], which is srutal/pajara without any intervals containing 3 in their prime factorization, so it becomes a 2.5.9 or 2.5.7.9 subgroup temperament. This means that the generator is twice that of srutal/pajara (210-220 cents rather than 105-110), since odd numbers of generators are only needed for intervals with 3. So this is a basically "whole tone" scale, but made uneven so some 2-step intervals are 5/4 and others are 9/7.