Subgroup temperaments - Revision history

FloraC: Merge from ambulatory

2026-10-17T10:26:03Z

Merge from ambulatory

@@ Line 1,722: / Line 1,722: @@
 == 3/2.5/4.… subgroups ==
 === Poseidon ===
 {{Todo|inline=1|rename|resolve temperament name conflict}}

FloraC: Merge from auk

2026-10-16T13:59:01Z

Merge from auk

← Older revision		Revision as of 13:59, 16 October 2026
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	* Subgroup CWE: ~3/2 = 701.9550{{c}}, ~9/7 = 440.4501{{c}}		* Subgroup CWE: ~3/2 = 701.9550{{c}}, ~9/7 = 440.4501{{c}}

	[[Optimal ET sequence]]: [[8edf\|8]], [[27edf\|28]], [[35edf\|35]], [[43edf\|43]], [[51edf\|*51]], [[110edf\|110…]]		Optimal ET sequence: [[8edf\|8]], [[27edf\|28]], [[35edf\|35]], [[43edf\|43]], [[51edf\|*51]], [[110edf\|110…]]
	: <nowiki/>* Wart for 3/2		: <nowiki/>* Wart for 3/2

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	== Other 3/2-repeating subgroups ==		== Other 3/2-repeating subgroups ==
	=== Auk ===		=== Auk ===
			Auk tempers out the comma [[87808/85293]] in the 3/2.7.13 subgroup, equating three instances of [[7/1]] (fifth-reduced to [[112/81]]) with [[13/1]] (fifth-reduced to 832/729). The generator when reduced is a superfourth in the neighborhood of 550 cents, representing 112/81. It possesses nontrivial mos of the families {{MOS scalesig\|1L 3s<3/2>}} ("neptunian"), {{MOS scalesig\|4L 1s<3/2>}}, {{MOS scalesig\|5L 4s<3/2>}}, and {{MOS scalesig\|9L 5s<3/2>}}. [[9edf]] is a particularly good equal temperament that [[support]]s it. It contains tempered versions of the chords 1–832/729–112/81 (fifth-reduction of 1:7:13) and 1–63/52–112/81 (fifth-reduction of 1/(13:7:1)).

	[[Subgroup]]: 3/2.7.13		[[Subgroup]]: 3/2.7.13

FloraC: Fixes. Recategorize

2026-10-16T13:48:55Z

Fixes. Recategorize

← Older revision		Revision as of 13:48, 16 October 2026
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	* [[Tp tuning\|Subgroup]] [[CWE]]: ~5/2 = 1586.3137{{c}}, ~44/25 = 991.6570{{c}}		* [[Tp tuning\|Subgroup]] [[CWE]]: ~5/2 = 1586.3137{{c}}, ~44/25 = 991.6570{{c}}

	[[Optimal ET sequence]]: [[~~8edf~~\|8]], [[~~83edf~~\|83…]], [[~~91edf~~\|91…]], …, [[~~155edf~~\|155…]], [[~~163edf~~\|*163…]]		[[Optimal ET sequence]]: [[8ed5/2\|8]], [[83ed5/2\|83…]], [[91ed5/2\|91…]], …, [[155ed5/2\|155…]], [[163ed5/2\|*163…]]
	: <nowiki/>* Wart for 5/2		: <nowiki/>* Wart for 5/2

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	<references/>		<references/>

	[[Category:~~Subgroup~~ temperaments\| ]] ~~<!-- main article -->~~		[[Category:Catalogs of subgroup temperaments]]
	[[Category:~~Temperament collections]]~~		[[Category:Catalogs of rank-2 temperaments]]
	~~[[Category:Rank~~ 2]]
	~~{{Todo\| review \| cleanup }}~~

FloraC: Add some info from 3L 2s (3/2-equivalent)

2026-10-16T13:41:54Z

Add some info from 3L 2s (3/2-equivalent)

← Older revision		Revision as of 13:41, 16 October 2026
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	=== Semiwolf ===		=== Semiwolf ===
			Semiwolf provides an interpretation for the {{MOS scalesig\|link=true\|3L 2s<3/2>}} [[mos scale]]. The name ''semiwolf'' comes from two [[~]][[7/6]] generators approximating a [[27/20]] wolf fourth. This is further extended to the 11-limit in two interpretations: semilupine, where 2 major 2-mossteps (LL) equal 11/9, and hemilycan. where 1 major and 2 minor 2-mossteps (sLs) equal 11/9. Basic [[8edf]] fits both extensions.

	[[Subgroup]]: 3/2.5/2.7/4		[[Subgroup]]: 3/2.5/2.7/4

FloraC: Cleanup (19/19)

2026-10-16T13:34:31Z

Cleanup (19/19)

← Older revision		Revision as of 13:34, 16 October 2026
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	=== Hemihemi ===		=== Hemihemi ===
			Named by {{u\|CompactStar}} in 2023, hemihemi tempers out 10976/10935, the [[hemimage comma]], in the 3/2.5/2.7/2 subgroup.

	[[Subgroup]]: 3/2.5/2.7/2		[[Subgroup]]: 3/2.5/2.7/2

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	=== Halftone ===		=== Halftone ===
	{{Main\| Halftone }}		{{Main\| Halftone }}

			Named by {{u\|CompactStar}} in 2023, halftone tempers out 9604/9375 in the 3/2.5/2.7/2 subgroup.

	[[Subgroup]]: 3/2.5/2.7/2		[[Subgroup]]: 3/2.5/2.7/2
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	== 3/2.5/4.… subgroups ==		== 3/2.5/4.… subgroups ==
	=== Poseidon ===		=== Poseidon ===
	~~'''This~~ temperament ~~will be subjected to renaming due to a~~ conflict~~.'''~~		{{Todo\|inline=1\|rename\|resolve temperament name conflict}}

	[[Subgroup]]: 3/2.5/4.11/8		[[Subgroup]]: 3/2.5/4.11/8

	[[Comma list]]: 121/120		[[Comma list]]: 121/120 ({{monzo\| -1 -1 2 }})

	{{Mapping\|legend=3\| 1 1 1 \| 0 2 -1 }}]		{{Mapping\|legend=3\| 1 1 1 \| 0 -2 -1 }}]
			: mapping generators: ~3/2, ~12/11

	: [[~~gencom~~]]: [3/2 12/11~~; 121/120]~~		[[Optimal tuning]]s:
			* [[Tp tuning\|Subgroup]] [[WE]]: ~3/2 = 701.8452{{c}}, ~12/11 = 157.8337{{c}}
			* [[Tp tuning\|Subgroup]] [[CWE]]: ~3/2 = 701.9550{{c}}, ~12/11 = 157.8574{{c}}

	[[Optimal ~~tuning~~]] ([[~~Tp tuning~~\|~~subgroup POTE~~]]~~): ~3/2~~, ~~~12/11 = 158.29~~		[[Optimal ET sequence]]: [[4edf\|4]], [[5edf\|5]], [[9edf\|9]], [[22edf\|22]], [[31edf\|31]], [[40edf\|40]]
			: <nowiki/>* Wart for 3/2
	~~{{Optimal ET sequence~~\|~~legend=1~~\|9, ~~5, 13,~~ 22, ~~14,~~ 31, ~~17, 6~~[~~+5/4~~]~~, 23, 40, 35, 21[-5~~/~~4], 19[+5~~/~~4], 49}}~~

	== Other 3/2-repeating subgroups ==		== Other 3/2-repeating subgroups ==
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	{{Mapping\|legend=3\| 1 0 -8 \| 0 1 3 }}		{{Mapping\|legend=3\| 1 0 -8 \| 0 1 3 }}
			: mapping generators: ~3/2, ~7

	: ~~sval mapping generators~~: ~3/2, ~7		[[Optimal tuning]]s:
			* [[Tp tuning\|Subgroup]] [[WE]]: ~3/2 = 706.5197{{c}}, ~112/81 = 538.8834{{c}}
			* [[Tp tuning\|Subgroup]] [[CWE]]: ~3/2 = 701.9550{{c}}, ~112/81 = 543.8799{{c}}

	[[Optimal ~~tuning~~]] ~~(subgroup~~ [[~~CTE~~]]~~): ~3/2 = 1\1edf, ~28/9 = 1950.859~~		[[Optimal ET sequence]]: [[4edf\|4]], [[13edf\|13]], [[17edf\|17df]], [[21edf\|21df]], [[38edf\|38ddff]], [[59edf\|59ddffff]]
			: <nowiki/>* Wart for 3/2
	~~Supporting ETs: 5~~, ~~6[+~~13], ~~7[-7, -13~~], 9, 11[~~+13], 13, 14, 17~~[~~-7, -13],~~ ~~19[+13~~]~~, 21[-7, -13~~], 22[~~-7], 23~~[~~+13],~~ ~~25[-7, -13~~]~~, 31[-7~~]
	: <nowiki />* ~~wart~~ for 3/2

	=== Doubleton ===		=== Doubleton ===
	[[Subgroup]]: 3/2.7.13		[[Subgroup]]: 3/2.7.13

	[[Comma list]]: 1352/1323		[[Comma list]]: 1352/1323 ({{monzo\| -3 -2 2 }})

	{{Mapping\|legend=3\| 2 0 3 \| 0 1 1 }}		{{Mapping\|legend=3\| 2 0 3 \| 0 1 1 }}
			: mapping generators: ~26/21, ~7

	: ~~sval mapping generators~~: ~26/21, ~7		[[Optimal tuning]]s:
			* [[Tp tuning\|Subgroup]] [[WE]]: ~26/21 = 352.6518{{c}}, ~112/81 = 552.6017{{c}}
			* [[Tp tuning\|Subgroup]] [[CWE]]: ~26/21 = 350.9775{{c}}, ~112/81 = 565.7161{{c}}

	[[Optimal ~~tuning~~]] ~~(subgroup~~ [[~~CTE~~]]~~): ~26/21 = 1\2edf, ~28/9 = 1971.772~~		[[Optimal ET sequence]]: [[4edf\|4]], [[10edf\|10]], [[14edf\|14f]], [[46edf\|46dfff]]
			: <nowiki/>* Wart for 3/2
	~~Supporting ETs: 6~~, 10~~, 16, 14[-13~~], 8[~~+7], 22, 18~~[~~-13],~~ ~~26, 24[-13~~]~~, 28[+7~~], 20[~~+7], 36~~[~~-13],~~ ~~12[+7, +13~~]~~, 34[-13~~]
	: <nowiki />* ~~wart~~ for 3/2

	== 5/2-repeating subgroups ==		== 5/2-repeating subgroups ==
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	[[Subgroup]]: 5/2.7.11		[[Subgroup]]: 5/2.7.11

	[[Comma list]]: {{monzo\| 11 1 -5 }}		[[Comma list]]: 341796875/329832448 ({{monzo\| 11 1 -5 }})

	{{Mapping\|legend=3\| 1 ~~4 3~~ \| 0 -5 -1 }}		{{Mapping\|legend=3\| 1 -1 2 \| 0 5 1 }}
			: mapping generators: ~5/2, ~44/25

	: [[~~gencom~~]]: [5/2 ~~125~~/~~88; 341796875/329832448]~~		[[Optimal tuning]]s:
			* [[Tp tuning\|Subgroup]] [[WE]]: ~5/2 = 1582.2958{{c}}, ~44/25 = 990.1327{{c}}
	[[~~Optimal~~ tuning]] ([[~~Tp tuning\|subgroup POTE~~]]): ~5/2 = 1586.3137, ~~~125~~/88 = ~~593~~.~~6668~~		* [[Tp tuning\|Subgroup]] [[CWE]]: ~5/2 = 1586.3137{{c}}, ~44/25 = 991.6570{{c}}

	~~Supporting ETs~~: 5[~~-7],~~ 8, ~~19[+7~~], ~~21[-7~~], 27[~~+7], 29~~[~~-7],~~ ~~35[+7~~]~~, 43[+7~~], 37[~~-7], 51~~[~~+7, +11],~~ ~~45[-7~~]~~, 59[+7, +11~~]		[[Optimal ET sequence]]: [[8edf\|8]], [[83edf\|83…]], [[91edf\|91…]], …, [[155edf\|155…]], [[163edf\|*163…]]
	: <nowiki />* ~~wart~~ for 5/2		: <nowiki/>* Wart for 5/2

	== See also ==		== See also ==

FloraC: Cleanup (18/)

2026-10-16T13:08:51Z

Cleanup (18/)

Show changes

FloraC: Cleanup (17/)

2026-10-16T10:44:05Z

Cleanup (17/)

← Older revision		Revision as of 10:44, 16 October 2026
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	== 2.25/7.… subgroups ==		== 2.25/7.… subgroups ==
	=== Supramine ===		=== Supramine ===
	~~This extension~~ approximates the 14:17:19:23:25 pentad in just six generator steps, at the cost of some accuracy. 25edo remains a strong tuning.		Supramine extends [[#Supramin\|supramin]] and approximates the 14:17:19:23:25 pentad in just six generator steps, at the cost of some accuracy. 25edo remains a strong tuning.

	Subgroup: 2.17/7.19/7.23/7		[[Subgroup]]: 2.25/7.17/7.19/7.23/7

	Comma list: [[323/322]], [[392/391]]		[[Comma list]]: 323/322, 392/391, 476/475

	~~Subgroup-val mapping:~~ {{Mapping\| 1 0 4 3 \| 0 1 -2 -1 }}		{{Mapping\|legend=3\| 1 -2 0 4 3 \| 0 3 1 -2 -1 }}
			: mapping generators: ~2, ~17/14

	Optimal ~~tunings~~:		[[Optimal tuning]]s:
	* ~~Subgroup WE: ~2 = 1199.871{{c}}, ~17/14 = 336.243{{c}}~~		* [[Tp tuning\|Subgroup]] [[WE]]: ~2 = 1199.7558{{c}}, ~17/14 = 335.4278{{c}}
	* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 336.296{{c}}		* [[Tp tuning\|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~17/14 = 335.4885{{c}}

	~~{{Optimal ET sequence\|legend=0~~\| ~~7, 18, 25 }}~~

	~~Badness (Sintel): 0.029~~

	~~==== 2.25/7.17/7.19/7.23/7 subgroup ====~~

	Subgroup~~: 2.25/7.17/7.19/7.23/7~~

	~~Comma list: [[323/322~~]], [[~~392/391]], [[476/475~~]]

	~~Subgroup-val mapping: {{Mapping\| 1 -2 0 4 3 \| 0 3 1 -2 -1 }}~~

	~~Optimal tunings:~~
	* Subgroup WE: ~2 = 1199.~~757~~{{c}}, ~17/14 = 335.~~428~~{{c}}
	* Subgroup CWE: ~2 = 1200.~~000~~{{c}}, ~17/14 = 335.~~479~~{{c}}

	{{Optimal ET sequence\|legend=0\| 7, 18, 25 }}		{{Optimal ET sequence\|legend=1\| 7, 18, 25, 68, 93 }}

	Badness (Sintel): 0.053		[[Badness]] (Sintel): 0.053

	== 2.11/5.… subgroups ==		== 2.11/5.… subgroups ==
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	{{See also\| 5L 3s/Temperaments #Petrtri }}		{{See also\| 5L 3s/Temperaments #Petrtri }}

	Petrtri can be described as 3 &~~amp;~~ 5 temperament in the 2.11/5.13/5 subgroup.		Petrtri can be described as 3 & 5 temperament in the 2.11/5.13/5 subgroup and tempers out [[2200/2197]]. It restricts [[#Tridec\|tridec]]. The tonic and the first two generator steps make a [[petrmic triad]], hence the name.

	[[Subgroup]]: 2.11/5.13/5		[[Subgroup]]: 2.11/5.13/5

	[[Comma list]]: [[2200/2197]]		[[Comma list]]: 2200/2197

	{{Mapping\|legend=3\| 1 0 1\| 0 3 1 }}		{{Mapping\|legend=3\| 1 0 1 \| 0 3 1 }}

	{{Mapping\|legend=5\| 1 0 -1/3 0 -1/3 2/3 \| 0 0 -4/3 0 5/3 -1/3 }}		{{Mapping\|legend=5\| 1 0 -1/3 0 -1/3 2/3 \| 0 0 -4/3 0 5/3 -1/3 }}
	: ~~[[gencom]]~~: [2 13/10~~; 2200/2197]~~		: mapping generators: ~2, ~13/10

	[[Optimal tuning]] ([[Tp tuning\|~~subgroup~~]] [[~~POTE~~]]): ~2 = ~~1\1~~, ~13/10 = 455.~~012~~		[[Optimal tuning]]s:
			* [[Tp tuning\|Subgroup]] [[WE]]: ~2 = 1199.9572{{c}}, ~13/10 = 454.9958{{c}}
			* [[Tp tuning\|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~13/10 = 455.0039{{c}}

	{{Optimal ET sequence\|legend=1\| 21, 29, 153, 182, 211, 240, 269~~, 298, 327, 356, 385~~, 509, ~~741c~~, ~~1126c~~ }}		{{Optimal ET sequence\|legend=1\| 5, 8, 21, 29, 153, 182, 211, 240, 269, 509, 778, 1287 }}

	[[Tp tuning #T2 tuning\|RMS error]]: 0.0749 cents		[[Tp tuning #T2 tuning\|RMS error]]: 0.0749 cents
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	== Other 2-repeating subgroups ==		== Other 2-repeating subgroups ==
	=== Supramin ===		=== Supramin ===
	~~This~~ is a remarkable low-complexity microtemperament that contains the 14:17:19 triad within just four generator steps. An excellent tuning is [[25edo]], which provides an accurate yet tone-efficient tuning of this temperament. It was named by ~~[[User:Overthink~~\|Overthink]] in 2026 after the fact that the generator is a [[17/14]] supraminor third, two of which reach [[28/19]]. It is related to [[cohemimabila]].		Supramin restricts [[#Supramine\|supramine]] and is a remarkable low-complexity microtemperament that contains the 14:17:19 triad within just four generator steps. An excellent tuning is [[25edo]], which provides an accurate yet tone-efficient tuning of this temperament. It was named by {{u\|Overthink}} in 2026 after the fact that the generator is a [[17/14]] supraminor third, two of which reach [[28/19]]. It is related to [[Mabila family #Cohemimabila\|cohemimabila]].

	[[Subgroup]]: 2.17/7.19/7		[[Subgroup]]: 2.17/7.19/7

	[[Comma list]]: [[5491/5488]] ({{~~Monzo~~\| -4 2 1 }})		[[Comma list]]: 5491/5488 ({{monzo\| -4 2 1 }})

	{{Mapping\|legend=3\| 1 0 4 \| 0 1 -2 }}		{{Mapping\|legend=3\| 1 0 4 \| 0 1 -2 }}
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	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[Tp tuning\|Subgroup]] [[WE]]: ~2 = 1200.~~022~~{{c}}, ~17/14 = 335.~~793~~{{c}}		* [[Tp tuning\|Subgroup]] [[WE]]: ~2 = 1200.0220{{c}}, ~17/14 = 335.7933{{c}}
	* [[Tp tuning\|Subgroup]] [[CWE]]: ~2 = 1200.~~000~~{{c}}, ~17/14 = 335.~~785~~{{c}}		* [[Tp tuning\|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~17/14 = 335.7852{{c}}

	{{Optimal ET sequence\|legend=1\| 7, 18, 25 }}		{{Optimal ET sequence\|legend=1\| 7, 18, 25, 118, 143, 168, 193, 218, 461, 679, 1576, 2255* }}

	[[Badness]] (Sintel): 0.005		[[Badness]] (Sintel): 0.005

	== 3/2.5/2.… subgroups ==		== 3/2.5/2.… subgroups ==
	{{Main\|Half-prime subgroup}}		{{Main\| Half-prime subgroup }}

	=== Hemihemi ===		=== Hemihemi ===

FloraC: Cleanup (16/). Thanks, Eliora

2026-10-16T10:07:00Z

Cleanup (16/). Thanks, Eliora

Show changes

FloraC at 09:48, 16 October 2026

2026-10-16T09:48:26Z

Show changes

Eliora: /* Gariberttet */ oops

2026-10-15T20:46:55Z

Gariberttet: oops

← Older revision		Revision as of 20:46, 15 October 2026
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	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[WE]]: ~1199.998, ~25/21 = 293.739		* Subgroup [[WE]]: ~1199.998, ~25/21 = 293.739
	: error map: {{val \| -0.002, -3.141, 1.826 }}		: error map: {{val \| -0.002, -3.141, 1.826 }}
	* [[CWE]]: ~1200.000, ~25/21 = 293.739		* Subgroup [[CWE]]: ~1200.000, ~25/21 = 293.739
	: error map: {{val \| 0.000, -3.140, 1.827 }}		: error map: {{val \| 0.000, -3.140, 1.827 }}