No-twos subgroup temperaments - Revision history

Overthink: /* BPS */ directly link instead of redirect

2026-05-02T03:53:17Z

BPS: directly link instead of redirect

← Older revision		Revision as of 03:53, 2 May 2026
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	== BPS ==		== BPS ==
	{{main\|~~Bohlen–Pierce–Stearns~~}}		{{main\|BPS}}

	For extensions to this temperament that include the octave, see [[Sensamagic clan]]. Non-octave extensions will be documented below.		For extensions to this temperament that include the octave, see [[Sensamagic clan]]. Non-octave extensions will be documented below.

Xenllium: /* 3.5.7 subgroup temperaments */

2026-03-08T08:17:36Z

3.5.7 subgroup temperaments

Show changes

Xenllium: /* 3.5.7 subgroup temperaments */

2026-03-08T07:04:22Z

3.5.7 subgroup temperaments

← Older revision		Revision as of 07:04, 8 March 2026
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	Badness (Sintel): 0.841		Badness (Sintel): 0.841

			== Bohlenic ==
			This temperament is identical to [[13edt]] (equal-tempered [[Bohlen–Pierce scale]]), but has an independent generator for 11.

			[[Subgroup]]: 3.5.7.11

			[[Comma list]]: 245/243, 3125/3087

			{{Mapping\|legend=2\| 13 19 23 0 \| 0 0 0 1 }}

			: sval mapping generators: ~27/25, ~11

			[[Optimal tuning]]s:
			* [[CTE]]: ~27/25 = 146.304¢ (1 ⧵ b13), ~11 = 4151.318¢
			* [[CWE]]: ~27/25 = 146.304¢ (1 ⧵ b13), ~11 = 4147.705¢

			[[Optimal ET sequence]]: [[13edt\|b13]], [[26edt\|b26]], [[39edt\|b39]]

			[[Badness]] (Sintel): 0.499

			=== Full no-twos 13-limit ===
			[[Subgroup]]: 3.5.7.11.13

			[[Comma list]]: 245/243, 275/273, 847/845

			{{Mapping\|legend=2\| 13 19 23 0 2 \| 0 0 0 1 1 }}

			[[Optimal tuning]]s:
			* [[CTE]]: ~27/25 = 146.304¢ (1 ⧵ b13), ~11 = 4149.733¢
			* [[CWE]]: ~27/25 = 146.304¢ (1 ⧵ b13), ~11 = 4146.033¢

			[[Optimal ET sequence]]: [[13edt\|b13]], [[26edt\|b26]], [[39edt\|b39]]

			[[Badness]] (Sintel): 0.365

	== Tuning diagrams ==		== Tuning diagrams ==

FloraC: Text replacement - "Category:Temperament collections↵Category:Pages with mostly numerical content" to "Category:Temperament collections"

2025-11-27T12:29:25Z

Text replacement - "Category:Temperament collections↵Category:Pages with mostly numerical content" to "Category:Temperament collections"

← Older revision		Revision as of 12:29, 27 November 2025
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	[[Category:Temperament collections]]		[[Category:Temperament collections]]
	~~[[Category:Pages with mostly numerical content]]~~
	[[Category:Non-octave temperaments]]		[[Category:Non-octave temperaments]]

Lériendil: fixed adhara mappings

2025-09-04T02:55:12Z

fixed adhara mappings

Lériendil: Text replacement - "Category:Temperament collections" to "Category:Temperament collections Category:Pages with mostly numerical content"

2025-06-24T00:36:26Z

Text replacement - "Category:Temperament collections" to "Category:Temperament collections Category:Pages with mostly numerical content"

← Older revision		Revision as of 00:36, 24 June 2025
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	[[Category:Temperament collections]]		[[Category:Temperament collections]]
			[[Category:Pages with mostly numerical content]]
	[[Category:Non-octave temperaments]]		[[Category:Non-octave temperaments]]

FloraC: Text replacement - "Badness (Dirichlet): " to "Badness (Sintel): "

2025-06-06T16:40:16Z

Text replacement - "Badness (Dirichlet): " to "Badness (Sintel): "

← Older revision		Revision as of 16:40, 6 June 2025
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	[[Optimal ET sequence]]: [[2edt\|b2]], [[11edt\|b11]], [[13edt\|b13]]		[[Optimal ET sequence]]: [[2edt\|b2]], [[11edt\|b11]], [[13edt\|b13]]

	Badness (~~Dirichlet~~): 0.535		Badness (Sintel): 0.535

	=== Polturus ===		=== Polturus ===
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	[[EDT]]s: 15, 13e, 28e, 43dee		[[EDT]]s: 15, 13e, 28e, 43dee

	Badness (~~Dirichlet~~): 2.507		Badness (Sintel): 2.507

	== BPS ==		== BPS ==
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	[[Optimal ET sequence]]: [[4edt\|b4]], [[9edt\|b9]], [[13edt\|b13]], [[56edt\|b56]], [[69edt\|b69]], [[82edt\|b82]], [[95edt\|b95]]		[[Optimal ET sequence]]: [[4edt\|b4]], [[9edt\|b9]], [[13edt\|b13]], [[56edt\|b56]], [[69edt\|b69]], [[82edt\|b82]], [[95edt\|b95]]

	Badness (~~Dirichlet~~): 0.066		Badness (Sintel): 0.066

	=== Alhena ===		=== Alhena ===
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	(Note that é is used as the wart for 11/2.)		(Note that é is used as the wart for 11/2.)

	Badness (~~Dirichlet~~): 0.187		Badness (Sintel): 0.187

	=== Mintra ===		=== Mintra ===
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	[[Support]]ing [[ET]]s: 39, 17, 56, 22, 5, 95, 12, 61, 73, 134, 27c, 151e, 100, 90		[[Support]]ing [[ET]]s: 39, 17, 56, 22, 5, 95, 12, 61, 73, 134, 27c, 151e, 100, 90

	Badness (~~Dirichlet~~): 0.302		Badness (Sintel): 0.302

	==== Tridecimal Mintra ====		==== Tridecimal Mintra ====
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	[[Support]]ing [[ET]]s: 39, 17, 22, 56, 5f, 61, 95, 100, 134, 73f, 139cf, 83cf, 173e, 178cef		[[Support]]ing [[ET]]s: 39, 17, 22, 56, 5f, 61, 95, 100, 134, 73f, 139cf, 83cf, 173e, 178cef

	Badness (~~Dirichlet~~): 0.373		Badness (Sintel): 0.373

	=== Dubhe ===		=== Dubhe ===
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	[[Support]]ing [[ET]]s: 26, 9, 17, 43, 69, 8, 35, 95, 61, 60, 121, 25g, 112, 44		[[Support]]ing [[ET]]s: 26, 9, 17, 43, 69, 8, 35, 95, 61, 60, 121, 25g, 112, 44

	Badness (~~Dirichlet~~): 0.177		Badness (Sintel): 0.177

	== Canopus ==		== Canopus ==
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	[[Optimal ET sequence]]: [[13edt\|b13]], [[62edt\|b62]], [[75edt\|b75]], [[88edt\|b88]], [[101edt\|b101]], [[114edt\|b114]], [[355edt\|b355]], [[469edt\|b469]], [[583edt\|b583]], [[697edt\|b697]]		[[Optimal ET sequence]]: [[13edt\|b13]], [[62edt\|b62]], [[75edt\|b75]], [[88edt\|b88]], [[101edt\|b101]], [[114edt\|b114]], [[355edt\|b355]], [[469edt\|b469]], [[583edt\|b583]], [[697edt\|b697]]

	Badness (~~Dirichlet~~): 0.100		Badness (Sintel): 0.100

	=== Suhail ===		=== Suhail ===
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	[[Optimal ET sequence]]: [[39edt\|b39]], [[114edt\|b114]], [[153edt\|b153]], [[498edt\|b498cf]], [[651edt\|b651cf]] <!-- b804cff is not a GPV -->		[[Optimal ET sequence]]: [[39edt\|b39]], [[114edt\|b114]], [[153edt\|b153]], [[498edt\|b498cf]], [[651edt\|b651cf]] <!-- b804cff is not a GPV -->

	Badness (~~Dirichlet~~): 0.330		Badness (Sintel): 0.330

	== Izar ==		== Izar ==
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	[[Optimal ET sequence]]: [[13edt\|b13]], [[141edt\|b141]], [[154edt\|b154]], ... [[258edt\|b258]], [[271edt\|b271]], [[800edt\|b800]], [[1071edt\|b1071]], [[1342edt\|b1342]], [[1613edt\|b1613]], [[4568edt\|b4568]], [[6181edt\|b6181]]		[[Optimal ET sequence]]: [[13edt\|b13]], [[141edt\|b141]], [[154edt\|b154]], ... [[258edt\|b258]], [[271edt\|b271]], [[800edt\|b800]], [[1071edt\|b1071]], [[1342edt\|b1342]], [[1613edt\|b1613]], [[4568edt\|b4568]], [[6181edt\|b6181]]

	Badness (~~Dirichlet~~): 0.017		Badness (Sintel): 0.017

	== Nekkar ==		== Nekkar ==
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	[[Support]]ing [[ET]]s: 22, 49, 5c, 71, 27, 17c, 120, 93, 76c, 32cc, 169d, 115, 191d, 164d		[[Support]]ing [[ET]]s: 22, 49, 5c, 71, 27, 17c, 120, 93, 76c, 32cc, 169d, 115, 191d, 164d

	Badness (~~Dirichlet~~): 17.120		Badness (Sintel): 17.120

	=== 3.5.7.11 subgroup ===		=== 3.5.7.11 subgroup ===
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	[[Support]]ing [[ET]]s: 22, 49, 71, 5c, 27, 120, 93, 17c, 76c, 169d, 191d, 115, 164d, 125cd		[[Support]]ing [[ET]]s: 22, 49, 71, 5c, 27, 120, 93, 17c, 76c, 169d, 191d, 115, 164d, 125cd

	Badness (~~Dirichlet~~): 1.375		Badness (Sintel): 1.375

	=== 3.5.7.11.13 subgroup ===		=== 3.5.7.11.13 subgroup ===
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	[[Support]]ing [[ET]]s: 22, 5c, 27, 49, 71f, 17cf		[[Support]]ing [[ET]]s: 22, 5c, 27, 49, 71f, 17cf

	Badness (~~Dirichlet~~): 1.723		Badness (Sintel): 1.723

	== Procyon ==		== Procyon ==
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	[[Support]]ing [[ET]]s: b13, b157, b144, b170, b131, b183, b118, b14, b105, b12c, b196, b92, b27, b79		[[Support]]ing [[ET]]s: b13, b157, b144, b170, b131, b183, b118, b14, b105, b12c, b196, b92, b27, b79

	Badness (~~Dirichlet~~): 0.200		Badness (Sintel): 0.200

	=== Erigone ===		=== Erigone ===
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	[[Optimal ET sequence]]: [[6edt\|b6]], [[7edt\|b7]], [[13edt\|b13]], [[71edt\|b71]], [[84edt\|b84]], [[97edt\|b97]], [[110edt\|b110]], [[123edt\|b123]], [[136edt\|b136]]		[[Optimal ET sequence]]: [[6edt\|b6]], [[7edt\|b7]], [[13edt\|b13]], [[71edt\|b71]], [[84edt\|b84]], [[97edt\|b97]], [[110edt\|b110]], [[123edt\|b123]], [[136edt\|b136]]

	Badness (~~Dirichlet~~): 0.213		Badness (Sintel): 0.213

	=== Remus ===		=== Remus ===
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	[[Support]]ing [[ET]]s: 39, 7, 32, 71, 110, 46, 149, 188, 181		[[Support]]ing [[ET]]s: 39, 7, 32, 71, 110, 46, 149, 188, 181

	Badness (~~Dirichlet~~): 0.286		Badness (Sintel): 0.286

	=== Mizar ===		=== Mizar ===
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	[[Support]]ing [[ET]]s: 26, 7, 19, 45, 71, 97, 33, 123, 12d, 149, 59d, 175, 64d, 85cd		[[Support]]ing [[ET]]s: 26, 7, 19, 45, 71, 97, 33, 123, 12d, 149, 59d, 175, 64d, 85cd

	Badness (~~Dirichlet~~): 0.383		Badness (Sintel): 0.383

	==== Full no-twos 17-limit ====		==== Full no-twos 17-limit ====
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	[[Support]]ing [[ET]]s: 26, 71, 45, 19, 97f, 116d		[[Support]]ing [[ET]]s: 26, 71, 45, 19, 97f, 116d

	Badness (~~Dirichlet~~): 0.841		Badness (Sintel): 0.841

	== Tuning diagrams ==		== Tuning diagrams ==

FloraC: Text replacement - "Badness (Dirichlet): " to "Badness (Sintel): "

2025-06-06T16:40:05Z

Text replacement - "Badness (Dirichlet): " to "Badness (Sintel): "

← Older revision		Revision as of 16:40, 6 June 2025
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	[[Badness]] (Smith): 0.00271		[[Badness]] (Smith): 0.00271

	[[Badness]] (~~Dirichlet~~): 0.21396		[[Badness]] (Sintel): 0.21396

	==== Hemigone ====		==== Hemigone ====
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	[[Badness]] (Smith): 0.00501		[[Badness]] (Smith): 0.00501

	[[Badness]] (~~Dirichlet~~): 0.45479		[[Badness]] (Sintel): 0.45479

	==== <small><small>(no-2s)</small> </small>23-limit ====		==== <small><small>(no-2s)</small> </small>23-limit ====
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	[[Badness]] (Smith): 0.00539		[[Badness]] (Smith): 0.00539

	[[Badness]] (~~Dirichlet~~): 0.54174		[[Badness]] (Sintel): 0.54174

	== Sirius ==		== Sirius ==

Lériendil at 00:57, 29 May 2025

2025-05-29T00:57:06Z

← Older revision		Revision as of 00:57, 29 May 2025
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	[[Badness]] (Dirichlet): 0.21396		[[Badness]] (Dirichlet): 0.21396

	=== Hemigone ===		==== Hemigone ====
	By tempering out [[3971/3969]], erigone's tritave-augmented generator ([[49/11]]) is split into two [[19/9]]s. Then, [[17/1]] is approximated at [[39/35]] below [[19/1]] (tempering out [[665/663]]).		By tempering out [[3971/3969]], erigone's tritave-augmented generator ([[49/11]]) is split into two [[19/9]]s. Then, [[17/1]] is approximated at [[39/35]] below [[19/1]] (tempering out [[665/663]]).

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	[[Badness]] (Dirichlet): 0.45479		[[Badness]] (Dirichlet): 0.45479

	=== <small><small>(no-2s)</small> </small>23-limit ===		==== <small><small>(no-2s)</small> </small>23-limit ====
	[[2277/2275]] may be used in the same way to extend the simpler [[#Erigone\|erigone]] to the 3.5.7.11.13.23 subgroup.		[[2277/2275]] may be used in the same way to extend the simpler [[#Erigone\|erigone]] to the 3.5.7.11.13.23 subgroup.

Lériendil: Text replacement - "[[Shibboleth family" to "[[Gamelismic clan"

2025-05-06T17:25:13Z

Text replacement - "[[Shibboleth family" to "[[Gamelismic clan"

← Older revision		Revision as of 17:25, 6 May 2025
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	{{main\|Arcturus}}		{{main\|Arcturus}}

	As for extensions of this temperament that include the prime 2, see [[Trienstonic clan #Opossum\|opossum]], [[Jubilismic clan #Crepuscular\|crepuscular]], [[Kleismic family #Catalan\|catalan]], [[Tetracot family #Bunya\|bunya]], [[Sensamagic clan #Bohpier\|bohpier]], and [[~~Shibboleth family~~ #Superkleismic\|superkleismic]].		As for extensions of this temperament that include the prime 2, see [[Trienstonic clan #Opossum\|opossum]], [[Jubilismic clan #Crepuscular\|crepuscular]], [[Kleismic family #Catalan\|catalan]], [[Tetracot family #Bunya\|bunya]], [[Sensamagic clan #Bohpier\|bohpier]], and [[Gamelismic clan #Superkleismic\|superkleismic]].

	Subgroup: 3.5.7		Subgroup: 3.5.7

← Older revision		Revision as of 02:55, 4 September 2025
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	[[Comma list]]: 3213/3211, 3971/3969, 14161/14157, 41553/41503		[[Comma list]]: 3213/3211, 3971/3969, 14161/14157, 41553/41503

	[[Sval]] [[mapping]]: [{{val\| 1 2 2 2 2 2}}, {{val\| 0 -15 12 22 38 -21}}]		[[Sval]] [[mapping]]: [{{val\| 1 2 2 2 2 3}}, {{val\| 0 -15 12 22 38 -21}}]

	Sval mapping generators: ~3, ~119/117		Sval mapping generators: ~3, ~119/117
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	[[Comma list]]: 513/512, 729/728, 833/832, 969/968, 3971/3969		[[Comma list]]: 513/512, 729/728, 833/832, 969/968, 3971/3969

	[[Sval]] [[mapping]]: [{{val\| 1 2 2 2 2 2}}, {{val\| 0 -15 -7 12 22 38 -21}}]		[[Sval]] [[mapping]]: [{{val\| 1 2 2 2 2 2 3}}, {{val\| 0 -15 -7 12 22 38 -21}}]

	Sval mapping generators: ~3, ~64/63		Sval mapping generators: ~3, ~64/63
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	==== 3.5.7.8.11.13.17.19.23 subgroup ====		==== 3.5.7.8.11.13.17.19.23 subgroup ====
	At ~~slightly~~ lower accuracy, [[Procyon]] can be added to the Adhara structure, thereby spanning the entire triple-octave 23-limit.		At the cost of lower accuracy, [[Procyon]] can be added to the Adhara structure, thereby spanning the entire triple-octave 23-limit.

	[[Subgroup]]: 3.5.7.8.11.13.17.19.23		[[Subgroup]]: 3.5.7.8.11.13.17.19.23
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	[[Comma list]]: 361/360, 441/440, 513/512, 729/728, 833/832, 969/968, 1127/1125		[[Comma list]]: 361/360, 441/440, 513/512, 729/728, 833/832, 969/968, 1127/1125

	[[Sval]] [[mapping]]: [{{val\| 1 2 2 2 2 2}}, {{val\| 0 -35 -15 -7 12 22 38 -21 -75}}]		[[Sval]] [[mapping]]: [{{val\| 1 2 2 2 2 2 2 3 4}}, {{val\| 0 -35 -15 -7 12 22 38 -21 -75}}]

	Sval mapping generators: ~3, ~64/63		Sval mapping generators: ~3, ~64/63