Limmic temperaments - Revision history

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

2025-11-27T12:30:24Z

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

← Older revision		Revision as of 12:30, 27 November 2025
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	[[Category:Temperament collections]]		[[Category:Temperament collections]]
	~~[[Category:Pages with mostly numerical content]]~~
	[[Category:Limmic temperaments]] <!-- main article -->		[[Category:Limmic temperaments]] <!-- main article -->
	[[Category:Rank 2]]		[[Category:Rank 2]]
	[[Category:Blackwood]]		[[Category:Blackwood]]

FloraC: /* Blackweed */ this is a restriction of undecimal blackwood

2025-07-20T10:42:36Z

Blackweed: this is a restriction of undecimal blackwood

← Older revision		Revision as of 10:42, 20 July 2025
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	== Blackweed ==		== Blackweed ==
	Blackweed is a ~~variant~~ of blackwood as it tempers out 256/243 alike but in the 2.3.11/7 [[subgroup]]. 20edo is close to the optimum, which has 4\20 as the period and 420{{c}} as the generator.		Blackweed is a [[restriction]] of undecimal blackwood as it tempers out 256/243 alike but in the 2.3.11/7 [[subgroup]]. 20edo is close to the optimum, which has 4\20 as the period and 420{{c}} as the generator.

	[[Subgroup]]: 2.3.11/7		[[Subgroup]]: 2.3.11/7

FloraC: - CTE & POTE tunings

2025-07-20T10:40:10Z

- CTE & POTE tunings

← Older revision		Revision as of 10:40, 20 July 2025
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	* [[CWE]]: ~8/7 = 240.000{{c}}, ~5/4 = 395.126{{c}}		* [[CWE]]: ~8/7 = 240.000{{c}}, ~5/4 = 395.126{{c}}
	: error map: {{val\| 0.000 +18.045 +8.812 }}		: error map: {{val\| 0.000 +18.045 +8.812 }}
	~~<!-- * [[CTE]]: ~9/8 = 240.000{{c}}, ~5/4 = 386.314{{c}}~~
	~~: [[error map]]: {{val\| 0.000 +18.045 0.000 }}~~
	* [[POTE]]: ~9/8 = 240.000{{c}}, ~5/4 = 399.594{{c}}
	~~: error map: {{val\| 0.000 +18.045 +13.280 }} -->~~

	{{Optimal ET sequence\|legend=1\| 5, 10, 15 }}		{{Optimal ET sequence\|legend=1\| 5, 10, 15 }}
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	* [[CWE]]: ~8/7 = 240.000{{c}}, ~5/4 = 391.098{{c}}		* [[CWE]]: ~8/7 = 240.000{{c}}, ~5/4 = 391.098{{c}}
	: error map: {{val\| 0.000 +18.045 +4.784 -8.826 }}		: error map: {{val\| 0.000 +18.045 +4.784 -8.826 }}
	~~<!-- * [[CTE]]: ~8/7 = 240.000{{c}}, ~5/4 = 386.313{{c}}~~
	~~: [[error map]]: {{val\| 0.000 +18.045 0.000 -8.826 }}~~
	* [[POTE]]: ~8/7 = 240.000{{c}}, ~5/4 = 392.767{{c}}
	~~: error map: {{val\| 0.000 +18.045 +6.454 -8.826 }} -->~~

	{{Optimal ET sequence\|legend=1\| 5, 10, 15, 40b }}		{{Optimal ET sequence\|legend=1\| 5, 10, 15, 40b }}
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	* WE: ~8/7 = 239.341{{c}}, ~5/4 = 393.864{{c}}		* WE: ~8/7 = 239.341{{c}}, ~5/4 = 393.864{{c}}
	* CWE: ~8/7 = 240.000{{c}}, ~5/4 = 394.655{{c}}		* CWE: ~8/7 = 240.000{{c}}, ~5/4 = 394.655{{c}}
	~~<!-- * POTE: ~8/7 = 240.000{{c}}, ~5/4 = 394.948{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 5, 10, 15, 40be }}		{{Optimal ET sequence\|legend=0\| 5, 10, 15, 40be }}
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	* WE: ~8/7 = 239.187{{c}}, ~5/4 = 389.713{{c}}		* WE: ~8/7 = 239.187{{c}}, ~5/4 = 389.713{{c}}
	* CWE: ~8/7 = 240.000{{c}}, ~5/4 = 390.282{{c}}		* CWE: ~8/7 = 240.000{{c}}, ~5/4 = 390.282{{c}}
	~~<!-- * POTE: ~8/7 = 240.000{{c}}, ~5/4 = 391.037{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 5, 10, 15, 25e }}		{{Optimal ET sequence\|legend=0\| 5, 10, 15, 25e }}
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	* WE: ~8/7 = 239.389{{c}}, ~5/4 = 397.056{{c}}		* WE: ~8/7 = 239.389{{c}}, ~5/4 = 397.056{{c}}
	* CWE: ~8/7 = 240.000{{c}}, ~5/4 = 396.599{{c}}		* CWE: ~8/7 = 240.000{{c}}, ~5/4 = 396.599{{c}}
	~~<!-- * POTE: ~8/7 = 240.000{{c}}, ~5/4 = 398.070{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 5e, 10e, 15 }}		{{Optimal ET sequence\|legend=0\| 5e, 10e, 15 }}
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	* WE: ~8/7 = 239.196{{c}}, ~5/4 = 395.483{{c}}		* WE: ~8/7 = 239.196{{c}}, ~5/4 = 395.483{{c}}
	* CWE: ~8/7 = 240.000{{c}}, ~5/4 = 394.759{{c}}		* CWE: ~8/7 = 240.000{{c}}, ~5/4 = 394.759{{c}}
	~~<!-- * POTE: ~8/7 = 240.000{{c}}, ~5/4 = 396.812{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 5e, 10e, 15 }}		{{Optimal ET sequence\|legend=0\| 5e, 10e, 15 }}
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	* WE: ~8/7 = 239.058{{c}}, ~5/4 = 373.292{{c}}		* WE: ~8/7 = 239.058{{c}}, ~5/4 = 373.292{{c}}
	* CWE: ~8/7 = 240.000{{c}}, ~5/4 = 371.659{{c}}		* CWE: ~8/7 = 240.000{{c}}, ~5/4 = 371.659{{c}}
	~~<!-- * POTE: ~8/7 = 240.000{{c}}, ~5/4 = 374.763{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 5e, 10 }}		{{Optimal ET sequence\|legend=0\| 5e, 10 }}
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	* [[Tp tuning\|subgroup]] [[CWE]]: ~8/7 = 240.000{{c}}, ~11/7 = 784.967{{c}}		* [[Tp tuning\|subgroup]] [[CWE]]: ~8/7 = 240.000{{c}}, ~11/7 = 784.967{{c}}
	: error map: {{val\| 0.000 +18.045 +2.475 }}		: error map: {{val\| 0.000 +18.045 +2.475 }}
	~~<!-- * ([[Tp tuning\|subgroup POTE]]): ~9/8 = 240.0000{{c}}, ~11/7 = 786.2215{{c}} -->~~

	{{Optimal ET sequence\|legend=1\| 15, 20, 35b, 55b }}		{{Optimal ET sequence\|legend=1\| 15, 20, 35b, 55b }}

FloraC: Switch to Sintel's badness, WE & CWE tunings, per community consensus

2025-07-20T10:38:57Z

Switch to Sintel's badness, WE & CWE tunings, per community consensus

← Older revision		Revision as of 10:38, 20 July 2025
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	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[CTE]]: ~9/8 = 240.000, ~5/4 = 386.314		* [[WE]]: ~8/7 = 238.851{{c}}, ~5/4 = 397.681{{c}}
			: [[error map]]: {{val\| -5.746 +8.852 -0.124 }}
			* [[CWE]]: ~8/7 = 240.000{{c}}, ~5/4 = 395.126{{c}}
			: error map: {{val\| 0.000 +18.045 +8.812 }}
			<!-- * [[CTE]]: ~9/8 = 240.000{{c}}, ~5/4 = 386.314{{c}}
	: [[error map]]: {{val\| 0.000 +18.045 0.000 }}		: [[error map]]: {{val\| 0.000 +18.045 0.000 }}
	* [[POTE]]: ~9/8 = 240.000, ~5/4 = 399.594		* [[POTE]]: ~9/8 = 240.000{{c}}, ~5/4 = 399.594{{c}}
	: error map: {{val\| 0.000 +18.045 +13.280 }}		: error map: {{val\| 0.000 +18.045 +13.280 }} -->

	{{Optimal ET sequence\|legend=1\| 5, 10, 15 }}		{{Optimal ET sequence\|legend=1\| 5, 10, 15 }}

	[[Badness]] (~~Smith~~): 0.~~063760~~		[[Badness]] (Sintel): 1.50

	=== 7-limit ===		=== 7-limit ===
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	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[CTE]]: ~8/7 = 240.000, ~5/4 = 386.313		* [[WE]]: ~8/7 = 239.426{{c}}, ~5/4 = 391.828{{c}}
			: [[error map]]: {{val\| -2.870 +13.453 -0.225 -16.861 }}
			* [[CWE]]: ~8/7 = 240.000{{c}}, ~5/4 = 391.098{{c}}
			: error map: {{val\| 0.000 +18.045 +4.784 -8.826 }}
			<!-- * [[CTE]]: ~8/7 = 240.000{{c}}, ~5/4 = 386.313{{c}}
	: [[error map]]: {{val\| 0.000 +18.045 0.000 -8.826 }}		: [[error map]]: {{val\| 0.000 +18.045 0.000 -8.826 }}
	* [[POTE]]: ~8/7 = 240.000, ~5/4 = 392.767		* [[POTE]]: ~8/7 = 240.000{{c}}, ~5/4 = 392.767{{c}}
	: error map: {{val\| 0.000 +18.045 +6.454 -8.826 }}		: error map: {{val\| 0.000 +18.045 +6.454 -8.826 }} -->

	{{Optimal ET sequence\|legend=1\| 5, 10, 15, 40b }}		{{Optimal ET sequence\|legend=1\| 5, 10, 15, 40b }}

	[[Badness]] (~~Smith~~): 0.~~025640~~		[[Badness]] (Sintel): 0.649

	==== Undecimal blackwood ====		==== Undecimal blackwood ====
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	Mapping: {{mapping\| 5 8 0 14 29 \| 0 0 1 0 -1 }}		Mapping: {{mapping\| 5 8 0 14 29 \| 0 0 1 0 -1 }}

	Optimal ~~tuning (~~POTE): ~8/7 = 240.000, ~5/4 = 394.948		Optimal tunings:
			* WE: ~8/7 = 239.341{{c}}, ~5/4 = 393.864{{c}}
			* CWE: ~8/7 = 240.000{{c}}, ~5/4 = 394.655{{c}}
			<!-- * POTE: ~8/7 = 240.000{{c}}, ~5/4 = 394.948{{c}} -->

	{{Optimal ET sequence\|legend=0\| 5, 10, 15, 40be~~, 55be, 70bde, 85bcde~~ }}		{{Optimal ET sequence\|legend=0\| 5, 10, 15, 40be }}

	Badness (~~Smith~~): 0.~~024641~~		Badness (Sintel): 0.815

	===== 13-limit =====		===== 13-limit =====
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	Mapping: {{mapping\| 5 8 0 14 29 7 \| 0 0 1 0 -1 1 }}		Mapping: {{mapping\| 5 8 0 14 29 7 \| 0 0 1 0 -1 1 }}

	Optimal ~~tuning (~~POTE): ~8/7 = 240.000, ~5/4 = 391.037		Optimal tunings:
			* WE: ~8/7 = 239.187{{c}}, ~5/4 = 389.713{{c}}
			* CWE: ~8/7 = 240.000{{c}}, ~5/4 = 390.282{{c}}
			<!-- * POTE: ~8/7 = 240.000{{c}}, ~5/4 = 391.037{{c}} -->

	{{Optimal ET sequence\|legend=0\| 5, 10, 15, 25e~~, 40bef~~ }}		{{Optimal ET sequence\|legend=0\| 5, 10, 15, 25e }}

	Badness (~~Smith~~): 0.~~020498~~		Badness (Sintel): 0.847

	==== Farrier ====		==== Farrier ====
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	Mapping: {{mapping\| 5 8 0 14 -6 \| 0 0 1 0 2 }}		Mapping: {{mapping\| 5 8 0 14 -6 \| 0 0 1 0 2 }}

	Optimal ~~tuning (~~POTE): ~8/7 = 240.000, ~5/4 = 398.070		Optimal tunings:
			* WE: ~8/7 = 239.389{{c}}, ~5/4 = 397.056{{c}}
			* CWE: ~8/7 = 240.000{{c}}, ~5/4 = 396.599{{c}}
			<!-- * POTE: ~8/7 = 240.000{{c}}, ~5/4 = 398.070{{c}} -->

	{{Optimal ET sequence\|legend=0\| 5e, 10e, 15 }}		{{Optimal ET sequence\|legend=0\| 5e, 10e, 15 }}

	Badness (~~Smith~~): 0.~~029200~~		Badness (Sintel): 0.965

	===== 13-limit =====		===== 13-limit =====
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	Mapping: {{mapping\| 5 8 0 14 -6 7 \| 0 0 1 0 2 1 }}		Mapping: {{mapping\| 5 8 0 14 -6 7 \| 0 0 1 0 2 1 }}

	Optimal ~~tuning (~~POTE): ~8/7 = 240.000, ~5/4 = 396.812		Optimal tunings:
			* WE: ~8/7 = 239.196{{c}}, ~5/4 = 395.483{{c}}
			* CWE: ~8/7 = 240.000{{c}}, ~5/4 = 394.759{{c}}
			<!-- * POTE: ~8/7 = 240.000{{c}}, ~5/4 = 396.812{{c}} -->

	{{Optimal ET sequence\|legend=0\| 5e, 10e, 15 }}		{{Optimal ET sequence\|legend=0\| 5e, 10e, 15 }}

	Badness (~~Smith~~): 0.~~022325~~		Badness (Sintel): 0.922

	==== Ferrum ====		==== Ferrum ====
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	Mapping: {{mapping\| 5 8 0 14 6 \| 0 0 1 0 1 }}		Mapping: {{mapping\| 5 8 0 14 6 \| 0 0 1 0 1 }}

	Optimal ~~tuning (~~POTE): ~8/7 = 240.000, ~5/4 = 374.763		Optimal tunings:
			* WE: ~8/7 = 239.058{{c}}, ~5/4 = 373.292{{c}}
			* CWE: ~8/7 = 240.000{{c}}, ~5/4 = 371.659{{c}}
			<!-- * POTE: ~8/7 = 240.000{{c}}, ~5/4 = 374.763{{c}} -->

	{{Optimal ET sequence\|legend=0\| 5e, 10 }}		{{Optimal ET sequence\|legend=0\| 5e, 10 }}

	Badness (~~Smith~~): 0.~~030883~~		Badness (Sintel): 1.02

	== Blackweed ==		== Blackweed ==
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	: sval mapping generators: ~9/8, ~11/7		: sval mapping generators: ~9/8, ~11/7

	[[Optimal tuning]] ([[Tp tuning\|subgroup POTE]]): ~9/8 = 240.0000, ~11/7 = 786.2215		[[Optimal tuning]]s:
			* [[Tp tuning\|subgroup]] [[WE]]: ~8/7 = 238.851{{c}}, ~11/7 = 782.457{{c}}
			: [[error map]]: {{val\| -5.746 +8.852 -0.035 }}
			* [[Tp tuning\|subgroup]] [[CWE]]: ~8/7 = 240.000{{c}}, ~11/7 = 784.967{{c}}
			: error map: {{val\| 0.000 +18.045 +2.475 }}
			<!-- * ([[Tp tuning\|subgroup POTE]]): ~9/8 = 240.0000{{c}}, ~11/7 = 786.2215{{c}} -->

	{{Optimal ET sequence\|legend=1\| 15, 20, 35b }}		{{Optimal ET sequence\|legend=1\| 15, 20, 35b, 55b }}

	[[Category:Temperament collections]]		[[Category:Temperament collections]]

VectorGraphics at 20:31, 28 June 2025

2025-06-28T20:31:40Z

← Older revision		Revision as of 20:31, 28 June 2025
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	}}		}}
	{{Technical data page}}		{{Technical data page}}
	'''Limmic temperaments''' are [[temperament]]s that [[temper out]] the Pythagorean limma, [[256/243]]. As a consequence, [[3/2]] is always represented by 3\5, 720 [[cent]]s assuming pure octaves. While quite sharp, this is close enough to a just fifth to serve as a fifth, and some people are fond of it.		'''Limmic temperaments''' are [[temperament]]s that [[temper out]] the Pythagorean limma, [[256/243]]. As a consequence, [[3/2]] is always represented by 3\5, 720 [[cent]]s assuming pure octaves. While quite sharp, this is close enough to a just fifth to serve as a fifth, and some people are fond of it. All temperaments shown here are pentaploid acot.

	== Blackwood ==		== Blackwood ==

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

2025-06-24T00:39:58Z

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

← Older revision		Revision as of 00:39, 24 June 2025
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	[[Category:Temperament collections]]		[[Category:Temperament collections]]
			[[Category:Pages with mostly numerical content]]
	[[Category:Limmic temperaments]] <!-- main article -->		[[Category:Limmic temperaments]] <!-- main article -->
	[[Category:Rank 2]]		[[Category:Rank 2]]
	[[Category:Blackwood]]		[[Category:Blackwood]]

VectorGraphics at 05:17, 12 June 2025

2025-06-12T05:17:08Z

← Older revision		Revision as of 05:17, 12 June 2025
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	{{Mapping\|legend=1\| 5 8 0 14 \| 0 0 1 0 }}		{{Mapping\|legend=1\| 5 8 0 14 \| 0 0 1 0 }}

	~~{{Multival\|legend=1\| 0 5 0 8 0 -14 }}~~

	[[Optimal tuning]]s:		[[Optimal tuning]]s:

FloraC: /* Blackweed */ style

2025-05-12T09:44:37Z

Blackweed: style

← Older revision		Revision as of 09:44, 12 May 2025
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	== Blackweed ==		== Blackweed ==
	Blackweed is a variant of blackwood as it tempers out 256/243 alike but in the 2.3.11/7 [[subgroup]]. 20edo is close to the optimum, which has 4\20 as the period and ~~420¢~~ as the generator.		Blackweed is a variant of blackwood as it tempers out 256/243 alike but in the 2.3.11/7 [[subgroup]]. 20edo is close to the optimum, which has 4\20 as the period and 420{{c}} as the generator.

	[[Subgroup]]: 2.3.11/7		[[Subgroup]]: 2.3.11/7

	[[Comma list]]: {{monzo\| 8 -5 }} = 256/243		[[Comma list]]: {{monzo\| 8 -5 }} (256/243)

	{{Mapping\|legend=2\| 5 8 0 \| 0 0 1 }}		{{Mapping\|legend=2\| 5 8 0 \| 0 0 1 }}
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	: sval mapping generators: ~9/8, ~11/7		: sval mapping generators: ~9/8, ~11/7

	[[Optimal tuning]] ([[Tp tuning\|subgroup POTE]]): ~9/8 = 240.~~000~~, ~11/7 = 786.2215		[[Optimal tuning]] ([[Tp tuning\|subgroup POTE]]): ~9/8 = 240.0000, ~11/7 = 786.2215

	{{Optimal ET sequence\|legend=1\| 15, 20, 35b }}		{{Optimal ET sequence\|legend=1\| 15, 20, 35b }}

FloraC: ... since blacksmith is formally deprecated now

2025-05-12T09:44:07Z

... since *blacksmith* is formally deprecated now

← Older revision		Revision as of 09:44, 12 May 2025
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	[[Badness]] (Smith): 0.063760		[[Badness]] (Smith): 0.063760

	=== 7-limit ~~(blacksmith)~~ ===		=== 7-limit ===
	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

FloraC: Undo revision 194120 by Xenllium (talk). Stop messing with the backslashes. Set the font in your browser

2025-04-26T16:33:29Z

Undo revision 194120 by Xenllium (talk). Stop messing with the backslashes. Set the font in your browser

← Older revision		Revision as of 16:33, 26 April 2025
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	}}		}}
	{{Technical data page}}		{{Technical data page}}
	'''Limmic temperaments''' are [[temperament]]s that [[temper out]] the Pythagorean limma, [[256/243]]. As a consequence, [[3/2]] is always represented by 3~~<span style="font-family:'Arial', sans-serif">~~\~~</span>~~5, 720 [[cent]]s assuming pure octaves. While quite sharp, this is close enough to a just fifth to serve as a fifth, and some people are fond of it.		'''Limmic temperaments''' are [[temperament]]s that [[temper out]] the Pythagorean limma, [[256/243]]. As a consequence, [[3/2]] is always represented by 3\5, 720 [[cent]]s assuming pure octaves. While quite sharp, this is close enough to a just fifth to serve as a fifth, and some people are fond of it.

	== Blackwood ==		== Blackwood ==
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	Blackwood is the 5edo [[circle of fifths]] with an independent dimension for the harmonic 5. It can be described as the {{nowrap\| 5 & 10 }} temperament. [[15edo]] is an obvious tuning.		Blackwood is the 5edo [[circle of fifths]] with an independent dimension for the harmonic 5. It can be described as the {{nowrap\| 5 & 10 }} temperament. [[15edo]] is an obvious tuning.

	The only extension to the 7-limit that makes any sense is to map the [[7/4\|harmonic seventh]] to 4~~<span style="font-family:'Arial', sans-serif">~~\~~</span>~~5, tempering out [[28/27]], [[49/48]], and [[64/63]]. This is known as ''blacksmith'' in earlier materials, including [[Graham Breed]]'s temperament finder.		The only extension to the 7-limit that makes any sense is to map the [[7/4\|harmonic seventh]] to 4\5, tempering out [[28/27]], [[49/48]], and [[64/63]]. This is known as ''blacksmith'' in earlier materials, including [[Graham Breed]]'s temperament finder.

	=== 5-limit ===		=== 5-limit ===
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	{{Mapping\|legend=1\| 5 8 0 \| 0 0 1 }}		{{Mapping\|legend=1\| 5 8 0 \| 0 0 1 }}

	: ~~Mapping~~ generators: ~9/8, ~5		: mapping generators: ~9/8, ~5

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
Line 119:		Line 119:

	== Blackweed ==		== Blackweed ==
	Blackweed is a variant of blackwood as it tempers out 256/243 alike but in the 2.3.11/7 [[subgroup]]. 20edo is close to the optimum, which has 4~~<span style="font-family:'Arial', sans-serif">~~\~~</span>~~20 as the period and 420¢ as the generator.		Blackweed is a variant of blackwood as it tempers out 256/243 alike but in the 2.3.11/7 [[subgroup]]. 20edo is close to the optimum, which has 4\20 as the period and 420¢ as the generator.

	[[Subgroup]]: 2.3.11/7		[[Subgroup]]: 2.3.11/7