Dual-fifth temperaments - Revision history

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

2025-11-27T12:19:39Z

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

← Older revision		Revision as of 12:19, 27 November 2025
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	[[Category:Subgroup temperaments]]		[[Category:Subgroup temperaments]]
	[[Category:Temperament collections]]		[[Category:Temperament collections]]
	~~[[Category:Pages with mostly numerical content]]~~

Xenllium at 03:32, 29 June 2025

2025-06-29T03:32:21Z

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 For example, "dual-3 sixix" is a 2.3⁻.9.5 temperament with an optimal [[generator]] around 335.8¢ (optimizing only the 2.9.5 portion of the subgroup). Two generators up make the flat fifth, and five generators down make the flat fourth. Hence 3 generators down represent [[9/8]] and 6 generators down represent [[5/4]]. Hence dual-3 sixix tempers out [[81/80]] in the 2.9.5 subgroup, but only every third interval in the sixix generator chains represents a JI interval.
 Alternatively, dual-fifth temperaments can be analyzed in a more conventional way as [[subgroup temperament]]s, where one of the fifths is mapped to [[3/2]] and the other is mapped to a nearby [[wolf interval|wolf fifth]] (such as [[64/43]], which is convenient since 2.3.43 is the same subgroup as 2.3.64/43).
 == Dual-3 A-Team ==
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 {{Optimal ET sequence|legend=1| 18, 28 }}
 == Megapyth ==
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 Mapping: [{{val| 1 0 0 -4 6 }}, {{val| 0 1 0 3 0 }}, {{val| 0 0 1 1 -2 }}]
-{{Optimal ET sequence|legend=1|30c, 47b, 52b}}
+{{Optimal ET sequence|legend=1| 30c, 47b, 52b }}
 == Travesty ==

Xenllium at 02:31, 29 June 2025

2025-06-29T02:31:59Z

← Older revision		Revision as of 02:31, 29 June 2025
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	Alternatively, dual-fifth temperaments can be analyzed in a more conventional way as [[subgroup temperament]]s, where one of the fifths is mapped to [[3/2]] and the other is mapped to a nearby [[wolf interval\|wolf fifth]] (such as [[64/43]], which is convenient since 2.3.43 is the same subgroup as 2.3.64/43).		Alternatively, dual-fifth temperaments can be analyzed in a more conventional way as [[subgroup temperament]]s, where one of the fifths is mapped to [[3/2]] and the other is mapped to a nearby [[wolf interval\|wolf fifth]] (such as [[64/43]], which is convenient since 2.3.43 is the same subgroup as 2.3.64/43).

			== Dual-3 Spell ==
			[[Subgroup]]: 2.3⁻.3⁺.5

			[[Comma\|Comma basis]]: [[25/24\|{{monzo\| -3 0 -1 2 }}]], [[81/80\|{{monzo\| -4 2 2 -1 }}]]

			Mapping: [{{val\| 1 2 1 2 }}, {{val\| 0 -3 4 2 }}]

			2.9.5 [[POTE]] generator: ~(3⁻×3⁺)/8 = 192.4773

			{{Optimal ET sequence\|legend=1\| 6, 13, 19 }}

	== Dual-3 A-Team ==		== Dual-3 A-Team ==
	[[Subgroup]]: 2.3⁻.3⁺.5		[[Subgroup]]: 2.3⁻.3⁺.5

	[[Comma]]s: [[81/80\|{{monzo\| -4 2 2 -1 }}]], {{monzo\| -8 1 4 0 }}		[[Comma\|Comma basis]]: [[81/80\|{{monzo\| -4 2 2 -1 }}]], [[256/243\|{{monzo\| 8 -1 -4 0 }}]]

	Mapping: [{{val\| 1 0 2 0 }}, {{val\| 0 4 -1 6 }}]		Mapping: [{{val\| 1 0 2 0 }}, {{val\| 0 4 -1 6 }}]

	2.9.5 [[POTE]] generator: ~~464~~.~~1591~~		2.9.5 [[POTE]] generator: ~3⁺/2 = 735.8409

	{{Optimal ET sequence\|legend=1\| 13, 18, 31 }}		{{Optimal ET sequence\|legend=1\| 13, 18, 31 }}
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	[[Subgroup]]: 2.3⁻.3⁺.5		[[Subgroup]]: 2.3⁻.3⁺.5

	[[Comma]]s: [[81/80\|{{monzo\| -4 2 2 -1 }}]], {{monzo\| 2 -~~3 0~~ 1 }}		[[Comma\|Comma basis]]: [[25/24\|{{monzo\| -3 1 -2 2 }}]], [[81/80\|{{monzo\| -4 2 2 -1 }}]]

	Mapping: [{{val\| 1 0 1 -2 }}, {{val\| 0 2 1 6 }}]		Mapping: [{{val\| 1 1 3 4 }}, {{val\| 0 2 -5 -6 }}]

	2.9.5 [[POTE]] generator: 335.8409		2.9.5 [[POTE]] generator: ~(2×3⁺)/5 = 335.8409

	{{Optimal ET sequence\|legend=1\| 18, 25, 43 }}		{{Optimal ET sequence\|legend=1\| 18, 25, 43 }}
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	[[Subgroup]]: 2.3⁻.3⁺.5		[[Subgroup]]: 2.3⁻.3⁺.5

	2~~.9.5~~ [[~~POTE~~]] ~~generator~~: ~~730.0679~~		[[Comma\|Comma basis]]: [[25/24\|{{monzo\| -3 1 -2 2 }}]], [[256/243\|{{monzo\| 8 -1 -4 0 }}]]

			Mapping: [{{val\| 2 8 2 1 }}, {{val\| 0 -4 1 3 }}]

	~~{{Optimal ET sequence\|legend~~=~~1\|18, 28}}~~		2.9.5 [[POTE]] generator: ~3⁺/2 = 730.0679

	{{~~Todo~~\|~~review\|inline~~=1\|~~comment=explicitly write out commas and vals for Dual-3 Octokaidecal~~}}		{{Optimal ET sequence\|legend=1\| 18, 28 }}

	== Megapyth ==		== Megapyth ==
	[[Subgroup]]: 2.3⁻.3⁺.5.7		[[Subgroup]]: 2.3⁻.3⁺.5.7

	[[Comma]]s: {{monzo\| -4 3 1 -1 0 }}, {{monzo\| 6 0 -2 0 -1 }}, {{monzo\| -5 -1 6 0 0 }}		[[Comma\|Comma basis]]: {{monzo\| -4 3 1 -1 0 }}, {{monzo\| 6 0 -2 0 -1 }}, {{monzo\| -5 -1 6 0 0 }}

	Mapping: [{{val\| 1 1 1 0 4 }}, {{val\| 0 6 1 19 -2 }}]		Mapping: [{{val\| 1 1 1 0 4 }}, {{val\| 0 6 1 19 -2 }}]
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	2.9.5.7 [[POTE]] generator: ~3⁺/2 = 715.319		2.9.5.7 [[POTE]] generator: ~3⁺/2 = 715.319

	{{Optimal ET sequence\|legend=1\|47b, 52b}}		{{Optimal ET sequence\|legend=1\| 47b, 52b }}

	== Duofamity (Rank-3) ==		== Duofamity (Rank-3) ==
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	Least-squares for 5/4, 9/8, and 8/7: 690.155, 715.325		Least-squares for 5/4, 9/8, and 8/7: 690.155, 715.325

	[[Comma]]s: {{monzo\| -4 3 1 -1 0 }}, {{monzo\| 6 0 -2 0 -1 }}		[[Comma\|Comma basis]]: {{monzo\| -4 3 1 -1 0 }}, {{monzo\| 6 0 -2 0 -1 }}

	Mapping: [{{val\| 1 0 0 -4 6 }}, {{val\| 0 1 0 3 0 }}, {{val\| 0 0 1 1 -2 }}]		Mapping: [{{val\| 1 0 0 -4 6 }}, {{val\| 0 1 0 3 0 }}, {{val\| 0 0 1 1 -2 }}]
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	[[Subgroup]]: 2.3.5.97		[[Subgroup]]: 2.3.5.97

	[[Comma]]s: [[177147/163840]], 2619/2560		[[Comma\|Comma basis]]: [[177147/163840]], 2619/2560

	{{Mapping\|legend=2\|1 1 -4 2\|0 1 11 8}}		{{Mapping\|legend=2\|1 1 -4 2\|0 1 11 8}}
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	== [[Ripple]] ==		== [[Ripple]] ==
	Subgroup: 2.3⁻.9.5.7.11		Subgroup: 2.3⁻.3⁺.5.7.11

	See [[ripple]].		See [[ripple]].

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

2025-06-24T00:25:49Z

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

← Older revision		Revision as of 00:25, 24 June 2025
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	[[Category:Subgroup temperaments]]		[[Category:Subgroup temperaments]]
	[[Category:Temperament collections]]		[[Category:Temperament collections]]
			[[Category:Pages with mostly numerical content]]

FloraC: Text replacement - "{{Technical data page}}

" to "{{Technical data page}}"

2025-04-13T13:46:28Z

Text replacement - "{{Technical data page}}<br><br>" to "{{Technical data page}}"

← Older revision		Revision as of 13:46, 13 April 2025
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	{{Technical data page}}~~<br><br>~~		{{Technical data page}}
	Unlike conventional temperaments, [[dual-fifth]] temperaments do not attempt to optimize every [[interval]] to low-limit [[JI]], but treat the "sharp 3" (3⁺) and the "flat 3" (3⁻) as distinct dimensions. The sharp 3 and the flat 3 are not meant to represent JI intervals by themselves, but satisfy 3⁺ × 3⁻ = 9 (representing [[9/1]] in JI); hence 2.3⁻.9 and 2.3⁻.3⁺ are the same [[subgroup]].		Unlike conventional temperaments, [[dual-fifth]] temperaments do not attempt to optimize every [[interval]] to low-limit [[JI]], but treat the "sharp 3" (3⁺) and the "flat 3" (3⁻) as distinct dimensions. The sharp 3 and the flat 3 are not meant to represent JI intervals by themselves, but satisfy 3⁺ × 3⁻ = 9 (representing [[9/1]] in JI); hence 2.3⁻.9 and 2.3⁻.3⁺ are the same [[subgroup]].

BudjarnLambeth: {{Technical data page}}

2025-04-13T05:42:10Z

{{Technical data page}}<br><br>

← Older revision		Revision as of 05:42, 13 April 2025
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			{{Technical data page}}<br><br>
	Unlike conventional temperaments, [[dual-fifth]] temperaments do not attempt to optimize every [[interval]] to low-limit [[JI]], but treat the "sharp 3" (3⁺) and the "flat 3" (3⁻) as distinct dimensions. The sharp 3 and the flat 3 are not meant to represent JI intervals by themselves, but satisfy 3⁺ × 3⁻ = 9 (representing [[9/1]] in JI); hence 2.3⁻.9 and 2.3⁻.3⁺ are the same [[subgroup]].		Unlike conventional temperaments, [[dual-fifth]] temperaments do not attempt to optimize every [[interval]] to low-limit [[JI]], but treat the "sharp 3" (3⁺) and the "flat 3" (3⁻) as distinct dimensions. The sharp 3 and the flat 3 are not meant to represent JI intervals by themselves, but satisfy 3⁺ × 3⁻ = 9 (representing [[9/1]] in JI); hence 2.3⁻.9 and 2.3⁻.3⁺ are the same [[subgroup]].

Godtone: note ripple

2025-03-14T01:19:55Z

note ripple

← Older revision		Revision as of 01:19, 14 March 2025
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	[[Support]]ing [[ET]]s: 7, 40, 33, 26[+5], 47, 19[+5, +97], 54[-3], 61[-3, -5, -97], 68[-3, -5, -97]		[[Support]]ing [[ET]]s: 7, 40, 33, 26[+5], 47, 19[+5, +97], 54[-3], 61[-3, -5, -97], 68[-3, -5, -97]

			== [[Ripple]] ==
			Subgroup: 2.3⁻.9.5.7.11

			See [[ripple]].

	[[Category:Dual-fifth]]		[[Category:Dual-fifth]]
	[[Category:Subgroup temperaments]]		[[Category:Subgroup temperaments]]
	[[Category:Temperament collections]]		[[Category:Temperament collections]]

BudjarnLambeth: /* Dual-3 Octokaidecal */ moved from todo to todo:review

2024-03-01T06:03:21Z

Dual-3 Octokaidecal: moved from todo to todo:review

← Older revision		Revision as of 06:03, 1 March 2024
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	{{Optimal ET sequence\|legend=1\|18, 28}}		{{Optimal ET sequence\|legend=1\|18, 28}}

	{{Todo\|inline=1\|comment=explicitly write out commas and vals for Dual-3 Octokaidecal}}		{{Todo\|review\|inline=1\|comment=explicitly write out commas and vals for Dual-3 Octokaidecal}}

	== Megapyth ==		== Megapyth ==

BudjarnLambeth: Categories

2024-01-31T11:21:40Z

BudjarnLambeth: Categories

2024-01-31T11:18:18Z

← Older revision		Revision as of 11:21, 31 January 2024
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	[[Category:Dual-fifth]]		[[Category:Dual-fifth]]
	~~[[Category:Lists of temperaments]]~~
	[[Category:Subgroup temperaments]]		[[Category:Subgroup temperaments]]
	[[Category:Temperament collections]]		[[Category:Temperament collections]]

← Older revision		Revision as of 11:18, 31 January 2024
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	[[Category:Dual-fifth]]		[[Category:Dual-fifth]]
	[[Category:Lists of temperaments]]		[[Category:Lists of temperaments]]
			[[Category:Subgroup temperaments]]
			[[Category:Temperament collections]]

Dual-fifth temperaments - Revision history

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

Xenllium at 03:32, 29 June 2025

Xenllium at 02:31, 29 June 2025

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

FloraC: Text replacement - "{{Technical data page}}" to "{{Technical data page}}"

BudjarnLambeth: {{Technical data page}}

Godtone: note ripple

BudjarnLambeth: /* Dual-3 Octokaidecal */ moved from todo to todo:review

BudjarnLambeth: Categories

BudjarnLambeth: Categories

FloraC: Text replacement - "{{Technical data page}}

" to "{{Technical data page}}"