Dual-fifth temperaments: Difference between revisions
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{{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]]. | |||
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 == | == Dual-3 A-Team == | ||
Subgroup: 2.3⁻. | [[Subgroup]]: 2.3⁻.3⁺.5 | ||
Comma | [[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 | Mapping: [{{val| 1 0 2 0 }}, {{val| 0 4 -1 6 }}] | ||
2.9.5 POTE generator: | 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 }} | ||
== Dual-3 Sixix == | == Dual-3 Sixix == | ||
Subgroup: 2.3⁻. | [[Subgroup]]: 2.3⁻.3⁺.5 | ||
Comma | [[Comma|Comma basis]]: [[25/24|{{monzo| -3 1 -2 2 }}]], [[81/80|{{monzo| -4 2 2 -1 }}]] | ||
Mapping: [{{val| 1 | 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 }} | ||
== Dual-3 Octokaidecal == | == Dual-3 Octokaidecal == | ||
Subgroup: 2.3⁻. | [[Subgroup]]: 2.3⁻.3⁺.5 | ||
[[Comma|Comma basis]]: [[25/24|{{monzo| -3 1 -2 2 }}]], [[256/243|{{monzo| 8 -1 -4 0 }}]] | |||
2 | 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 | ||
{{Optimal ET sequence|legend=1| 18, 28 }} | |||
=== 7-limit === | |||
Subgroup: 2.3⁻.3⁺.5.7 | |||
Comma basis: [[25/24|{{monzo| -3 1 -2 2 0 }}]], [[49/48|{{monzo| -4 1 -2 0 2 }}]], [[64/63|{{monzo| 6 -1 -1 0 -1 }}]] | |||
Mapping: [{{val| 2 8 2 1 2 }}, {{val| 0 -4 1 3 3 }}] | |||
2.9.5.7 POTE generator: ~3⁺/2 = 728.6349 | |||
{{Optimal ET sequence|legend=0| 18, 28 }} | |||
== Megapyth == | == Megapyth == | ||
Subgroup: 2. | [[Subgroup]]: 2.3⁻.3⁺.5.7 | ||
[[Comma|Comma basis]]: {{monzo| -4 3 1 -1 0 }}, {{monzo| 6 0 -2 0 -1 }}, {{monzo| -5 -1 6 0 0 }} | |||
2 | Mapping: [{{val| 1 1 1 0 4 }}, {{val| 0 6 1 19 -2 }}] | ||
2.9.5.7 [[POTE]] generator: ~3⁺/2 = 715.319 | |||
{{Optimal ET sequence|47b, 52b}} | {{Optimal ET sequence|legend=1| 47b, 52b }} | ||
== Duofamity (Rank-3) == | == Duofamity (Rank-3) == | ||
Subgroup: 2. | [[Subgroup]]: 2.3⁻.3⁺.5.7 | ||
Generators: 2, | Generators: 2, 3⁻, 3⁺ | ||
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]] | [[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 }}] | |||
{{Optimal ET sequence|30c, 47b, 52b}} | {{Optimal ET sequence|legend=1| 30c, 47b, 52b }} | ||
== Travesty == | == Travesty == | ||
[[Subgroup]]: 2.3.5.97 | [[Subgroup]]: 2.3.5.97 | ||
[[Comma | [[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}} | ||
| Line 60: | Line 84: | ||
[[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⁻.3⁺.5.7.11 | |||
See [[ripple]]. | |||
[[Category:Dual-fifth]] | [[Category:Dual-fifth]] | ||
[[Category: | [[Category:Subgroup temperaments]] | ||
[[Category:Temperament collections]] | |||
[[Category:Pages with mostly numerical content]] | |||