Dual-fifth temperaments: Difference between revisions
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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]] | [[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: | 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]] | [[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 }} | ||
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[[Subgroup]]: 2.3⁻.3⁺.5 | [[Subgroup]]: 2.3⁻.3⁺.5 | ||
2 | [[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 }}] | |||
2.9.5 [[POTE]] generator: ~3⁺/2 = 730.0679 | |||
{{ | {{Optimal ET sequence|legend=1| 18, 28 }} | ||
== Megapyth == | == Megapyth == | ||
[[Subgroup]]: 2.3⁻.3⁺.5.7 | [[Subgroup]]: 2.3⁻.3⁺.5.7 | ||
[[Comma]] | [[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]] | [[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]] | [[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⁻. | Subgroup: 2.3⁻.3⁺.5.7.11 | ||
See [[ripple]]. | See [[ripple]]. | ||