Dual-fifth temperaments: Difference between revisions

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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 ==

[[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 }}]

2.9.5 [[POTE]] generator: ~~464~~.~~1591~~

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

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[[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

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[[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 }}]

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

2.9.5 [[POTE]] generator: ~3⁺/2 = 730.0679

=== 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

== Megapyth ==

[[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 }}]

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2.9.5.7 [[POTE]] generator: ~3⁺/2 = 715.319

== Duofamity (Rank-3) ==

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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 }}]

== Travesty ==

[[Subgroup]]: 2.3.5.97

[[Comma]]s: [[177147/163840]], 2619/2560

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

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== [[Ripple]] ==

Subgroup: 2.3⁻.9.5.7.11

Subgroup: 2.3⁻.3⁺.5.7.11

See [[ripple]].

@@ Line 4: / Line 4: @@
 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 ==
 [[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 }}]
-.9.5 [[POTE]] generator: 464.1591
+.9.5 [[POTE]] generator: ~3⁺/2 = 735.8409
 {{Optimal ET sequence|legend=1| 13, 18, 31 }}
@@ Line 20: / Line 20: @@
 [[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 }}]
-.9.5 [[POTE]] generator: 335.8409
+.9.5 [[POTE]] generator: ~(2×3⁺)/5 = 335.8409
 {{Optimal ET sequence|legend=1| 18, 25, 43 }}
@@ Line 31: / Line 31: @@
 [[Subgroup]]: 2.3⁻.3⁺.5
-.9.5 [[POTE]] generator: 730.0679
+[[Comma|Comma basis]]: [[25/24|{{monzo| -3 1 -2 2 }}]], [[256/243|{{monzo| 8 -1 -4 0 }}]]
-{{Optimal ET sequence|legend=1|18, 28}}
+Mapping: [{{val| 2 8 2 1 }}, {{val| 0 -4 1 3 }}]
-{{Todo|review|inline=1|comment=explicitly write out commas and vals for Dual-3 Octokaidecal}}
+.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 }}]
+.9.5.7 POTE generator: ~3⁺/2 = 728.6349
+{{Optimal ET sequence|legend=0| 18, 28 }}
 == Megapyth ==
 [[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 }}]
@@ Line 46: / Line 59: @@
 .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) ==
@@ Line 55: / Line 68: @@
 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 }}]
-{{Optimal ET sequence|legend=1|30c, 47b, 52b}}
+{{Optimal ET sequence|legend=1| 30c, 47b, 52b }}
 == Travesty ==
 [[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}}
@@ Line 73: / Line 86: @@
 == [[Ripple]] ==
-Subgroup: 2.3⁻.9.5.7.11
+Subgroup: 2.3⁻.3⁺.5.7.11
 See [[ripple]].