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 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 ==

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=== 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 ==

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

== Travesty ==

@@ 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 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 }}]
-.9.5 [[POTE]] generator: ~(3⁻×3⁺)/8 = 192.4773
-{{Optimal ET sequence|legend=1| 6, 13, 19 }}
 == Dual-3 A-Team ==
@@ Line 49: / Line 38: @@
 {{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 ==
@@ Line 72: / Line 72: @@
 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 ==