Dual-fifth temperaments
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
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 temperaments, where one of the fifths is mapped to 3/2 and the other is mapped to a nearby 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 basis: [-4 2 2 -1⟩, [8 -1 -4 0⟩
Mapping: [⟨1 0 2 0], ⟨0 4 -1 6]]
2.9.5 POTE generator: ~3⁺/2 = 735.8409
Optimal ET sequence: 13, 18, 31
Dual-3 Sixix
Subgroup: 2.3⁻.3⁺.5
Comma basis: [-3 1 -2 2⟩, [-4 2 2 -1⟩
Mapping: [⟨1 1 3 4], ⟨0 2 -5 -6]]
2.9.5 POTE generator: ~(2×3⁺)/5 = 335.8409
Optimal ET sequence: 18, 25, 43
Dual-3 Octokaidecal
Subgroup: 2.3⁻.3⁺.5
Comma basis: [-3 1 -2 2⟩, [8 -1 -4 0⟩
Mapping: [⟨2 8 2 1], ⟨0 -4 1 3]]
2.9.5 POTE generator: ~3⁺/2 = 730.0679
7-limit
Subgroup: 2.3⁻.3⁺.5.7
Comma basis: [-3 1 -2 2 0⟩, [-4 1 -2 0 2⟩, [6 -1 -1 0 -1⟩
Mapping: [⟨2 8 2 1 2], ⟨0 -4 1 3 3]]
2.9.5.7 POTE generator: ~3⁺/2 = 728.6349
Megapyth
Subgroup: 2.3⁻.3⁺.5.7
Comma basis: [-4 3 1 -1 0⟩, [6 0 -2 0 -1⟩, [-5 -1 6 0 0⟩
Mapping: [⟨1 1 1 0 4], ⟨0 6 1 19 -2]]
2.9.5.7 POTE generator: ~3⁺/2 = 715.319
Duofamity (Rank-3)
Subgroup: 2.3⁻.3⁺.5.7
Generators: 2, 3⁻, 3⁺
Least-squares for 5/4, 9/8, and 8/7: 690.155, 715.325
Comma basis: [-4 3 1 -1 0⟩, [6 0 -2 0 -1⟩
Mapping: [⟨1 0 0 -4 6], ⟨0 1 0 3 0], ⟨0 0 1 1 -2]]
Optimal ET sequence: 30c, 47b, 52b
Travesty
Subgroup: 2.3.5.97
Comma basis: 177147/163840, 2619/2560
Sval mapping: [⟨1 1 -4 2], ⟨0 1 11 8]]
CTE generator: ~3/2 = 689.886
Supporting ETs: 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.