# Dual-fifth 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

Commas: [-4 2 2 -1⟩, [-8 1 4 0⟩

Mapping: [⟨1 0 2 0], ⟨0 4 -1 6]]

2.9.5 POTE generator: 464.1591

Optimal ET sequence: 13, 18, 31

## Dual-3 Sixix

Subgroup: 2.3⁻.3⁺.5

Commas: [-4 2 2 -1⟩, [2 -3 0 1⟩

Mapping: [⟨1 0 1 -2], ⟨0 2 1 6]]

2.9.5 POTE generator: 335.8409

Optimal ET sequence: 18, 25, 43

## Dual-3 Octokaidecal

Subgroup: 2.3⁻.3⁺.5

2.9.5 POTE generator: 730.0679

Todo: reviewexplicitly write out commas and vals for Dual-3 Octokaidecal |

## Megapyth

Subgroup: 2.3⁻.3⁺.5.7

Commas: [-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

Commas: [-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

Commas: 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]