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).