Dual-fifth tuning

A dual-fifth system is a(n octave-equivalent) tuning system with two sizes of fifths, major fifth and minor fifth instead of a single perfect fifth, and accordingly two sizes fourths, major fourth and minor fourth instead of a single perfect fourth.

Dual-fifth edos

18edo is usually considered the quintessential dual-fifth edo by people who work in dual-fifth systems (which are admittedly few), which has a 733.3¢ sharp fifth 31.4c sharp from pure 3/2, and a 666.7¢ flat fifth almost equally off as the sharp one, 35.3 cents flat.

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.

18edo is notable for supporting both dual-3 sixix and dual-3 A-Team with the 2.3⁻.3⁺.5 val ⟨18 28 29 42].