Dual-fifth tuning

A dual-fifth tuning system is a tuning system, often octave-equivalent, with two sizes of fifths, major fifth and minor fifth instead of a single perfect fifth, and accordingly two sizes of fourths, major fourth and minor fourth instead of a single perfect fourth. The opposite of dual-fifth may be called plain-fifth.

Dual-fifth edos

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

Some other edos which have been studied as dual-fifth are:

• 13edo (major fifth +36.5¢ from just, minor fifth -55.8¢ from just)
• 23edo (major fifth +28.5¢ from just, minor fifth -23.7¢ from just)
• 25edo (major fifth +18.0¢ from just, minor fifth -30.0¢ from just)
• 35edo
• 47edo
• 59edo
• 100edo
• 112edo

We may, heuristically, define dual-fifth edos as those whose relative error of the third harmonic is greater than 1/3. In that case 1/3 of all edos will be dual-fifth and the other 2/3 will be plain-fifth.

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].

Multiple-fifth temperaments

By extension, it is also possible to consider a multiple fifth temperament where

[math]\prod_{N=1}^{n} 3^{(N)} = 3^n[/math].

That is, all the different mappings of 3 align eventually at a 3ⁿ interval.

For example, 91edo has 3 usable fifths with their own functions - 52\91 (3^-), 53\91 (3), and 54\91 (3⁺). Thus, if used this way they do not represent distinct dimensions, but rather correspon to 3 × 3^- × 3⁺ = 27/1.