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^{[by whom?]} 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.

Another tuning system which can be regarded as "quintessential" dual-fifth system is 35edo, since its fifths of 685.71¢ (derived from 7edo) and 720¢ (derived from 5edo) correspond to the bounds of the tuning range for the diatonic scale, the predominating scale in the world musical practice. Edos like 18edo and 25edo have intervals that are more considered as mavila generators or subminor sixths, and not every musical approach treats them as fifths or approximants of 3/2.

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)
• 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]\displaystyle{ \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.