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 scales
Sixix[7] can be regarded as the scale which is the most authentic representation of the "dual-fifth" phenomenon via its modes, since it features both sharp and flat fifth on different modes, and the interval in this case occupies 5 staff positions. For example, in 25edo, sixix can take form of 4 3 4 3 4 3 4, where five staff positions occupy 18\25 (sharp fifth), but if the mode is 3 4 3 4 3 4 4, then five staff positions are equal to 17\25 (flat fifth). However, it should be noted that in better tunings of sixix, the flat fifth is Mavila-like in quality while the sharp fifth is a comparatively accurate superpyth diatonic fifth.
Dual-fifth edos
35edo is the equal temperament which can be said to most authentically represent the concept of "dual-fifth", since its fifths of 20\35 and 21\35 correspond to the bounds of the tuning range for the diatonic scale where the term fifth in the standard Western practice originates from. 35edo is the largest edo without a diatonic scale, and it is therefore the smallest whose sharp and flat fifth can be equally treated as being approximants of five staff positions of the diatonic scale.
Although edos like 18edo, 23edo and 25edo have been extensively studied as dual-fifth, their corresponding dual-fifth intervals that are also often considered as antidiatonic generators or subminor sixths, and not every musical approach treats them as approximants of 3/2 or intervals playing the role of the fifth.
Perhaps a more familiar dual-fifth system to many is 18edo. It is the first system that has intervals that are close enough to 3/2 that they can be regarded as sharp and flat fifth, but also far enough to sound different. 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.
For a list of edos which could be considered dual-fifth, see:
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].
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).
For a list of dual-fifth temperaments and their properties, see:
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 correspond to 3 × 3- × 3+ = 27/1.