Dual-fifth tuning: Difference between revisions
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A '''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 == | == 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. | [[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 | Some other edos which have been studied as dual-fifth are: | ||
* [[13edo]] (major fifth +36.5¢ from just, minor fifth -55.8¢ from just) | * [[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) | * [[23edo]] (major fifth +28.5¢ from just, minor fifth -23.7¢ from just) | ||
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* [[100edo]] | * [[100edo]] | ||
* [[112edo]] | * [[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 == | == Dual-fifth temperaments == | ||
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18edo is notable for supporting both dual-3 sixix and dual-3 A-Team with the 2.3⁻.3⁺.5 val {{val|18 28 29 42}}. | 18edo is notable for supporting both dual-3 sixix and dual-3 A-Team with the 2.3⁻.3⁺.5 val {{val|18 28 29 42}}. | ||
== Multiple fifth temperaments == | == Multiple-fifth temperaments == | ||
By extension, it | By extension, it is also possible to consider a multiple fifth temperament where | ||
:<math>\prod_{N=1}^{ | :<math>\prod_{N=1}^{n} 3^{(N)} = 3^n</math>. | ||
That is, all the different mappings of 3 align eventually at a 3<sup>n</sup> interval. | That is, all the different mappings of 3 align eventually at a 3<sup>''n''</sup> interval. | ||
For example, [[91edo]] has 3 usable fifths with their own functions - 52\91 ( | For example, [[91edo]] has 3 usable fifths with their own functions - 52\91 (3<sup>-</sup>), 53\91 (3), and 54\91 (3<sup>+</sup>). Thus, if used this way they do not represent distinct dimensions, but rather correspon to 3 × 3<sup>-</sup> × 3<sup>+</sup> = 27/1. | ||
[[Category:Dual-fifth| ]] | [[Category:Dual-fifth| ]] <!-- main article --> | ||
[[Category:Terms]] | [[Category:Terms]] | ||
Revision as of 09:34, 30 December 2022
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]\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.