Dual-fifth tuning: Difference between revisions

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[[18edo]] is usually considered the quintessential dual-fifth edo by people who work in dual-fifth systems, which has a 733.3c sharp fifth 31.4c sharp from pure [[3/2], and a 666.7c flat fifth 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 usually 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⁻.9.5" and 2.3⁺.3⁻.5 are equivalent subgroups.

Unlike conventional temperaments, "[[dual-fifth temperaments]]" do not attempt to optimize every interval to low-limit JI, but usually 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⁻.9.5" and 2.3⁺.3⁻.5 are equivalent subgroups.

For example, "dual-3 sixix" is a 2.3⁻.9.5 temperament with an optimal generator around 335.8c (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.

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 [[18edo]] is usually considered the quintessential dual-fifth edo by people who work in dual-fifth systems, which has a 733.3c sharp fifth 31.4c sharp from pure [[3/2], and a 666.7c flat fifth 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 usually 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⁻.9.5" and 2.3⁺.3⁻.5 are equivalent subgroups.
+Unlike conventional temperaments, "[[dual-fifth temperaments]]" do not attempt to optimize every interval to low-limit JI, but usually 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⁻.9.5" and 2.3⁺.3⁻.5 are equivalent subgroups.
 For example, "dual-3 sixix" is a 2.3⁻.9.5 temperament with an optimal generator around 335.8c (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.