Dual-fifth tuning: Difference between revisions

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A '''dual-fifth system''' is a(n octave-equivalent) tuning system with two sizes of fifths, '''major fifth''' and '''minor fifth''' instead of a single perfect fifth, and accordingly two sizes fourths, '''major fourth''' and '''minor fourth''' instead of a single perfect fourth.

== Dual-fifth edos ==

[[18edo]] is usually considered the quintessential dual-fifth edo by people who work in dual-fifth systems (which are admittedly few), which has a 733.3c sharp fifth 31.4c sharp from pure [[3/2]], and a 666.7c flat fifth almost equally off as the sharp one, 35.3 cents flat.

[[18edo]] is usually considered the quintessential dual-fifth edo by people who work in dual-fifth systems (which are admittedly few), which has a 733.3¢ sharp fifth 31.4c sharp from pure [[3/2]], and a 666.7¢ flat fifth almost equally off as the sharp one, 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 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 335.8c (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.

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 {{val|18 28 29 42}}.

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 A '''dual-fifth system''' is a(n octave-equivalent) tuning system with two sizes of fifths, '''major fifth''' and '''minor fifth''' instead of a single perfect fifth, and accordingly two sizes fourths, '''major fourth''' and '''minor fourth''' instead of a single perfect fourth.
 == Dual-fifth edos ==
-[[18edo]] is usually considered the quintessential dual-fifth edo by people who work in dual-fifth systems (which are admittedly few), which has a 733.3c sharp fifth 31.4c sharp from pure [[3/2]], and a 666.7c flat fifth almost equally off as the sharp one, 35.3 cents flat.
+[[18edo]] is usually considered the quintessential dual-fifth edo by people who work in dual-fifth systems (which are admittedly few), which has a 733.3¢ sharp fifth 31.4c sharp from pure [[3/2]], and a 666.7¢ flat fifth almost equally off as the sharp one, 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 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 335.8c (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.
+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.
 edo 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}}.