Dual-fifth tuning: Difference between revisions

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== Dual-fifth edos ==

[[18edo]] is ~~usually considered<sup>[by whom?]<~~/~~sup> 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 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.

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.

Another notable 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:

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* [[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 ==

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 == Dual-fifth edos ==
-[[18edo]] is usually considered<sup>[by whom?]</sup> 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 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.
-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.
+Another notable 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:
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 * [[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 ==