Dual-fifth tuning: Difference between revisions

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== Dual-fifth edos ==

[[~~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~~.

[[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 appoximants of five staff positions of the diatonic scale.

~~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.

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 [[2L 5s|mavila]] generators or subminor sixths, and not every musical approach treats them as approximants of 3/2 or intervals playing the role of the Western 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.

Some other edos which have been studied as dual-fifth are:

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 == Dual-fifth edos ==
-[[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.
+[[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 appoximants of five staff positions of the diatonic scale.
-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.
+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 [[2L 5s|mavila]] generators or subminor sixths, and not every musical approach treats them as approximants of 3/2 or intervals playing the role of the Western 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.
 Some other edos which have been studied as dual-fifth are: