User:Overthink/Draft edits: Difference between revisions

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= 56edo =

== Theory ==

56edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th in the "shrub region" between those of [[17edo]] and [[22edo]]. ~~Because it contains 28edo's major third and also~~ has ~~a step size very close~~ to the ~~syntonic comma, 56edo contains very accurate approximations~~ of ~~both the classic major third~~ [[5/4]] ~~and the Pythagorean major third~~ [[81/64]]. ~~Unfortunately~~, ~~this "Pythagorean major third"~~ is not ~~the major third as is stacked by fifths in 56edo. However, this interval represents the pythagorean major third consistently in~~ [[~~224edo~~]]~~, which is the quadruple of 56edo~~.

56edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th in the "shrub region" between those of [[17edo]] and [[22edo]]. It has unambiguous approximations to prime harmonics up to [[19/1|19]], but due to the sharpness of its harmonic [[3/1|3]], several intervals of [[9/1|9]] are inconsistent. Therefore, 56edo is not very popular compared to edos like [[53edo|53]] and [[58edo|58]].

56edo has unambiguous approximations to prime harmonics up to [[19/1|19]]. However, the harmonic [[3/1|3]] is quite sharp, leading harmonic [[9/1|9]] to be even more so, and causing intervals like [[10/9]], [[9/7]], and [[13/9]] to be inconsistent. Therefore, 56edo is not very popular compared to edos like [[53edo|53]] and [[58edo|58]].

One step of 56edo is the closest direct approximation to the syntonic comma, [[81/80]], with the number of directly approximated syntonic commas per octave being 55.7976. [[Barium]] temperament realizes this proximity through regular temperament theory, and is supported by notable edos like [[224edo]], [[1848edo]], and [[2520edo]], which is a highly composite edo. Because it contains 28edo's major third and also has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third [[5/4]] and the Pythagorean major third [[81/64]]. Unfortunately, this "Pythagorean major third" is not the major third as is stacked by fifths in 56edo, which is in fact a supermajor third of 428.6 [[cent]]s. However, this interval represents the pythagorean major third consistently in [[224edo]], which is the quadruple of 56edo.

One step of 56edo is the closest direct approximation to the syntonic comma, [[81/80]], with the number of directly approximated syntonic commas per octave being 55.7976. ~~(However, note that by [[patent val]] mapping, 56edo actually maps the syntonic comma inconsistently, to two steps.)~~ [[Barium]] temperament realizes this proximity through regular temperament theory, and is supported by notable edos like [[224edo]], [[1848edo]], and [[2520edo]], which is a highly composite edo.

~~56edo can be used~~ to ~~tune [[hemithirds]]~~, [[~~superkleismic]], [[sycamore~~]] and [[~~keen~~]] ~~temperaments~~, ~~and using {{val| 56 89 130 158 }} (56d)~~ as ~~the equal temperament val~~, ~~for~~ [[~~pajara~~]]. ~~It provides~~ the [[~~optimal patent val~~]] ~~for 7-~~, ~~11- and 13-limit [[sycamore]], and the 11-limit 56d val~~ is ~~close to~~ the ~~[[POTE tuning]] for undecimal pajara~~.

=== Prime harmonics ===

=== As a tuning of other temperaments ===

56edo can be used to tune [[hemithirds]], [[superkleismic]], [[sycamore]] and [[keen]] temperaments, and using {{val| 56 89 130 158 }} (56d) as the equal temperament val, for [[pajara]]. It provides the [[optimal patent val]] for 7-, 11- and 13-limit [[sycamore]], and the 11-limit 56d val is close to the [[POTE tuning]] for undecimal pajara.

=== Subsets and supersets ===

@@ Line 3: / Line 3: @@
 = 56edo =
 == Theory ==
-edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th in the "shrub region" between those of [[17edo]] and [[22edo]]. Because it contains 28edo's major third and also has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third [[5/4]] and the Pythagorean major third [[81/64]]. Unfortunately, this "Pythagorean major third" is not the major third as is stacked by fifths in 56edo. However, this interval represents the pythagorean major third consistently in [[224edo]], which is the quadruple of 56edo.
+edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th in the "shrub region" between those of [[17edo]] and [[22edo]]. It has unambiguous approximations to prime harmonics up to [[19/1|19]], but due to the sharpness of its harmonic [[3/1|3]], several intervals of [[9/1|9]] are inconsistent. Therefore, 56edo is not very popular compared to edos like [[53edo|53]] and [[58edo|58]].
-edo has unambiguous approximations to prime harmonics up to [[19/1|19]]. However, the harmonic [[3/1|3]] is quite sharp, leading harmonic [[9/1|9]] to be even more so, and causing intervals like [[10/9]], [[9/7]], and [[13/9]] to be inconsistent. Therefore, 56edo is not very popular compared to edos like [[53edo|53]] and [[58edo|58]].
+One step of 56edo is the closest direct approximation to the syntonic comma, [[81/80]], with the number of directly approximated syntonic commas per octave being 55.7976. [[Barium]] temperament realizes this proximity through regular temperament theory, and is supported by notable edos like [[224edo]], [[1848edo]], and [[2520edo]], which is a highly composite edo. Because it contains 28edo's major third and also has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third [[5/4]] and the Pythagorean major third [[81/64]]. Unfortunately, this "Pythagorean major third" is not the major third as is stacked by fifths in 56edo, which is in fact a supermajor third of 428.6 [[cent]]s. However, this interval represents the pythagorean major third consistently in [[224edo]], which is the quadruple of 56edo.
-One step of 56edo is the closest direct approximation to the syntonic comma, [[81/80]], with the number of directly approximated syntonic commas per octave being 55.7976. (However, note that by [[patent val]] mapping, 56edo actually maps the syntonic comma inconsistently, to two steps.) [[Barium]] temperament realizes this proximity through regular temperament theory, and is supported by notable edos like [[224edo]], [[1848edo]], and [[2520edo]], which is a highly composite edo.
-edo can be used to tune [[hemithirds]], [[superkleismic]], [[sycamore]] and [[keen]] temperaments, and using {{val| 56 89 130 158 }} (56d) as the equal temperament val, for [[pajara]]. It provides the [[optimal patent val]] for 7-, 11- and 13-limit [[sycamore]], and the 11-limit 56d val is close to the [[POTE tuning]] for undecimal pajara.
 === Prime harmonics ===
 {{Harmonics in equal|56}}
+=== As a tuning of other temperaments ===
+edo can be used to tune [[hemithirds]], [[superkleismic]], [[sycamore]] and [[keen]] temperaments, and using {{val| 56 89 130 158 }} (56d) as the equal temperament val, for [[pajara]]. It provides the [[optimal patent val]] for 7-, 11- and 13-limit [[sycamore]], and the 11-limit 56d val is close to the [[POTE tuning]] for undecimal pajara.
 === Subsets and supersets ===