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]]. 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 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 decent 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]] or [[58edo]].

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. [[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, the Pythagorean major third is represented as such consistently in [[224edo]], which is the quadruple of 56edo.

=== Prime harmonics ===

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=== 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.

As an equal temperament, 56et most notably tempers out [[2048/2025]], the diaschisma, as well as the [[shibboleth comma]] in the [[5-limit]]. Using the patent val, it tempers out [[686/675]], [[875/864]], and [[1029/1024]] in the [[7-limit]], [[100/99]], [[245/242]], and [[385/384]] in the [[11-limit]], and [[91/90]] and [[169/168]] in the 13-limit. It supports 7- and 11-limit keen, and its 13- and 17-limit extension [[keenic]]. It also supports [[hemithirds]], [[superkleismic]], and [[sycamore]] in various limits, being an especially optimal tuning for sycamore in the 11-, and 13-limits. It is also a very sharp tuning of [[slendric]], mapping 7/6 to a wide semifourth of 257.1{{c}}, and 9/7 inconsistently to a 450{{c}} [[Interseptimal interval|naiadic]].

Another interesting val to consider is 56d ({{Val|56 89 130 '''158'''}}), which maps 7/4 sharply to around 986{{c}}. This mapping tempers out [[50/49]] and [[64/63]], providing an alternative to [[22edo]] for [[pajara]]. It improves accuracy of the 3rd harmonic and makes the 5th harmonic basically just, especially improving [[6/5]] and [[10/9]], which are quite out of tune in 22edo. Its 7th harmonic is sharper than 22edo's, and combined with the fact that the 3rd harmonic is sharp, one may want to [[Octave stretch|compress the octave]], using tunings such as [[145ed6]] or [[201ed12]]. It is also an excellent tuning for the 11-limit version of pajara, which tempers out [[99/98]] and [[100/99]].

=== Subsets and supersets ===

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 = 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]]. 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 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 decent 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]] or [[58edo]].
-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. [[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, the Pythagorean major third is represented as such consistently in [[224edo]], which is the quadruple of 56edo.
 === Prime harmonics ===
@@ Line 11: / Line 11: @@
 === 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.
+As an equal temperament, 56et most notably tempers out [[2048/2025]], the diaschisma, as well as the [[shibboleth comma]] in the [[5-limit]]. Using the patent val, it tempers out [[686/675]], [[875/864]], and [[1029/1024]] in the [[7-limit]], [[100/99]], [[245/242]], and [[385/384]] in the [[11-limit]], and [[91/90]] and [[169/168]] in the 13-limit. It supports 7- and 11-limit keen, and its 13- and 17-limit extension [[keenic]]. It also supports [[hemithirds]], [[superkleismic]], and [[sycamore]] in various limits, being an especially optimal tuning for sycamore in the 11-, and 13-limits. It is also a very sharp tuning of [[slendric]], mapping 7/6 to a wide semifourth of 257.1{{c}}, and 9/7 inconsistently to a 450{{c}} [[Interseptimal interval|naiadic]].
+Another interesting val to consider is 56d ({{Val|56 89 130 '''158'''}}), which maps 7/4 sharply to around 986{{c}}. This mapping tempers out [[50/49]] and [[64/63]], providing an alternative to [[22edo]] for [[pajara]]. It improves accuracy of the 3rd harmonic and makes the 5th harmonic basically just, especially improving [[6/5]] and [[10/9]], which are quite out of tune in 22edo. Its 7th harmonic is sharper than 22edo's, and combined with the fact that the 3rd harmonic is sharp, one may want to [[Octave stretch|compress the octave]], using tunings such as [[145ed6]] or [[201ed12]]. It is also an excellent tuning for the 11-limit version of pajara, which tempers out [[99/98]] and [[100/99]].
 === Subsets and supersets ===