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 that is a convergent towards the [[Metallic harmonic series|bronze metallic mean]], following [[17edo]] and preceding [[185edo]]. 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 has unambiguous approximations to prime harmonics up to [[19/1|19]], and possibly up to [[29/1|29]]. However 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. (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 ===

=== Subsets and supersets ===

Since 56 factors into {{nowrap|2<sup>3</sup> × 7}}, 56edo has subset edos {{EDOs| 2, 4, 7, 8, 14, 28 }}.

= Main page =

== Welcome to the Xenharmonic Wiki! ==

@@ Line 327: / Line 327: @@
 | Octave
 |}
+= 56edo =
+== Theory ==
+edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th that is a convergent towards the [[Metallic harmonic series|bronze metallic mean]], following [[17edo]] and preceding [[185edo]]. 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 has unambiguous approximations to prime harmonics up to [[19/1|19]], and possibly up to [[29/1|29]]. However 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. (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}}
+=== Subsets and supersets ===
+Since 56 factors into {{nowrap|2<sup>3</sup> &times; 7}}, 56edo has subset edos {{EDOs| 2, 4, 7, 8, 14, 28 }}.
 = Main page =
 == Welcome to the Xenharmonic Wiki! ==