56edo: Difference between revisions

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56 equal divisions of the octave (56edo), or 56-tone equal temperament (56tet), 56 equal temperament (56et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 56 equal parts of about 21.4 ¢ each, a size close to the syntonic comma [[81/80]].

== Theory ==

It shares it's near perfect 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]].

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 family #Septimal sycamore|sycamore]], and the 11-limit 56d val is close to the [[POTE tuning]] for 11-limit pajara~~. 56edo can be used to tune [[Barium]] temperament, which sets 56 syntonic commas to the octave~~.

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 family #Septimal sycamore|sycamore]], and the 11-limit 56d val is close to the [[POTE tuning]] for 11-limit pajara.

=== Prime harmonics ===

=== Subsets and supersets ===

56edo has subset edos {{EDOs|1, 2, 4, 7, 8, 14, 28}}.

One step of 56edo is the closest direct approximation to the syntonic comma, [[81/80]], with the unrounded value 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.

== Intervals ==

The following table assumes the [[patent val]] {{val| 56 89 130 157 194 207 }}. Other approaches are possible.

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 {{Infobox ET}}
-equal divisions of the octave (56edo), or 56-tone equal temperament (56tet), 56 equal temperament (56et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 56 equal parts of about 21.4 ¢ each, a size close to the syntonic comma [[81/80]].
+{{EDO intro|56}}
 == Theory ==
 It shares it's near perfect 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]].
-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 family #Septimal sycamore|sycamore]], and the 11-limit 56d val is close to the [[POTE tuning]] for 11-limit pajara. 56edo can be used to tune [[Barium]] temperament, which sets 56 syntonic commas to the octave.
+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 family #Septimal sycamore|sycamore]], and the 11-limit 56d val is close to the [[POTE tuning]] for 11-limit pajara.
+=== Prime harmonics ===
 {{harmonics in equal|56}}
+=== Subsets and supersets ===
+edo has subset edos {{EDOs|1, 2, 4, 7, 8, 14, 28}}.
+One step of 56edo is the closest direct approximation to the syntonic comma, [[81/80]], with the unrounded value 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.
 == Intervals ==
 The following table assumes the [[patent val]] {{val| 56 89 130 157 194 207 }}. Other approaches are possible.