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{{Infobox ET}}
{{Infobox ET}}
{{ED intro}}  
{{ED intro}}


== Theory ==
== 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.  
56edo shares its near perfect quality of the [[5/4|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 of [[prime harmonic]]s up to [[19/1|19]], but due to the sharpness of its harmonic [[3/1|3]], several intervals of [[9/1|9]] are [[consistency|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. (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.
=== Prime harmonics ===
{{Harmonics in equal|56}}
 
=== As a tuning of other temperaments ===
In the 5-limit, 56et most notably tempers out the [[diaschisma]], as well as the [[shibboleth comma]]. 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 the diaschismic extension [[keen]] in the 7- and 11-limit, 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 also supports a very sharp tuning of [[slendric]], mapping 7/6 to an [[Ultramajor and inframinor|inframinor]] third of 257.1[[Cent|{{c}}]], and mapping 9/7 inconsistently to an ultramajor third of 450{{c}}.
 
Another interesting val to consider is 56d ({{Val|56 89 130 '''158''' 194}}), which maps 7/4 sharply to around 986{{c}}. This mapping tempers out [[50/49]] and [[64/63]] in the 7-limit, 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 approximated 7th harmonic is sharper than 22edo's, and combined with the fact that the 3rd harmonic is sharp, one may wish 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 additionally tempers out [[99/98]], [[100/99]], [[176/175]], and [[896/891]]. Finally, it gives an excellent tuning for the [[2.3.7.11 subgroup|2.3.7.11-subgroup]] [[supra]] temperament tempering out [[64/63]] and [[99/98]].


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.
=== Miscellaneous properties ===


=== Prime harmonics ===
One step of 56edo is the closest to the syntonic comma, [[81/80]], of any integer edo's step size by [[direct approximation]], 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 has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third [[5/4]] at 18\56, and the Pythagorean major third [[81/64]] at 19\56. Unfortunately, 56edo does not map the Pythagorean major third 19\56, but instead inconsistently to 20\56, a supermajor third of 428.6{{c}}. However, the Pythagorean major third is mapped to 19\56 consistently in [[224edo]], which is the quadruple of 56edo.
{{Harmonics in equal|56}}
 
The perfect fifth generates a [[5L 2s|diatonic]] scale with a [[step ratio]] that is a convergent towards the [[Metallic harmonic series|bronze metallic mean]], following [[17edo]] and preceding [[185edo]].


=== Subsets and supersets ===
=== 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 }}.
Since 56 factors into {{nowrap|2<sup>3</sup> &times; 7}}, 56edo has subset edos {{EDOs| 2, 4, 7, 8, 14, and 28}}.


== Intervals ==
== Intervals ==
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=== Ups and downs notation ===
=== Ups and downs notation ===
56edo can be notated using [[ups and downs notation|ups and downs]]. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.
56edo can be notated using [[ups and downs notation|ups and downs]]. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.
{{Ups and downs sharpness}}
{{Ups and downs sharpness}}
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| [[Sevond]]
| [[Sevond]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


== Scales ==
== Scales ==
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== Music ==
== Music ==
=== Modern renderings ===
; {{W|The Beatles}}
* [https://www.youtube.com/shorts/WsvSVp3xyr8 "I Will" from ''The Beatles''] (1968) – covered by [[Bryan Deister]] (2026)
; {{W|Susumu Hirasawa}}
* [https://www.youtube.com/watch?v=mGcPxb-ESAQ "Parade" from ''Paprika OST''] (2006) – covered by Bryan Deister (2026)
; LSPLASH
* [https://www.youtube.com/watch?v=xkfao6yGKGE "Curious Light" from ''DOORS OST''] (2023) – covered by Bryan Deister (2025)
=== 21st century ===
; [[Bryan Deister]]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/o0imqFPDh9k ''56edo''] (2023)
* [https://www.youtube.com/shorts/o0imqFPDh9k ''56edo''] (2023)
* [https://www.youtube.com/watch?v=xkfao6yGKGE ''Curious Light - DOORS (microtonal cover in 56edo)''] (2025)
* [https://www.youtube.com/watch?v=qzMOnS-lgWs ''Waltz in 56edo''] (2025)
* [https://www.youtube.com/watch?v=qzMOnS-lgWs ''Waltz in 56edo''] (2025)


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