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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|56}}  
{{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]].
 
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 11-limit pajara.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|56}}
{{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]].
=== Miscellaneous properties ===
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.
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}}.
 
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 ==
== Intervals ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
! &#35;
! #
! Cents
! Cents
! Approximate ratios*
! Approximate ratios*
Line 24: Line 31:
|-
|-
| 0
| 0
| 0.000
| 0.0
| [[1/1]]
| [[1/1]]
| {{UDnote|step=0}}
| {{UDnote|step=0}}
|-
|-
| 1
| 1
| 21.429
| 21.4
| ''[[49/48]]'', [[64/63]]
| ''[[49/48]]'', [[55/54]], [[56/55]], [[64/63]]
| {{UDnote|step=1}}
| {{UDnote|step=1}}
|-
|-
| 2
| 2
| 42.857
| 42.9
| ''[[28/27]]'', [[50/49]], ''[[81/80]]''
| ''[[28/27]]'', [[40/39]], [[45/44]], [[50/49]], ''[[81/80]]''
| {{UDnote|step=2}}
| {{UDnote|step=2}}
|-
|-
| 3
| 3
| 64.286
| 64.3
| [[25/24]], ''[[36/35]]'', ''[[33/32]]''
| [[25/24]], ''[[36/35]]'', ''[[33/32]]''
| {{UDnote|step=3}}
| {{UDnote|step=3}}
|-
|-
| 4
| 4
| 85.714
| 85.7
| [[21/20]], [[22/21]]
| [[19/18]], [[20/19]], [[21/20]], [[22/21]]
| {{UDnote|step=4}}
| {{UDnote|step=4}}
|-
|-
| 5
| 5
| 107.143
| 107.1
| [[16/15]]
| [[16/15]], [[17/16]], [[18/17]]
| {{UDnote|step=5}}
| {{UDnote|step=5}}
|-
|-
| 6
| 6
| 128.571
| 128.6
| [[15/14]], [[13/12]], [[14/13]]
| [[15/14]], [[13/12]], [[14/13]]
| {{UDnote|step=6}}
| {{UDnote|step=6}}
|-
|-
| 7
| 7
| 150.000
| 150.0
| [[12/11]]
| [[12/11]]
| {{UDnote|step=7}}
| {{UDnote|step=7}}
|-
|-
| 8
| 8
| 171.429
| 171.4
| ''[[10/9]]'', [[11/10]]
| ''[[10/9]]'', [[11/10]], [[21/19]]
| {{UDnote|step=8}}
| {{UDnote|step=8}}
|-
|-
| 9
| 9
| 192.857
| 192.9
| [[28/25]]
| [[19/17]], [[28/25]]
| {{UDnote|step=9}}
| {{UDnote|step=9}}
|-
|-
| 10
| 10
| 214.286
| 214.3
| [[9/8]]
| [[9/8]], [[17/15]]
| {{UDnote|step=10}}
| {{UDnote|step=10}}
|-
|-
| 11
| 11
| 235.714
| 235.7
| [[8/7]]
| [[8/7]]
| {{UDnote|step=11}}
| {{UDnote|step=11}}
|-
|-
| 12
| 12
| 257.143
| 257.1
| [[7/6]], [[15/13]]
| [[7/6]]
| {{UDnote|step=12}}
| {{UDnote|step=12}}
|-
|-
| 13
| 13
| 278.571
| 278.6
| [[75/64]], [[13/11]]
| [[13/11]], [[20/17]]
| {{UDnote|step=13}}
| {{UDnote|step=13}}
|-
|-
| 14
| 14
| 300.000
| 300.0
| [[25/21]]
| [[19/16]], [[25/21]]
| {{UDnote|step=14}}
| {{UDnote|step=14}}
|-
|-
| 15
| 15
| 321.429
| 321.4
| [[6/5]]
| [[6/5]]
| {{UDnote|step=15}}
| {{UDnote|step=15}}
|-
|-
| 16
| 16
| 342.857
| 342.9
| [[11/9]], [[39/32]]
| [[11/9]], [[17/14]]
| {{UDnote|step=16}}
| {{UDnote|step=16}}
|-
|-
| 17
| 17
| 364.286
| 364.3
| [[27/22]], [[16/13]], [[26/21]]
| [[16/13]], [[21/17]], [[26/21]]
| {{UDnote|step=17}}
| {{UDnote|step=17}}
|-
|-
| 18
| 18
| 385.714
| 385.7
| [[5/4]]
| [[5/4]]
| {{UDnote|step=18}}
| {{UDnote|step=18}}
|-
|-
| 19
| 19
| 407.143
| 407.1
| [[14/11]]
| [[14/11]], [[19/12]], [[24/19]]
| {{UDnote|step=19}}
| {{UDnote|step=19}}
|-
|-
| 20
| 20
| 428.571
| 428.6
| [[32/25]], [[33/26]]
| [[32/25]], [[33/26]]
| {{UDnote|step=20}}
| {{UDnote|step=20}}
|-
|-
| 21
| 21
| 450.000
| 450.0
| ''[[9/7]]'', [[13/10]]
| ''[[9/7]]'', [[13/10]]
| {{UDnote|step=21}}
| {{UDnote|step=21}}
|-
|-
| 22
| 22
| 471.429
| 471.4
| [[21/16]]
| [[17/13]], [[21/16]]
| {{UDnote|step=22}}
| {{UDnote|step=22}}
|-
|-
| 23
| 23
| 492.857
| 492.9
| [[4/3]]
| [[4/3]]
| {{UDnote|step=23}}
| {{UDnote|step=23}}
|-
|-
| 24
| 24
| 514.286
| 514.3
| [[35/26]]
| [[35/26]]
| {{UDnote|step=24}}
| {{UDnote|step=24}}
|-
|-
| 25
| 25
| 535.714
| 535.7
| ''[[27/20]]'', [[15/11]]
| [[15/11]], [[19/14]], [[26/19]], ''[[27/20]]''
| {{UDnote|step=25}}
| {{UDnote|step=25}}
|-
|-
| 26
| 26
| 557.143
| 557.1
| [[11/8]]
| [[11/8]]
| {{UDnote|step=26}}
| {{UDnote|step=26}}
|-
|-
| 27
| 27
| 578.571
| 578.6
| [[7/5]]
| [[7/5]]
| {{UDnote|step=27}}
| {{UDnote|step=27}}
|-
|-
| 28
| 28
| 600.000
| 600.0
| [[45/32]], [[64/45]]
| [[17/12]], [[24/17]]
| {{UDnote|step=28}}
| {{UDnote|step=28}}
|-
|-
Line 173: Line 180:
| …
| …
|}
|}
<nowiki />* The following table assumes the [[patent val]] {{val| 56 89 130 157 194 207 }}; other approaches are possible. Inconsistent intervals are marked in ''italics''.
<nowiki/>* The following table assumes the 19-limit [[patent val]]; other approaches are possible. Inconsistent intervals are marked in ''italics''.


== Notation ==
== 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.
{{Ups and downs sharpness}}
Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
{{Sharpness-sharp7}}
=== Sagittal notation ===
=== Sagittal notation ===
This notation uses the same sagittal sequence as [[63edo#Sagittal notation|63-EDO]].
This notation uses the same sagittal sequence as [[63edo#Sagittal notation|63-EDO]].
Line 203: Line 218:
</imagemap>
</imagemap>


=== Ups and downs notation ===
== Approximation to JI ==
Using [[Helmholtz–Ellis]] accidentals, 56edo can also be notated using [[ups and downs notation]]:
{{Q-odd-limit intervals}}
{{Sharpness-sharp7}}
Here, a sharp raises by seven steps, and a flat lowers by seven steps, so single, double, and triple arrows can be used to fill in the gap.


== Regular temperament properties ==
== Regular temperament properties ==
Line 223: Line 236:
| {{monzo| 89 -56 }}
| {{monzo| 89 -56 }}
| {{mapping| 56 89 }}
| {{mapping| 56 89 }}
| &minus;1.64
| −1.64
| 1.63
| 1.63
| 7.64
| 7.64
Line 230: Line 243:
| 2048/2025, 1953125/1889568
| 2048/2025, 1953125/1889568
| {{mapping| 56 89 130 }}
| {{mapping| 56 89 130 }}
| &minus;1.01
| −1.01
| 1.61
| 1.61
| 7.50
| 7.50
Line 237: Line 250:
| 686/675, 875/864, 1029/1024
| 686/675, 875/864, 1029/1024
| {{mapping| 56 89 130 157 }}
| {{mapping| 56 89 130 157 }}
| &minus;0.352
| −0.352
| 1.80
| 1.80
| 8.38
| 8.38
Line 244: Line 257:
| 100/99, 245/242, 385/384, 686/675
| 100/99, 245/242, 385/384, 686/675
| {{mapping| 56 89 130 157 194 }}
| {{mapping| 56 89 130 157 194 }}
| &minus;0.618
| −0.618
| 1.69
| 1.69
| 7.90
| 7.90
Line 251: Line 264:
| 91/90, 100/99, 169/168, 245/242, 385/384
| 91/90, 100/99, 169/168, 245/242, 385/384
| {{mapping| 56 89 130 157 194 207 }}
| {{mapping| 56 89 130 157 194 207 }}
| &minus;0.299
| −0.299
| 1.70
| 1.70
| 7.95
| 7.95
Line 320: Line 333:
| [[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


== Zeta properties ==
===Zeta peak index===
{| class="wikitable"
! colspan="3" |Tuning
! colspan="3" |Strength
! colspan="2" |Closest EDO
! colspan="2" |Integer limit
|-
!ZPI
!Steps per octave
!Step size (cents)
!Height
!Integral
!Gap
!EDO
!Octave (cents)
!Consistent
!Distinct
|-
|[[276zpi]]
|56.0083399588546
|21.4253805929895
|6.063216
|0.931117
|14.804703
|56edo
|1199.82131320741
|8
|8
|}
== Scales ==
== Scales ==
* [[Supra7]]
* [[Supra7]]
* [[Supra12]]  
* [[Supra12]]
* Subsets of [[echidnic]][16] (6u8d):
** Frankincense{{idio}} (this is the original/default tuning): 364.3 - 492.9 - 707.1 - 835.7 - 1200.0
** Quasi-[[equipentatonic]]: 257.1 - 492.9 - 707.1 - 964.3 - 1200.0
** Sakura-like scale containing [[phi]]: 107.1 - 492.9 - 707.1 - 835.7 - 1200.0
* Subsets of [[sevond]][14]
** Evened minor pentatonic (approximated from [[72edo]]): 321.4 - 492.9 - 685.7 - 1028.6 - 1200.0


== Instruments ==
[[Lumatone mapping for 56edo|Lumatone mappings for 56edo]] are available.
== 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]]
* [https://www.youtube.com/shorts/o0imqFPDh9k ''56edo''] (2023)
* [https://www.youtube.com/watch?v=qzMOnS-lgWs ''Waltz in 56edo''] (2025)
; [[Budjarn Lambeth]]
* [https://www.youtube.com/watch?v=VsBXIvBZY6A ''56edo Track (Echidnic16 Scale)''] (2025)
; [[Claudi Meneghin]]
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=xWKa59qDkXQ ''Prelude & Fugue in Pajara''] (2020) &ndash; in pajara, 56edo tuning
* [https://www.youtube.com/watch?v=xWKa59qDkXQ ''Prelude & Fugue in Pajara''] (2020) – in pajara, 56edo tuning
* [https://www.youtube.com/watch?v=3oO1SIVWBgI ''Mirror Canon in F''] (2020)
* [https://www.youtube.com/watch?v=3oO1SIVWBgI ''Mirror Canon in F''] (2020)
* [https://www.youtube.com/watch?v=s1h083BRWXU ''Canon 3-in-1 on a Ground''] (2020)
* [https://www.youtube.com/watch?v=s1h083BRWXU ''Canon 3-in-1 on a Ground''] (2020)


== See also ==
* [[Lumatone mapping for 56edo]]   
[[Category:Hemithirds]]
[[Category:Hemithirds]]
[[Category:Keen]]
[[Category:Keen]]
Retrieved from "https://en.xen.wiki/w/56edo"