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{{Infobox ET}}
{{Infobox ET}}
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]].
{{ED intro}}


== Theory ==
== 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 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 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 syntonci commas to the octave.
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 11-limit 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 }}.


{{harmonics in equal|56}}
== Intervals ==
== Intervals ==
The following table assumes the [[patent val]] {{val| 56 89 130 157 194 207 }}. Other approaches are possible.
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
! #
! #
! Cents
! Cents
! Approximate Ratios
! Approximate ratios*
! [[Ups and downs notation]]
! [[Ups and downs notation]]
|-
|-
| 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]]
| {{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}}
|-
|-
| 26
| 26
| 557.143
| 557.1
| [[11/8]]
| [[11/8]]
| {{UDnote|step=25}}
| {{UDnote|step=26}}
|-
|-
| 27
| 27
| 578.571
| 578.6
| [[7/5]]
| [[7/5]]
| {{UDnote|step=26}}
| {{UDnote|step=27}}
|-
|-
| 28
| 28
| 600.000
| 600.0
| [[45/32]], [[64/45]]
| [[17/12]], [[24/17]]
| {{UDnote|step=27}}
| {{UDnote|step=28}}
|-
|-
| …
| …
| …
| …
| …
| …
| …
|}
|}
<nowiki/>* The following table assumes the 19-limit [[patent val]]; other approaches are possible. Inconsistent intervals are marked in ''italics''.


== Commas ==
== Notation ==
* 5-limit commas: 2048/2025, {{monzo| -5 -10 9 }};
 
* 7-limit commas: 686/675, 875/864, 1029/1024
=== Ups and downs notation ===
* 11-limit commas: 100/99, 245/242, 385/384, 686/675
 
56edo can be notated using [[ups and downs notation|ups and downs]]. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.
{{Sharpness-sharp7a}}
 
Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
{{Sharpness-sharp7}}
 
=== Sagittal notation ===
This notation uses the same sagittal sequence as [[63edo#Sagittal notation|63-EDO]].
 
==== Evo flavor ====
<imagemap>
File:56-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 575 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[64/63]]
rect 120 80 220 106 [[81/80]]
rect 220 80 340 106 [[33/32]]
default [[File:56-EDO_Evo_Sagittal.svg]]
</imagemap>
 
==== Revo flavor ====
<imagemap>
File:56-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[64/63]]
rect 120 80 220 106 [[81/80]]
rect 220 80 340 106 [[33/32]]
default [[File:56-EDO_Revo_Sagittal.svg]]
</imagemap>
 
== Approximation to JI ==
{{Q-odd-limit intervals}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 89 -56 }}
| {{mapping| 56 89 }}
| −1.64
| 1.63
| 7.64
|-
| 2.3.5
| 2048/2025, 1953125/1889568
| {{mapping| 56 89 130 }}
| −1.01
| 1.61
| 7.50
|-
| 2.3.5.7
| 686/675, 875/864, 1029/1024
| {{mapping| 56 89 130 157 }}
| −0.352
| 1.80
| 8.38
|-
| 2.3.5.7.11
| 100/99, 245/242, 385/384, 686/675
| {{mapping| 56 89 130 157 194 }}
| −0.618
| 1.69
| 7.90
|-
| 2.3.5.7.11.13
| 91/90, 100/99, 169/168, 245/242, 385/384
| {{mapping| 56 89 130 157 194 207 }}
| −0.299
| 1.70
| 7.95
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
| 1
| 3\56
| 64.29
| 25/24
| [[Sycamore]]
|-
| 1
| 9\56
| 192.86
| 28/25
| [[Hemithirds]]
|-
| 1
| 11\56
| 235.71
| 8/7
| [[Slendric]]
|-
| 1
| 15\56
| 321.43
| 6/5
| [[Superkleismic]]
|-
| 1
| 25\56
| 535.71
| 15/11
| [[Maquila]] (56d) / [[maquiloid]] (56)
|-
| 2
| 11\56
| 235.71
| 8/7
| [[Echidnic]]
|-
| 2
| 23\56<br>(5\56)
| 492.86<br>(107.14)
| 4/3<br>(17/16)
| [[Keen]] / keenic
|-
| 4
| 23\56<br>(5\56)
| 492.86<br>(107.14)
| 4/3<br>(17/16)
| [[Bidia]] (7-limit)
|-
| 7
| 23\56<br>(1\56)
| 492.86<br>(21.43)
| 4/3<br>(250/243)
| [[Sevond]]
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== 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 ==
; [[Bryan Deister]]
* [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)


== See also ==
; [[Budjarn Lambeth]]
* [https://www.youtube.com/watch?v=VsBXIvBZY6A ''56edo Track (Echidnic16 Scale)''] (2025)
 
; [[Claudi Meneghin]]
* [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=s1h083BRWXU ''Canon 3-in-1 on a Ground''] (2020)


[[Lumatone mapping for 56edo]]   
[[Category:56edo| ]] <!-- main article -->
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Hemithirds]]
[[Category:Hemithirds]]
[[Category:Keen]]
[[Category:Keen]]
[[Category:Listen]]
[[Category:Pajara]]
[[Category:Pajara]]
[[Category:Superkleismic]]
[[Category:Superkleismic]]
[[Category:Sycamore]]
[[Category:Sycamore]]
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