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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-03-22 11:33:23 UTC</tt>.<br>
: The original revision id was <tt>497559418</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">//56edo// divides the octave into 56 parts of 21.429 cents each. It can be used to tune hemithirds, superkliesmic, sycamore and keen temperaments, and using &lt;56 89 130 158| as the equal temperament val, for pajara. It provides the optimal patent val for 7-, 11- and 13-limit [[Sycamore family#Sycamore|sycamore]], and the 11-limit 56d val is close to the POTE tuning for 11-limit pajara.


=Commas=  
== Theory ==
5-limit commas: 2048/2025, |-5 -10 9&gt;
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]].


7-limit commas: 686/675, 875/864, 1029/1024
=== Prime harmonics ===
{{Harmonics in equal|56}}


11-limit commas: 100/99, 245/242, 385/384, 686/675
=== 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}}.


=Some scales=  
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]].
[[supra7]]
 
[[supra12]]</pre></div>
=== Miscellaneous properties ===
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;56edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;em&gt;56edo&lt;/em&gt; divides the octave into 56 parts of 21.429 cents each. It can be used to tune hemithirds, superkliesmic, sycamore and keen temperaments, and using &amp;lt;56 89 130 158| as the equal temperament val, for pajara. It provides the optimal patent val for 7-, 11- and 13-limit &lt;a class="wiki_link" href="/Sycamore%20family#Sycamore"&gt;sycamore&lt;/a&gt;, and the 11-limit 56d val is close to the POTE tuning for 11-limit pajara.&lt;br /&gt;
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.
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Commas&lt;/h1&gt;
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]].
5-limit commas: 2048/2025, |-5 -10 9&amp;gt;&lt;br /&gt;
 
&lt;br /&gt;
=== Subsets and supersets ===
7-limit commas: 686/675, 875/864, 1029/1024&lt;br /&gt;
Since 56 factors into {{nowrap|2<sup>3</sup> &times; 7}}, 56edo has subset edos {{EDOs| 2, 4, 7, 8, 14, and 28}}.
&lt;br /&gt;
 
11-limit commas: 100/99, 245/242, 385/384, 686/675&lt;br /&gt;
== Intervals ==
&lt;br /&gt;
{| class="wikitable center-all right-2 left-3"
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Some scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Some scales&lt;/h1&gt;
|-
  &lt;a class="wiki_link" href="/supra7"&gt;supra7&lt;/a&gt;&lt;br /&gt;
! #
&lt;a class="wiki_link" href="/supra12"&gt;supra12&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
! Cents
! Approximate ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 21.4
| ''[[49/48]]'', [[55/54]], [[56/55]], [[64/63]]
| {{UDnote|step=1}}
|-
| 2
| 42.9
| ''[[28/27]]'', [[40/39]], [[45/44]], [[50/49]], ''[[81/80]]''
| {{UDnote|step=2}}
|-
| 3
| 64.3
| [[25/24]], ''[[36/35]]'', ''[[33/32]]''
| {{UDnote|step=3}}
|-
| 4
| 85.7
| [[19/18]], [[20/19]], [[21/20]], [[22/21]]
| {{UDnote|step=4}}
|-
| 5
| 107.1
| [[16/15]], [[17/16]], [[18/17]]
| {{UDnote|step=5}}
|-
| 6
| 128.6
| [[15/14]], [[13/12]], [[14/13]]
| {{UDnote|step=6}}
|-
| 7
| 150.0
| [[12/11]]
| {{UDnote|step=7}}
|-
| 8
| 171.4
| ''[[10/9]]'', [[11/10]], [[21/19]]
| {{UDnote|step=8}}
|-
| 9
| 192.9
| [[19/17]], [[28/25]]
| {{UDnote|step=9}}
|-
| 10
| 214.3
| [[9/8]], [[17/15]]
| {{UDnote|step=10}}
|-
| 11
| 235.7
| [[8/7]]
| {{UDnote|step=11}}
|-
| 12
| 257.1
| [[7/6]]
| {{UDnote|step=12}}
|-
| 13
| 278.6
| [[13/11]], [[20/17]]
| {{UDnote|step=13}}
|-
| 14
| 300.0
| [[19/16]], [[25/21]]
| {{UDnote|step=14}}
|-
| 15
| 321.4
| [[6/5]]
| {{UDnote|step=15}}
|-
| 16
| 342.9
| [[11/9]], [[17/14]]
| {{UDnote|step=16}}
|-
| 17
| 364.3
| [[16/13]], [[21/17]], [[26/21]]
| {{UDnote|step=17}}
|-
| 18
| 385.7
| [[5/4]]
| {{UDnote|step=18}}
|-
| 19
| 407.1
| [[14/11]], [[19/12]], [[24/19]]
| {{UDnote|step=19}}
|-
| 20
| 428.6
| [[32/25]], [[33/26]]
| {{UDnote|step=20}}
|-
| 21
| 450.0
| ''[[9/7]]'', [[13/10]]
| {{UDnote|step=21}}
|-
| 22
| 471.4
| [[17/13]], [[21/16]]
| {{UDnote|step=22}}
|-
| 23
| 492.9
| [[4/3]]
| {{UDnote|step=23}}
|-
| 24
| 514.3
| [[35/26]]
| {{UDnote|step=24}}
|-
| 25
| 535.7
| [[15/11]], [[19/14]], [[26/19]], ''[[27/20]]''
| {{UDnote|step=25}}
|-
| 26
| 557.1
| [[11/8]]
| {{UDnote|step=26}}
|-
| 27
| 578.6
| [[7/5]]
| {{UDnote|step=27}}
|-
| 28
| 600.0
| [[17/12]], [[24/17]]
| {{UDnote|step=28}}
|-
| …
| …
| …
| …
|}
<nowiki/>* The following table assumes the 19-limit [[patent val]]; other approaches are possible. Inconsistent intervals are marked in ''italics''.
 
== 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 ===
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 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 ==
* [[Supra7]]
* [[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 ==
=== 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]]
* [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)
 
[[Category:Hemithirds]]
[[Category:Keen]]
[[Category:Listen]]
[[Category:Pajara]]
[[Category:Superkleismic]]
[[Category:Sycamore]]
Retrieved from "https://en.xen.wiki/w/56edo"