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57edo: Difference between revisions

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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>2010-09-22 02:30:34 UTC</tt>.<br>
: The original revision id was <tt>164480301</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">//57edo// divides the octave into 57 parts of size 21.053. It can be used to tune mothra temperament.


5-limit commas: 81/80, 3125/3072
== Theory ==
57edo is an excellent tuning for the 2.5/3.7.11.13.17.19 [[just intonation subgroup]]. One way to describe 57edo is that it has a [[5-limit]] part consisting of three [[ring number|ring]]s of [[19edo]], plus a no-threes no-fives part which is much more accurate.


7-limit commas: 81/80, 3125/3072, 1029/1024
Using the full prime-limit [[patent val]], the equal temperament tempers out [[81/80]], [[1029/1024]], and [[3125/3072]] in the 7-limit; and [[99/98]], [[385/384]], [[441/440]], and [[625/616]] in the [[11-limit]]. A good generator to exploit the 2.5/3.7.11.13.17.19 aspect of 57 is the approximate [[11/8]], which is 26\57. This gives the [[19-limit]] 46 &amp; 57 temperament [[heinz]]. It can also be used to tune [[mothra]] as well as [[trismegistus]].


11-limit commas: 99/98, 385/384, 441/440, 625/616</pre></div>
=== Odd harmonics ===
<h4>Original HTML content:</h4>
{{Harmonics in equal|57}}
<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;57edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;em&gt;57edo&lt;/em&gt; divides the octave into 57 parts of size 21.053. It can be used to tune mothra temperament.&lt;br /&gt;
{{Harmonics in equal|57|intervals=odd|columns=11|prec=2|start=12|collapsed=true|title=Approximation of odd harmonics in 57edo (continued)}}
&lt;br /&gt;
 
5-limit commas: 81/80, 3125/3072&lt;br /&gt;
=== Subsets and supersets ===
&lt;br /&gt;
57edo contains [[3edo]] and [[19edo]] as subsets. Tripling 57edo yields [[171edo]], which corrects the 3rd and 5th harmonics.
7-limit commas: 81/80, 3125/3072, 1029/1024&lt;br /&gt;
 
&lt;br /&gt;
== Intervals ==
11-limit commas: 99/98, 385/384, 441/440, 625/616&lt;/body&gt;&lt;/html&gt;</pre></div>
{| class="wikitable center-1 right-2 center-4 center-5"
|-
! #
! [[Cent]]s
! Approximate ratios*
! [[Kite's ups and downs notation|Ups and downs notation]]<br>(Flat fifth 11\19)
! [[Kite's ups and downs notation|Ups and downs notation]]<br>(Sharp fifth 34\57)
|-
| 0
| 0.00
|[[1/1]]
| {{UDnote|step=0}}
| {{UDnote|fifth=34|step=0}}
|-
| 1
| 21.05
|
| {{UDnote|step=1}}
| {{UDnote|fifth=34|step=1}}
|-
| 2
| 42.11
|
| {{UDnote|step=2}}
| {{UDnote|fifth=34|step=2}}
|-
| 3
| 63.16
|[[29/28]]
| {{UDnote|step=3}}
| {{UDnote|fifth=34|step=3}}
|-
| 4
| 84.21
|[[20/19]], [[21/20]], [[22/21]]
| {{UDnote|step=4}}
| {{UDnote|fifth=34|step=4}}
|-
| 5
| 105.26
|[[17/16]], [[33/31]]
| {{UDnote|step=5}}
| {{UDnote|fifth=34|step=5}}
|-
| 6
| 126.32
|[[14/13]], ''[[16/15]]''
| {{UDnote|step=6}}
| {{UDnote|fifth=34|step=6}}
|-
| 7
| 147.37
|[[12/11]]
| {{UDnote|step=7}}
| {{UDnote|fifth=34|step=7}}
|-
| 8
| 168.42
|[[11/10]], [[32/29]]
| {{UDnote|step=8}}
| {{UDnote|fifth=34|step=8}}
|-
| 9
| 189.47
|[[19/17]], [[29/26]],<br>[[10/9]], ''[[9/8]]''
| {{UDnote|step=9}}
| {{UDnote|fifth=34|step=9}}
|-
| 10
| 210.53
|[[26/23]]
| {{UDnote|step=10}}
| {{UDnote|fifth=34|step=10}}
|-
| 11
| 231.58
|[[8/7]]
| {{UDnote|step=11}}
| {{UDnote|fifth=34|step=11}}
|-
| 12
| 252.63
|[[22/19]]
| {{UDnote|step=12}}
| {{UDnote|fifth=34|step=12}}
|-
| 13
| 273.68
|[[7/6]], [[34/29]]
| {{UDnote|step=13}}
| {{UDnote|fifth=34|step=13}}
|-
| 14
| 294.74
|[[19/16]]
| {{UDnote|step=14}}
| {{UDnote|fifth=34|step=14}}
|-
| 15
| 315.79
|[[6/5]]
| {{UDnote|step=15}}
| {{UDnote|fifth=34|step=15}}
|-
| 16
| 336.84
|[[17/14]], [[28/23]]
| {{UDnote|step=16}}
| {{UDnote|fifth=34|step=16}}
|-
| 17
| 357.89
|[[16/13]]
| {{UDnote|step=17}}
| {{UDnote|fifth=34|step=17}}
|-
| 18
| 378.95
|[[5/4]]
| {{UDnote|step=18}}
| {{UDnote|fifth=34|step=18}}
|-
| 19
| 400.00
|[[24/19]], [[29/23]]
| {{UDnote|step=19}}
| {{UDnote|fifth=34|step=19}}
|-
| 20
| 421.05
|[[14/11]], ''[[9/7]]''
| {{UDnote|step=20}}
| {{UDnote|fifth=34|step=20}}
|-
| 21
| 442.11
|[[22/17]], [[31/24]]
| {{UDnote|step=21}}
| {{UDnote|fifth=34|step=21}}
|-
| 22
| 463.16
|[[17/13]], [[21/16]], [[38/29]]
| {{UDnote|step=22}}
| {{UDnote|fifth=34|step=22}}
|-
| 23
| 484.21
|[[33/25]]
| {{UDnote|step=23}}
| {{UDnote|fifth=34|step=23}}
|-
| 24
| 505.26
|[[4/3]]
| {{UDnote|step=24}}
| {{UDnote|fifth=34|step=24}}
|-
| 25
| 526.32
|[[19/14]], [[42/31]], [[23/17]]
| {{UDnote|step=25}}
| {{UDnote|fifth=34|step=25}}
|-
| 26
| 547.37
|[[11/8]], [[26/19]]
| {{UDnote|step=26}}
| {{UDnote|fifth=34|step=26}}
|-
| 27
| 568.42
|[[25/18]], [[32/23]]
| {{UDnote|step=27}}
| {{UDnote|fifth=34|step=27}}
|-
| 28
| 589.47
|[[31/22]]
| {{UDnote|step=28}}
| {{UDnote|fifth=34|step=28}}
|-
| 29
| 610.53
|[[44/31]]
| {{UDnote|step=29}}
| {{UDnote|fifth=34|step=29}}
|-
| 30
| 631.58
|[[23/16]]
| {{UDnote|step=30}}
| {{UDnote|fifth=34|step=30}}
|-
| 31
| 652.63
|[[16/11]], [[19/13]]
| {{UDnote|step=31}}
| {{UDnote|fifth=34|step=31}}
|-
| 32
| 673.68
|[[28/19]], [[31/21]], [[34/23]]
| {{UDnote|step=32}}
| {{UDnote|fifth=34|step=32}}
|-
| 33
| 694.74
|[[3/2]]
| {{UDnote|step=33}}
| {{UDnote|fifth=34|step=33}}
|-
| 34
| 715.79
|[[50/33]]
| {{UDnote|step=34}}
| {{UDnote|fifth=34|step=34}}
|-
| 35
| 736.84
|[[26/17]], [[32/21]], [[29/19]]
| {{UDnote|step=35}}
| {{UDnote|fifth=34|step=35}}
|-
| 36
| 757.89
|[[17/11]], [[31/20]]
| {{UDnote|step=36}}
| {{UDnote|fifth=34|step=36}}
|-
| 37
| 778.95
|[[11/7]], ''[[14/9]]''
| {{UDnote|step=37}}
| {{UDnote|fifth=34|step=37}}
|-
| 38
| 800.00
|[[19/12]]
| {{UDnote|step=38}}
| {{UDnote|fifth=34|step=38}}
|-
| 39
| 821.05
|[[8/5]]
| {{UDnote|step=39}}
| {{UDnote|fifth=34|step=39}}
|-
| 40
| 842.11
|[[13/8]]
| {{UDnote|step=40}}
| {{UDnote|fifth=34|step=40}}
|-
| 41
| 863.16
|[[23/14]], [[28/17]], [[33/20]]
| {{UDnote|step=41}}
| {{UDnote|fifth=34|step=41}}
|-
| 42
| 884.21
|[[5/3]]
| {{UDnote|step=42}}
| {{UDnote|fifth=34|step=42}}
|-
| 43
| 905.26
|[[32/19]]
| {{UDnote|step=43}}
| {{UDnote|fifth=34|step=43}}
|-
| 44
| 926.32
|[[12/7]], [[29/17]]
| {{UDnote|step=44}}
| {{UDnote|fifth=34|step=44}}
|-
| 45
| 947.37
|[[19/11]]
| {{UDnote|step=45}}
| {{UDnote|fifth=34|step=45}}
|-
| 46
| 968.42
|[[7/4]]
| {{UDnote|step=46}}
| {{UDnote|fifth=34|step=46}}
|-
| 47
| 989.47
|[[23/13]]
| {{UDnote|step=47}}
| {{UDnote|fifth=34|step=47}}
|-
| 48
| 1010.53
|[[34/19]], [[52/29]],<br>[[9/5]], ''[[16/9]]''
| {{UDnote|step=48}}
| {{UDnote|fifth=34|step=48}}
|-
| 49
| 1031.58
|[[20/11]], [[29/16]]
| {{UDnote|step=49}}
| {{UDnote|fifth=34|step=49}}
|-
| 50
| 1052.63
|[[11/6]]
| {{UDnote|step=50}}
| {{UDnote|fifth=34|step=50}}
|-
| 51
| 1073.68
|[[13/7]], ''[[15/8]]''
| {{UDnote|step=51}}
| {{UDnote|fifth=34|step=51}}
|-
| 52
| 1094.74
|[[32/17]]
| {{UDnote|step=52}}
| {{UDnote|fifth=34|step=52}}
|-
| 53
| 1115.79
|[[19/10]], [[21/11]]
| {{UDnote|step=53}}
| {{UDnote|fifth=34|step=53}}
|-
| 54
| 1136.84
|[[56/29]]
| {{UDnote|step=54}}
| {{UDnote|fifth=34|step=54}}
|-
| 55
| 1157.89
|
| {{UDnote|step=55}}
| {{UDnote|fifth=34|step=55}}
|-
| 56
| 1178.95
|
| {{UDnote|step=56}}
| {{UDnote|fifth=34|step=56}}
|-
| 57
| 1200.00
|[[2/1]]
| {{UDnote|step=57}}
| {{UDnote|fifth=34|step=57}}
|}
<nowiki>*</nowiki> As a 2.3.5.7.11.13.17.19.23.29.31-subgroup temperament, in ''italics'' if inconsistent
 
== Notation ==
=== Stein–Zimmermann–Gould notation ===
57edo can be notated using [[Stein–Zimmermann–Gould notation]]:
{{Sharpness-sharp3-szg}}
 
Here, a sharp raises by three steps, and a flat lowers by three steps, so arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with double arrows.
 
=== Kite's ups and downs notation ===
Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.
{{Ups and downs sharpness}}
 
=== Sagittal notation ===
This notation uses the same sagittal sequence as edos [[50edo #Sagittal notation|50]], [[64edo #Sagittal notation|64]], and [[71edo #Sagittal notation|71b]], and is a superset of the notation for [[19edo #Sagittal notation|19edo]].
 
==== Evo flavor ====
<imagemap>
File:57-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 671 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[1053/1024]]
default [[File:57-EDO_Evo_Sagittal.svg]]
</imagemap>
 
==== Revo flavor ====
<imagemap>
File:57-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 647 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[1053/1024]]
default [[File:57-EDO_Revo_Sagittal.svg]]
</imagemap>
 
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo.
 
== Scales ==
* 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 1 - 3mos of type 18L&nbsp;21s (augene)
 
[[Category:Heinz]]
[[Category:Mothra]]
{{todo|add rank 2 temperaments table}}
 
== Instruments ==
 
Since 57edo contains [[19edo]] as a non-trivial subset, it would be possible to use three 19edo instruments tuned 1\57 apart from each other to play the full gamut of 57edo.
 
A [[Lumatone mapping for 57edo]] is available.
 
== Music ==
 
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=4NZvSgS3ryY ''57edo improv''] (2025)
* ''Prelude in 57edo'' (2025) (switches Lumatone mapping between sections)
** [https://www.youtube.com/shorts/FylRB5MGBz4 ''Part 1 (short clip)'']
** [https://www.youtube.com/shorts/yVv0s8t--pg ''Part 2 (short clip)'']
** [https://www.youtube.com/watch?v=RH2QkHqVb2c ''Whole composition''] (but no view of Lumatone)
* [https://www.youtube.com/shorts/MTMRlRxRbjQ ''57edo groove''] (2025)
Retrieved from "https://en.xen.wiki/w/57edo"