@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|57}}
+{{ED intro}}
 == Theory ==
-edo 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.
+edo 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.
 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]].
+edo is interesting in that it is a [[dual-fifth]] system where both fifths are somewhat plausible. The 57b val uses the alternative fifth of 715.789 cents. While not very accurate in the 5 and 7-limits (being even worse than 19edo), it does support some interesting temperaments like [[sixix]] and a near-perfect 1/2-comma [[archy]] (such that the diatonic minor seventh is almost exactly [[7/4]]).
 === Odd harmonics ===
 {{Harmonics in equal|57}}
+{{Harmonics in equal|57|intervals=odd|columns=11|prec=2|start=12|collapsed=true|title=Approximation of odd harmonics in 57edo (continued)}}
 === Subsets and supersets ===
-edo contains [[3edo]] and [[19edo]] as subsets.
+edo contains [[3edo]] and [[19edo]] as subsets. Tripling 57edo yields [[171edo]], which corrects the 3rd and 5th harmonics.
 == Intervals ==
+{| class="wikitable center-1 right-2 center-4 center-5"
-{| class="wikitable center-1 right-2 center-3 center-4"
 |-
 ! #
 ! [[Cent]]s
-! [[Ups and downs notation|Ups and Downs Notation]]<br>(Flat Fifth 11\19)
+! Approximate ratios*
-! [[Ups and downs notation|Ups and Downs Notation]]<br>(Sharp Fifth 34\57)
+! [[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}}
@@ Line 29: / Line 33: @@
 | 1
 | 21.05
+|
 | {{UDnote|step=1}}
 | {{UDnote|fifth=34|step=1}}
@@ Line 34: / Line 39: @@
 | 2
 | 42.11
+|
 | {{UDnote|step=2}}
 | {{UDnote|fifth=34|step=2}}
@@ Line 39: / Line 45: @@
 | 3
 | 63.16
+|[[29/28]]
 | {{UDnote|step=3}}
 | {{UDnote|fifth=34|step=3}}
@@ Line 44: / Line 51: @@
 | 4
 | 84.21
+|[[20/19]], [[21/20]], [[22/21]]
 | {{UDnote|step=4}}
 | {{UDnote|fifth=34|step=4}}
@@ Line 49: / Line 57: @@
 | 5
 | 105.26
+|[[17/16]], [[33/31]]
 | {{UDnote|step=5}}
 | {{UDnote|fifth=34|step=5}}
@@ Line 54: / Line 63: @@
 | 6
 | 126.32
+|[[14/13]], ''[[16/15]]''
 | {{UDnote|step=6}}
 | {{UDnote|fifth=34|step=6}}
@@ Line 59: / Line 69: @@
 | 7
 | 147.37
+|[[12/11]]
 | {{UDnote|step=7}}
 | {{UDnote|fifth=34|step=7}}
@@ Line 64: / Line 75: @@
 | 8
 | 168.42
+|[[11/10]], [[32/29]]
 | {{UDnote|step=8}}
 | {{UDnote|fifth=34|step=8}}
@@ Line 69: / Line 81: @@
 | 9
 | 189.47
+|[[19/17]], [[29/26]],<br>[[10/9]], ''[[9/8]]''
 | {{UDnote|step=9}}
 | {{UDnote|fifth=34|step=9}}
@@ Line 74: / Line 87: @@
 | 10
 | 210.53
+|[[26/23]]
 | {{UDnote|step=10}}
 | {{UDnote|fifth=34|step=10}}
@@ Line 79: / Line 93: @@
 | 11
 | 231.58
+|[[8/7]]
 | {{UDnote|step=11}}
 | {{UDnote|fifth=34|step=11}}
@@ Line 84: / Line 99: @@
 | 12
 | 252.63
+|[[22/19]]
 | {{UDnote|step=12}}
 | {{UDnote|fifth=34|step=12}}
@@ Line 89: / Line 105: @@
 | 13
 | 273.68
+|[[7/6]], [[34/29]]
 | {{UDnote|step=13}}
 | {{UDnote|fifth=34|step=13}}
@@ Line 94: / Line 111: @@
 | 14
 | 294.74
+|[[19/16]]
 | {{UDnote|step=14}}
 | {{UDnote|fifth=34|step=14}}
@@ Line 99: / Line 117: @@
 | 15
 | 315.79
+|[[6/5]]
 | {{UDnote|step=15}}
 | {{UDnote|fifth=34|step=15}}
@@ Line 104: / Line 123: @@
 | 16
 | 336.84
+|[[17/14]], [[28/23]]
 | {{UDnote|step=16}}
 | {{UDnote|fifth=34|step=16}}
@@ Line 109: / Line 129: @@
 | 17
 | 357.89
+|[[16/13]]
 | {{UDnote|step=17}}
 | {{UDnote|fifth=34|step=17}}
@@ Line 114: / Line 135: @@
 | 18
 | 378.95
+|[[5/4]]
 | {{UDnote|step=18}}
 | {{UDnote|fifth=34|step=18}}
@@ Line 119: / Line 141: @@
 | 19
 | 400.00
+|[[24/19]], [[29/23]]
 | {{UDnote|step=19}}
 | {{UDnote|fifth=34|step=19}}
@@ Line 124: / Line 147: @@
 | 20
 | 421.05
+|[[14/11]], ''[[9/7]]''
 | {{UDnote|step=20}}
 | {{UDnote|fifth=34|step=20}}
@@ Line 129: / Line 153: @@
 | 21
 | 442.11
+|[[22/17]], [[31/24]]
 | {{UDnote|step=21}}
 | {{UDnote|fifth=34|step=21}}
@@ Line 134: / Line 159: @@
 | 22
 | 463.16
+|[[17/13]], [[21/16]], [[38/29]]
 | {{UDnote|step=22}}
 | {{UDnote|fifth=34|step=22}}
@@ Line 139: / Line 165: @@
 | 23
 | 484.21
+|[[33/25]]
 | {{UDnote|step=23}}
 | {{UDnote|fifth=34|step=23}}
@@ Line 144: / Line 171: @@
 | 24
 | 505.26
+|[[4/3]]
 | {{UDnote|step=24}}
 | {{UDnote|fifth=34|step=24}}
@@ Line 149: / Line 177: @@
 | 25
 | 526.32
+|[[19/14]], [[42/31]], [[23/17]]
 | {{UDnote|step=25}}
 | {{UDnote|fifth=34|step=25}}
@@ Line 154: / Line 183: @@
 | 26
 | 547.37
+|[[11/8]], [[26/19]]
 | {{UDnote|step=26}}
 | {{UDnote|fifth=34|step=26}}
@@ Line 159: / Line 189: @@
 | 27
 | 568.42
+|[[25/18]], [[32/23]]
 | {{UDnote|step=27}}
 | {{UDnote|fifth=34|step=27}}
@@ Line 164: / Line 195: @@
 | 28
 | 589.47
+|[[31/22]]
 | {{UDnote|step=28}}
 | {{UDnote|fifth=34|step=28}}
@@ Line 169: / Line 201: @@
 | 29
 | 610.53
+|[[44/31]]
 | {{UDnote|step=29}}
 | {{UDnote|fifth=34|step=29}}
@@ Line 174: / Line 207: @@
 | 30
 | 631.58
+|[[23/16]]
 | {{UDnote|step=30}}
 | {{UDnote|fifth=34|step=30}}
@@ Line 179: / Line 213: @@
 | 31
 | 652.63
+|[[16/11]], [[19/13]]
 | {{UDnote|step=31}}
 | {{UDnote|fifth=34|step=31}}
@@ Line 184: / Line 219: @@
 | 32
 | 673.68
+|[[28/19]], [[31/21]], [[34/23]]
 | {{UDnote|step=32}}
 | {{UDnote|fifth=34|step=32}}
@@ Line 189: / Line 225: @@
 | 33
 | 694.74
+|[[3/2]]
 | {{UDnote|step=33}}
 | {{UDnote|fifth=34|step=33}}
@@ Line 194: / Line 231: @@
 | 34
 | 715.79
+|[[50/33]]
 | {{UDnote|step=34}}
 | {{UDnote|fifth=34|step=34}}
@@ Line 199: / Line 237: @@
 | 35
 | 736.84
+|[[26/17]], [[32/21]], [[29/19]]
 | {{UDnote|step=35}}
 | {{UDnote|fifth=34|step=35}}
@@ Line 204: / Line 243: @@
 | 36
 | 757.89
+|[[17/11]], [[31/20]]
 | {{UDnote|step=36}}
 | {{UDnote|fifth=34|step=36}}
@@ Line 209: / Line 249: @@
 | 37
 | 778.95
+|[[11/7]], ''[[14/9]]''
 | {{UDnote|step=37}}
 | {{UDnote|fifth=34|step=37}}
@@ Line 214: / Line 255: @@
 | 38
 | 800.00
+|[[19/12]]
 | {{UDnote|step=38}}
 | {{UDnote|fifth=34|step=38}}
@@ Line 219: / Line 261: @@
 | 39
 | 821.05
+|[[8/5]]
 | {{UDnote|step=39}}
 | {{UDnote|fifth=34|step=39}}
@@ Line 224: / Line 267: @@
 | 40
 | 842.11
+|[[13/8]]
 | {{UDnote|step=40}}
 | {{UDnote|fifth=34|step=40}}
@@ Line 229: / Line 273: @@
 | 41
 | 863.16
+|[[23/14]], [[28/17]], [[33/20]]
 | {{UDnote|step=41}}
 | {{UDnote|fifth=34|step=41}}
@@ Line 234: / Line 279: @@
 | 42
 | 884.21
+|[[5/3]]
 | {{UDnote|step=42}}
 | {{UDnote|fifth=34|step=42}}
@@ Line 239: / Line 285: @@
 | 43
 | 905.26
+|[[32/19]]
 | {{UDnote|step=43}}
 | {{UDnote|fifth=34|step=43}}
@@ Line 244: / Line 291: @@
 | 44
 | 926.32
+|[[12/7]], [[29/17]]
 | {{UDnote|step=44}}
 | {{UDnote|fifth=34|step=44}}
@@ Line 249: / Line 297: @@
 | 45
 | 947.37
+|[[19/11]]
 | {{UDnote|step=45}}
 | {{UDnote|fifth=34|step=45}}
@@ Line 254: / Line 303: @@
 | 46
 | 968.42
+|[[7/4]]
 | {{UDnote|step=46}}
 | {{UDnote|fifth=34|step=46}}
@@ Line 259: / Line 309: @@
 | 47
 | 989.47
+|[[23/13]]
 | {{UDnote|step=47}}
 | {{UDnote|fifth=34|step=47}}
@@ Line 264: / Line 315: @@
 | 48
 | 1010.53
+|[[34/19]], [[52/29]],<br>[[9/5]], ''[[16/9]]''
 | {{UDnote|step=48}}
 | {{UDnote|fifth=34|step=48}}
@@ Line 269: / Line 321: @@
 | 49
 | 1031.58
+|[[20/11]], [[29/16]]
 | {{UDnote|step=49}}
 | {{UDnote|fifth=34|step=49}}
@@ Line 274: / Line 327: @@
 | 50
 | 1052.63
+|[[11/6]]
 | {{UDnote|step=50}}
 | {{UDnote|fifth=34|step=50}}
@@ Line 279: / Line 333: @@
 | 51
 | 1073.68
+|[[13/7]], ''[[15/8]]''
 | {{UDnote|step=51}}
 | {{UDnote|fifth=34|step=51}}
@@ Line 284: / Line 339: @@
 | 52
 | 1094.74
+|[[32/17]]
 | {{UDnote|step=52}}
 | {{UDnote|fifth=34|step=52}}
@@ Line 289: / Line 345: @@
 | 53
 | 1115.79
+|[[19/10]], [[21/11]]
 | {{UDnote|step=53}}
 | {{UDnote|fifth=34|step=53}}
@@ Line 294: / Line 351: @@
 | 54
 | 1136.84
+|[[56/29]]
 | {{UDnote|step=54}}
 | {{UDnote|fifth=34|step=54}}
@@ Line 299: / Line 357: @@
 | 55
 | 1157.89
+|
 | {{UDnote|step=55}}
 | {{UDnote|fifth=34|step=55}}
@@ Line 304: / Line 363: @@
 | 56
 | 1178.95
+|
 | {{UDnote|step=56}}
 | {{UDnote|fifth=34|step=56}}
@@ Line 309: / Line 369: @@
 | 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 ===
+edo can be notated using [[Stein–Zimmermann–Gould notation]]:
+{{Sharpness-sharp3-szg}}
-===Sagittal notation===
+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.
-In the following diagrams, 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.
-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|19-EDO]].
+=== Kite's ups and downs notation ===
-====Evo flavor====
+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
@@ Line 330: / Line 399: @@
 </imagemap>
-====Revo flavor====
+==== Revo flavor ====
 <imagemap>
 File:57-EDO_Revo_Sagittal.svg
@@ Line 340: / Line 408: @@
 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)
-* 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 21s (augene)
 [[Category:Heinz]]
 [[Category:Mothra]]
-{{todo|add rank 2 temperaments table|add lumatone mapping}}
+{{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''] (2025-05-12)
+* ''Prelude in 57edo'' (2025) (Uses different temperaments in different sections)
+** [https://www.youtube.com/shorts/FylRB5MGBz4 ''Part 1 (short clip)''] (Lumatone view using [[5L&nbsp;2s|diatonic]] sharp fifth + [[mothra]] flat fifth mapping)
+** [https://www.youtube.com/shorts/yVv0s8t--pg ''Part 2 (short clip)''] (Lumatone view using [[lithium]] mapping)
+** [https://www.youtube.com/watch?v=RH2QkHqVb2c ''Whole composition''] (but no view of Lumatone)
+* [https://www.youtube.com/shorts/MTMRlRxRbjQ ''57edo groove''] (2025)
+* [https://www.youtube.com/shorts/TrgdgDduDIk ''57edo improv''] (2026-09-27)

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