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57edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|57}}
{{ED intro}}


== Theory ==
== Theory ==
57edo can be used to tune the [[mothra]] temperament, and 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. 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 & 57 temperament [[heinz]].
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.  


[[5-limit|5-limit]] [[comma]]s: [[81/80]], [[3125/3072]]
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 & 57 temperament [[heinz]]. It can also be used to tune [[mothra]] as well as [[trismegistus]].
 
[[7-limit|7-limit]] commas: 81/80, 3125/3072, [[1029/1024]]
 
[[11-limit|11-limit]] commas: [[99/98]], [[385/384]], [[441/440]], [[625/616]]


=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|57}}
{{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 ===
57edo contains [[3edo]] and [[19edo]] as subsets. Tripling 57edo yields [[171edo]], which corrects the 3rd and 5th harmonics.


== Intervals ==
== Intervals ==
 
{| class="wikitable center-1 right-2 center-4 center-5"
{| class="wikitable center-1 right-2 center-3 center-4"
|-
|-
! #
! #
! [[Cent]]s
! [[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
| 0.00
| 0.00
|[[1/1]]
| {{UDnote|step=0}}
| {{UDnote|step=0}}
| {{UDnote|fifth=34|step=0}}
| {{UDnote|fifth=34|step=0}}
Line 30: Line 31:
| 1
| 1
| 21.05
| 21.05
|
| {{UDnote|step=1}}
| {{UDnote|step=1}}
| {{UDnote|fifth=34|step=1}}
| {{UDnote|fifth=34|step=1}}
Line 35: Line 37:
| 2
| 2
| 42.11
| 42.11
|
| {{UDnote|step=2}}
| {{UDnote|step=2}}
| {{UDnote|fifth=34|step=2}}
| {{UDnote|fifth=34|step=2}}
Line 40: Line 43:
| 3
| 3
| 63.16
| 63.16
|[[29/28]]
| {{UDnote|step=3}}
| {{UDnote|step=3}}
| {{UDnote|fifth=34|step=3}}
| {{UDnote|fifth=34|step=3}}
Line 45: Line 49:
| 4
| 4
| 84.21
| 84.21
|[[20/19]], [[21/20]], [[22/21]]
| {{UDnote|step=4}}
| {{UDnote|step=4}}
| {{UDnote|fifth=34|step=4}}
| {{UDnote|fifth=34|step=4}}
Line 50: Line 55:
| 5
| 5
| 105.26
| 105.26
|[[17/16]], [[33/31]]
| {{UDnote|step=5}}
| {{UDnote|step=5}}
| {{UDnote|fifth=34|step=5}}
| {{UDnote|fifth=34|step=5}}
Line 55: Line 61:
| 6
| 6
| 126.32
| 126.32
|[[14/13]], ''[[16/15]]''
| {{UDnote|step=6}}
| {{UDnote|step=6}}
| {{UDnote|fifth=34|step=6}}
| {{UDnote|fifth=34|step=6}}
Line 60: Line 67:
| 7
| 7
| 147.37
| 147.37
|[[12/11]]
| {{UDnote|step=7}}
| {{UDnote|step=7}}
| {{UDnote|fifth=34|step=7}}
| {{UDnote|fifth=34|step=7}}
Line 65: Line 73:
| 8
| 8
| 168.42
| 168.42
|[[11/10]], [[32/29]]
| {{UDnote|step=8}}
| {{UDnote|step=8}}
| {{UDnote|fifth=34|step=8}}
| {{UDnote|fifth=34|step=8}}
Line 70: Line 79:
| 9
| 9
| 189.47
| 189.47
|[[19/17]], [[29/26]],<br>[[10/9]], ''[[9/8]]''
| {{UDnote|step=9}}
| {{UDnote|step=9}}
| {{UDnote|fifth=34|step=9}}
| {{UDnote|fifth=34|step=9}}
Line 75: Line 85:
| 10
| 10
| 210.53
| 210.53
|[[26/23]]
| {{UDnote|step=10}}
| {{UDnote|step=10}}
| {{UDnote|fifth=34|step=10}}
| {{UDnote|fifth=34|step=10}}
Line 80: Line 91:
| 11
| 11
| 231.58
| 231.58
|[[8/7]]
| {{UDnote|step=11}}
| {{UDnote|step=11}}
| {{UDnote|fifth=34|step=11}}
| {{UDnote|fifth=34|step=11}}
Line 85: Line 97:
| 12
| 12
| 252.63
| 252.63
|[[22/19]]
| {{UDnote|step=12}}
| {{UDnote|step=12}}
| {{UDnote|fifth=34|step=12}}
| {{UDnote|fifth=34|step=12}}
Line 90: Line 103:
| 13
| 13
| 273.68
| 273.68
|[[7/6]], [[34/29]]
| {{UDnote|step=13}}
| {{UDnote|step=13}}
| {{UDnote|fifth=34|step=13}}
| {{UDnote|fifth=34|step=13}}
Line 95: Line 109:
| 14
| 14
| 294.74
| 294.74
|[[19/16]]
| {{UDnote|step=14}}
| {{UDnote|step=14}}
| {{UDnote|fifth=34|step=14}}
| {{UDnote|fifth=34|step=14}}
Line 100: Line 115:
| 15
| 15
| 315.79
| 315.79
|[[6/5]]
| {{UDnote|step=15}}
| {{UDnote|step=15}}
| {{UDnote|fifth=34|step=15}}
| {{UDnote|fifth=34|step=15}}
Line 105: Line 121:
| 16
| 16
| 336.84
| 336.84
|[[17/14]], [[28/23]]
| {{UDnote|step=16}}
| {{UDnote|step=16}}
| {{UDnote|fifth=34|step=16}}
| {{UDnote|fifth=34|step=16}}
Line 110: Line 127:
| 17
| 17
| 357.89
| 357.89
|[[16/13]]
| {{UDnote|step=17}}
| {{UDnote|step=17}}
| {{UDnote|fifth=34|step=17}}
| {{UDnote|fifth=34|step=17}}
Line 115: Line 133:
| 18
| 18
| 378.95
| 378.95
|[[5/4]]
| {{UDnote|step=18}}
| {{UDnote|step=18}}
| {{UDnote|fifth=34|step=18}}
| {{UDnote|fifth=34|step=18}}
Line 120: Line 139:
| 19
| 19
| 400.00
| 400.00
|[[24/19]], [[29/23]]
| {{UDnote|step=19}}
| {{UDnote|step=19}}
| {{UDnote|fifth=34|step=19}}
| {{UDnote|fifth=34|step=19}}
Line 125: Line 145:
| 20
| 20
| 421.05
| 421.05
|[[14/11]], ''[[9/7]]''
| {{UDnote|step=20}}
| {{UDnote|step=20}}
| {{UDnote|fifth=34|step=20}}
| {{UDnote|fifth=34|step=20}}
Line 130: Line 151:
| 21
| 21
| 442.11
| 442.11
|[[22/17]], [[31/24]]
| {{UDnote|step=21}}
| {{UDnote|step=21}}
| {{UDnote|fifth=34|step=21}}
| {{UDnote|fifth=34|step=21}}
Line 135: Line 157:
| 22
| 22
| 463.16
| 463.16
|[[17/13]], [[21/16]], [[38/29]]
| {{UDnote|step=22}}
| {{UDnote|step=22}}
| {{UDnote|fifth=34|step=22}}
| {{UDnote|fifth=34|step=22}}
Line 140: Line 163:
| 23
| 23
| 484.21
| 484.21
|[[33/25]]
| {{UDnote|step=23}}
| {{UDnote|step=23}}
| {{UDnote|fifth=34|step=23}}
| {{UDnote|fifth=34|step=23}}
Line 145: Line 169:
| 24
| 24
| 505.26
| 505.26
|[[4/3]]
| {{UDnote|step=24}}
| {{UDnote|step=24}}
| {{UDnote|fifth=34|step=24}}
| {{UDnote|fifth=34|step=24}}
Line 150: Line 175:
| 25
| 25
| 526.32
| 526.32
|[[19/14]], [[42/31]], [[23/17]]
| {{UDnote|step=25}}
| {{UDnote|step=25}}
| {{UDnote|fifth=34|step=25}}
| {{UDnote|fifth=34|step=25}}
Line 155: Line 181:
| 26
| 26
| 547.37
| 547.37
|[[11/8]], [[26/19]]
| {{UDnote|step=26}}
| {{UDnote|step=26}}
| {{UDnote|fifth=34|step=26}}
| {{UDnote|fifth=34|step=26}}
Line 160: Line 187:
| 27
| 27
| 568.42
| 568.42
|[[25/18]], [[32/23]]
| {{UDnote|step=27}}
| {{UDnote|step=27}}
| {{UDnote|fifth=34|step=27}}
| {{UDnote|fifth=34|step=27}}
Line 165: Line 193:
| 28
| 28
| 589.47
| 589.47
|[[31/22]]
| {{UDnote|step=28}}
| {{UDnote|step=28}}
| {{UDnote|fifth=34|step=28}}
| {{UDnote|fifth=34|step=28}}
Line 170: Line 199:
| 29
| 29
| 610.53
| 610.53
|[[44/31]]
| {{UDnote|step=29}}
| {{UDnote|step=29}}
| {{UDnote|fifth=34|step=29}}
| {{UDnote|fifth=34|step=29}}
Line 175: Line 205:
| 30
| 30
| 631.58
| 631.58
|[[23/16]]
| {{UDnote|step=30}}
| {{UDnote|step=30}}
| {{UDnote|fifth=34|step=30}}
| {{UDnote|fifth=34|step=30}}
Line 180: Line 211:
| 31
| 31
| 652.63
| 652.63
|[[16/11]], [[19/13]]
| {{UDnote|step=31}}
| {{UDnote|step=31}}
| {{UDnote|fifth=34|step=31}}
| {{UDnote|fifth=34|step=31}}
Line 185: Line 217:
| 32
| 32
| 673.68
| 673.68
|[[28/19]], [[31/21]], [[34/23]]
| {{UDnote|step=32}}
| {{UDnote|step=32}}
| {{UDnote|fifth=34|step=32}}
| {{UDnote|fifth=34|step=32}}
Line 190: Line 223:
| 33
| 33
| 694.74
| 694.74
|[[3/2]]
| {{UDnote|step=33}}
| {{UDnote|step=33}}
| {{UDnote|fifth=34|step=33}}
| {{UDnote|fifth=34|step=33}}
Line 195: Line 229:
| 34
| 34
| 715.79
| 715.79
|[[50/33]]
| {{UDnote|step=34}}
| {{UDnote|step=34}}
| {{UDnote|fifth=34|step=34}}
| {{UDnote|fifth=34|step=34}}
Line 200: Line 235:
| 35
| 35
| 736.84
| 736.84
|[[26/17]], [[32/21]], [[29/19]]
| {{UDnote|step=35}}
| {{UDnote|step=35}}
| {{UDnote|fifth=34|step=35}}
| {{UDnote|fifth=34|step=35}}
Line 205: Line 241:
| 36
| 36
| 757.89
| 757.89
|[[17/11]], [[31/20]]
| {{UDnote|step=36}}
| {{UDnote|step=36}}
| {{UDnote|fifth=34|step=36}}
| {{UDnote|fifth=34|step=36}}
Line 210: Line 247:
| 37
| 37
| 778.95
| 778.95
|[[11/7]], ''[[14/9]]''
| {{UDnote|step=37}}
| {{UDnote|step=37}}
| {{UDnote|fifth=34|step=37}}
| {{UDnote|fifth=34|step=37}}
Line 215: Line 253:
| 38
| 38
| 800.00
| 800.00
|[[19/12]]
| {{UDnote|step=38}}
| {{UDnote|step=38}}
| {{UDnote|fifth=34|step=38}}
| {{UDnote|fifth=34|step=38}}
Line 220: Line 259:
| 39
| 39
| 821.05
| 821.05
|[[8/5]]
| {{UDnote|step=39}}
| {{UDnote|step=39}}
| {{UDnote|fifth=34|step=39}}
| {{UDnote|fifth=34|step=39}}
Line 225: Line 265:
| 40
| 40
| 842.11
| 842.11
|[[13/8]]
| {{UDnote|step=40}}
| {{UDnote|step=40}}
| {{UDnote|fifth=34|step=40}}
| {{UDnote|fifth=34|step=40}}
Line 230: Line 271:
| 41
| 41
| 863.16
| 863.16
|[[23/14]], [[28/17]], [[33/20]]
| {{UDnote|step=41}}
| {{UDnote|step=41}}
| {{UDnote|fifth=34|step=41}}
| {{UDnote|fifth=34|step=41}}
Line 235: Line 277:
| 42
| 42
| 884.21
| 884.21
|[[5/3]]
| {{UDnote|step=42}}
| {{UDnote|step=42}}
| {{UDnote|fifth=34|step=42}}
| {{UDnote|fifth=34|step=42}}
Line 240: Line 283:
| 43
| 43
| 905.26
| 905.26
|[[32/19]]
| {{UDnote|step=43}}
| {{UDnote|step=43}}
| {{UDnote|fifth=34|step=43}}
| {{UDnote|fifth=34|step=43}}
Line 245: Line 289:
| 44
| 44
| 926.32
| 926.32
|[[12/7]], [[29/17]]
| {{UDnote|step=44}}
| {{UDnote|step=44}}
| {{UDnote|fifth=34|step=44}}
| {{UDnote|fifth=34|step=44}}
Line 250: Line 295:
| 45
| 45
| 947.37
| 947.37
|[[19/11]]
| {{UDnote|step=45}}
| {{UDnote|step=45}}
| {{UDnote|fifth=34|step=45}}
| {{UDnote|fifth=34|step=45}}
Line 255: Line 301:
| 46
| 46
| 968.42
| 968.42
|[[7/4]]
| {{UDnote|step=46}}
| {{UDnote|step=46}}
| {{UDnote|fifth=34|step=46}}
| {{UDnote|fifth=34|step=46}}
Line 260: Line 307:
| 47
| 47
| 989.47
| 989.47
|[[23/13]]
| {{UDnote|step=47}}
| {{UDnote|step=47}}
| {{UDnote|fifth=34|step=47}}
| {{UDnote|fifth=34|step=47}}
Line 265: Line 313:
| 48
| 48
| 1010.53
| 1010.53
|[[34/19]], [[52/29]],<br>[[9/5]], ''[[16/9]]''
| {{UDnote|step=48}}
| {{UDnote|step=48}}
| {{UDnote|fifth=34|step=48}}
| {{UDnote|fifth=34|step=48}}
Line 270: Line 319:
| 49
| 49
| 1031.58
| 1031.58
|[[20/11]], [[29/16]]
| {{UDnote|step=49}}
| {{UDnote|step=49}}
| {{UDnote|fifth=34|step=49}}
| {{UDnote|fifth=34|step=49}}
Line 275: Line 325:
| 50
| 50
| 1052.63
| 1052.63
|[[11/6]]
| {{UDnote|step=50}}
| {{UDnote|step=50}}
| {{UDnote|fifth=34|step=50}}
| {{UDnote|fifth=34|step=50}}
Line 280: Line 331:
| 51
| 51
| 1073.68
| 1073.68
|[[13/7]], ''[[15/8]]''
| {{UDnote|step=51}}
| {{UDnote|step=51}}
| {{UDnote|fifth=34|step=51}}
| {{UDnote|fifth=34|step=51}}
Line 285: Line 337:
| 52
| 52
| 1094.74
| 1094.74
|[[32/17]]
| {{UDnote|step=52}}
| {{UDnote|step=52}}
| {{UDnote|fifth=34|step=52}}
| {{UDnote|fifth=34|step=52}}
Line 290: Line 343:
| 53
| 53
| 1115.79
| 1115.79
|[[19/10]], [[21/11]]
| {{UDnote|step=53}}
| {{UDnote|step=53}}
| {{UDnote|fifth=34|step=53}}
| {{UDnote|fifth=34|step=53}}
Line 295: Line 349:
| 54
| 54
| 1136.84
| 1136.84
|[[56/29]]
| {{UDnote|step=54}}
| {{UDnote|step=54}}
| {{UDnote|fifth=34|step=54}}
| {{UDnote|fifth=34|step=54}}
Line 300: Line 355:
| 55
| 55
| 1157.89
| 1157.89
|
| {{UDnote|step=55}}
| {{UDnote|step=55}}
| {{UDnote|fifth=34|step=55}}
| {{UDnote|fifth=34|step=55}}
Line 305: Line 361:
| 56
| 56
| 1178.95
| 1178.95
|
| {{UDnote|step=56}}
| {{UDnote|step=56}}
| {{UDnote|fifth=34|step=56}}
| {{UDnote|fifth=34|step=56}}
Line 310: Line 367:
| 57
| 57
| 1200.00
| 1200.00
|[[2/1]]
| {{UDnote|step=57}}
| {{UDnote|step=57}}
| {{UDnote|fifth=34|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 ==
== 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:Heinz]]
[[Category:Mothra]]
[[Category:Mothra]]
[[Category:Todo:add rank 2 temperaments table]]
{{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"