57edo: Difference between revisions

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


== Theory ==
== 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.  
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.  


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]].  
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]].  
57edo 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 ===
=== 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 ===
=== Subsets and supersets ===
57edo contains [[3edo]] and [[19edo]] as subsets.  
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 29: Line 33:
| 1
| 1
| 21.05
| 21.05
|
| {{UDnote|step=1}}
| {{UDnote|step=1}}
| {{UDnote|fifth=34|step=1}}
| {{UDnote|fifth=34|step=1}}
Line 34: Line 39:
| 2
| 2
| 42.11
| 42.11
|
| {{UDnote|step=2}}
| {{UDnote|step=2}}
| {{UDnote|fifth=34|step=2}}
| {{UDnote|fifth=34|step=2}}
Line 39: Line 45:
| 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 44: Line 51:
| 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 49: Line 57:
| 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 54: Line 63:
| 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 59: Line 69:
| 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 64: Line 75:
| 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 69: Line 81:
| 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 74: Line 87:
| 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 79: Line 93:
| 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 84: Line 99:
| 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 89: Line 105:
| 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 94: Line 111:
| 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 99: Line 117:
| 15
| 15
| 315.79
| 315.79
|[[6/5]]
| {{UDnote|step=15}}
| {{UDnote|step=15}}
| {{UDnote|fifth=34|step=15}}
| {{UDnote|fifth=34|step=15}}
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| 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}}
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| 17
| 17
| 357.89
| 357.89
|[[16/13]]
| {{UDnote|step=17}}
| {{UDnote|step=17}}
| {{UDnote|fifth=34|step=17}}
| {{UDnote|fifth=34|step=17}}
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| 18
| 18
| 378.95
| 378.95
|[[5/4]]
| {{UDnote|step=18}}
| {{UDnote|step=18}}
| {{UDnote|fifth=34|step=18}}
| {{UDnote|fifth=34|step=18}}
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| 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}}
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| 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}}
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| 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 134: Line 159:
| 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}}
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| 23
| 23
| 484.21
| 484.21
|[[33/25]]
| {{UDnote|step=23}}
| {{UDnote|step=23}}
| {{UDnote|fifth=34|step=23}}
| {{UDnote|fifth=34|step=23}}
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| 24
| 24
| 505.26
| 505.26
|[[4/3]]
| {{UDnote|step=24}}
| {{UDnote|step=24}}
| {{UDnote|fifth=34|step=24}}
| {{UDnote|fifth=34|step=24}}
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| 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 154: Line 183:
| 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 159: Line 189:
| 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}}
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| 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 169: Line 201:
| 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 174: Line 207:
| 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 179: Line 213:
| 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 184: Line 219:
| 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 189: Line 225:
| 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 194: Line 231:
| 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 199: Line 237:
| 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 204: Line 243:
| 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 209: Line 249:
| 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 214: Line 255:
| 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 219: Line 261:
| 39
| 39
| 821.05
| 821.05
|[[8/5]]
| {{UDnote|step=39}}
| {{UDnote|step=39}}
| {{UDnote|fifth=34|step=39}}
| {{UDnote|fifth=34|step=39}}
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| 40
| 40
| 842.11
| 842.11
|[[13/8]]
| {{UDnote|step=40}}
| {{UDnote|step=40}}
| {{UDnote|fifth=34|step=40}}
| {{UDnote|fifth=34|step=40}}
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| 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}}
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| 42
| 42
| 884.21
| 884.21
|[[5/3]]
| {{UDnote|step=42}}
| {{UDnote|step=42}}
| {{UDnote|fifth=34|step=42}}
| {{UDnote|fifth=34|step=42}}
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| 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 244: Line 291:
| 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 249: Line 297:
| 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 254: Line 303:
| 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 259: Line 309:
| 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 264: Line 315:
| 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 269: Line 321:
| 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 274: Line 327:
| 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 279: Line 333:
| 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 284: Line 339:
| 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 289: Line 345:
| 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 294: Line 351:
| 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 299: Line 357:
| 55
| 55
| 1157.89
| 1157.89
|
| {{UDnote|step=55}}
| {{UDnote|step=55}}
| {{UDnote|fifth=34|step=55}}
| {{UDnote|fifth=34|step=55}}
Line 304: Line 363:
| 56
| 56
| 1178.95
| 1178.95
|
| {{UDnote|step=56}}
| {{UDnote|step=56}}
| {{UDnote|fifth=34|step=56}}
| {{UDnote|fifth=34|step=56}}
Line 309: Line 369:
| 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 ==
== Notation ==
=== Stein–Zimmermann–Gould notation ===
57edo 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.
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]].
====Evo flavor====


=== 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>
<imagemap>
File:57-EDO_Evo_Sagittal.svg
File:57-EDO_Evo_Sagittal.svg
Line 328: Line 399:
</imagemap>
</imagemap>


====Revo flavor====
==== Revo flavor ====
 
<imagemap>
<imagemap>
File:57-EDO_Revo_Sagittal.svg
File:57-EDO_Revo_Sagittal.svg
Line 339: Line 409:
</imagemap>
</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.
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]]
{{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)

Latest revision as of 21:47, 14 October 2026

← 56edo 57edo 58edo →
Prime factorization 3 × 19
Step size 21.0526 ¢ 
Fifth 33\57 (694.737 ¢) (→ 11\19)
Semitones (A1:m2) 3:6 (63.16 ¢ : 126.3 ¢)
Dual sharp fifth 34\57 (715.789 ¢)
Dual flat fifth 33\57 (694.737 ¢) (→ 11\19)
Dual major 2nd 10\57 (210.526 ¢)
Consistency limit 7
Distinct consistency limit 7

57 equal divisions of the octave (abbreviated 57edo or 57ed2), also called 57-tone equal temperament (57tet) or 57 equal temperament (57et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 57 equal parts of about 21.1 ¢ each. Each step represents a frequency ratio of 21/57, or the 57th root of 2.

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 rings 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 & 57 temperament heinz. It can also be used to tune mothra as well as trismegistus.

57edo 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

Approximation of odd harmonics in 57edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -7.22 -7.37 -0.40 +6.62 -3.95 +1.58 +6.47 +0.31 -2.78 -7.62 +3.30
Relative (%) -34.3 -35.0 -1.9 +31.4 -18.8 +7.5 +30.7 +1.5 -13.2 -36.2 +15.7
Steps
(reduced) 		90
(33) 		132
(18) 		160
(46) 		181
(10) 		197
(26) 		211
(40) 		223
(52) 		233
(5) 		242
(14) 		250
(22) 		258
(30)
Approximation of odd harmonics in 57edo (continued)
Harmonic 25 27 29 31 33 35 37 39 41 43 45
Error Absolute (¢) +6.32 -0.60 +2.00 -8.19 +9.88 -7.77 +1.29 -5.64 -8.01 -6.25 -0.75
Relative (%) +30.0 -2.9 +9.5 -38.9 +47.0 -36.9 +6.1 -26.8 -38.0 -29.7 -3.6
Steps
(reduced) 		265
(37) 		271
(43) 		277
(49) 		282
(54) 		288
(3) 		292
(7) 		297
(12) 		301
(16) 		305
(20) 		309
(24) 		313
(28)

Subsets and supersets

57edo contains 3edo and 19edo as subsets. Tripling 57edo yields 171edo, which corrects the 3rd and 5th harmonics.

Intervals

# Cents Approximate ratios* Ups and downs notation
(Flat fifth 11\19) 		Ups and downs notation
(Sharp fifth 34\57)
0 0.00 1/1 D D
1 21.05 ^D, ^E♭♭♭ ^D, E♭
2 42.11 vD♯, vE♭♭ ^^D, ^E♭
3 63.16 29/28 D♯, E♭♭ ^3D, ^^E♭
4 84.21 20/19, 21/20, 22/21 ^D♯, ^E♭♭ ^4D, ^3E♭
5 105.26 17/16, 33/31 vD𝄪, vE♭ ^5D, ^4E♭
6 126.32 14/13, 16/15 D𝄪, E♭ v4D♯, v5E
7 147.37 12/11 ^D𝄪, ^E♭ v3D♯, v4E
8 168.42 11/10, 32/29 vD♯𝄪, vE vvD♯, v3E
9 189.47 19/17, 29/26,
10/9, 9/8 		E vD♯, vvE
10 210.53 26/23 ^E, ^F♭♭ D♯, vE
11 231.58 8/7 vE♯, vF♭ E
12 252.63 22/19 E♯, F♭ F
13 273.68 7/6, 34/29 ^E♯, ^F♭ ^F, G♭
14 294.74 19/16 vE𝄪, vF ^^F, ^G♭
15 315.79 6/5 F ^3F, ^^G♭
16 336.84 17/14, 28/23 ^F, ^G♭♭♭ ^4F, ^3G♭
17 357.89 16/13 vF♯, vG♭♭ ^5F, ^4G♭
18 378.95 5/4 F♯, G♭♭ v4F♯, v5G
19 400.00 24/19, 29/23 ^F♯, ^G♭♭ v3F♯, v4G
20 421.05 14/11, 9/7 vF𝄪, vG♭ vvF♯, v3G
21 442.11 22/17, 31/24 F𝄪, G♭ vF♯, vvG
22 463.16 17/13, 21/16, 38/29 ^F𝄪, ^G♭ F♯, vG
23 484.21 33/25 vF♯𝄪, vG G
24 505.26 4/3 G ^G, A♭
25 526.32 19/14, 42/31, 23/17 ^G, ^A♭♭♭ ^^G, ^A♭
26 547.37 11/8, 26/19 vG♯, vA♭♭ ^3G, ^^A♭
27 568.42 25/18, 32/23 G♯, A♭♭ ^4G, ^3A♭
28 589.47 31/22 ^G♯, ^A♭♭ ^5G, ^4A♭
29 610.53 44/31 vG𝄪, vA♭ v4G♯, v5A
30 631.58 23/16 G𝄪, A♭ v3G♯, v4A
31 652.63 16/11, 19/13 ^G𝄪, ^A♭ vvG♯, v3A
32 673.68 28/19, 31/21, 34/23 vG♯𝄪, vA vG♯, vvA
33 694.74 3/2 A G♯, vA
34 715.79 50/33 ^A, ^B♭♭♭ A
35 736.84 26/17, 32/21, 29/19 vA♯, vB♭♭ ^A, B♭
36 757.89 17/11, 31/20 A♯, B♭♭ ^^A, ^B♭
37 778.95 11/7, 14/9 ^A♯, ^B♭♭ ^3A, ^^B♭
38 800.00 19/12 vA𝄪, vB♭ ^4A, ^3B♭
39 821.05 8/5 A𝄪, B♭ ^5A, ^4B♭
40 842.11 13/8 ^A𝄪, ^B♭ v4A♯, v5B
41 863.16 23/14, 28/17, 33/20 vA♯𝄪, vB v3A♯, v4B
42 884.21 5/3 B vvA♯, v3B
43 905.26 32/19 ^B, ^C♭♭ vA♯, vvB
44 926.32 12/7, 29/17 vB♯, vC♭ A♯, vB
45 947.37 19/11 B♯, C♭ B
46 968.42 7/4 ^B♯, ^C♭ C
47 989.47 23/13 vB𝄪, vC ^C, D♭
48 1010.53 34/19, 52/29,
9/5, 16/9 		C ^^C, ^D♭
49 1031.58 20/11, 29/16 ^C, ^D♭♭♭ ^3C, ^^D♭
50 1052.63 11/6 vC♯, vD♭♭ ^4C, ^3D♭
51 1073.68 13/7, 15/8 C♯, D♭♭ ^5C, ^4D♭
52 1094.74 32/17 ^C♯, ^D♭♭ v4C♯, v5D
53 1115.79 19/10, 21/11 vC𝄪, vD♭ v3C♯, v4D
54 1136.84 56/29 C𝄪, D♭ vvC♯, v3D
55 1157.89 ^C𝄪, ^D♭ vC♯, vvD
56 1178.95 vC♯𝄪, vD C♯, vD
57 1200.00 2/1 D D

* 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:

Step offset 0 1 2 3 4 5 6 7
Sharp symbol
Flat symbol

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.

Step offset 0 1 2 3 4 5 6 7
Sharp symbol   
 
   
  
   
   
  
   
Flat symbol
 
   
  
   
   
  
   

Sagittal notation

This notation uses the same sagittal sequence as edos 50, 64, and 71b, and is a superset of the notation for 19edo.

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation1053/1024

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation1053/1024

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's 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 21s (augene)

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
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