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{{Infobox ET}}
{{Infobox ET}}
'''57edo''' divides the [[Octave|octave]] into 57 parts of size 21.053¢. It can be used to tune [[Mothra_temperament|mothra temperament]], and is an excellent tuning for the 2.5/3.7.11.13.17.19 [[just_intonation_subgroup|just intonation subgroup]]. One way to describe 57 is that it has a [[5-limit|5-limit]] part consisting of three versions of 19, 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|19-limit]] 46&57 temperament [[Sensipent_family|Heinz]].
{{ED intro}}


[[5-limit|5-limit]] [[comma]]s: [[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|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 & 57 temperament [[heinz]]. It can also be used to tune [[mothra]] as well as [[trismegistus]].


[[11-limit|11-limit]] commas: [[99/98]], [[385/384]], [[441/440]], [[625/616]]
=== Odd harmonics ===
{{Harmonics in equal|57}}


==Just approximation==
=== Subsets and supersets ===
57edo contains [[3edo]] and [[19edo]] as subsets.


{{Primes in edo|57}}
== Intervals ==
 
{| class="wikitable center-1 right-2 center-3 center-4"
==Intervals==
 
{| class="wikitable"
|-
|-
! [[Degree|Degree]]
! #
![[cent|Cents]]
! [[Cent]]s
! [[Ups and downs notation]] (flat fifth 11\19)
! [[Ups and downs notation|Ups and downs notation]]<br>(Flat Fifth 11\19)
! [[Ups and downs notation]] (sharp fifth 34\57)
! [[Ups and downs notation|Ups and downs notation]]<br>(Sharp Fifth 34\57)
|-
|-
| style="text-align:center;" | 0
| 0
| style="text-align:right;" | 0.0000
| 0.00
| {{Ups and downs note name|step=0}}
| {{UDnote|step=0}}
| {{Ups and downs note name|fifth=34|step=0}}
| {{UDnote|fifth=34|step=0}}
|-
|-
| style="text-align:center;" | 1
| 1
| style="text-align:right;" | 21.0526
| 21.05
| {{Ups and downs note name|step=1}}
| {{UDnote|step=1}}
| {{Ups and downs note name|fifth=34|step=1}}
| {{UDnote|fifth=34|step=1}}
|-
|-
| style="text-align:center;" | 2
| 2
| style="text-align:right;" | 42.1053
| 42.11
| {{Ups and downs note name|step=2}}
| {{UDnote|step=2}}
| {{Ups and downs note name|fifth=34|step=2}}
| {{UDnote|fifth=34|step=2}}
|-
|-
| style="text-align:center;" | 3
| 3
| style="text-align:right;" | 63.1579
| 63.16
| {{Ups and downs note name|step=3}}
| {{UDnote|step=3}}
| {{Ups and downs note name|fifth=34|step=3}}
| {{UDnote|fifth=34|step=3}}
|-
|-
| style="text-align:center;" | 4
| 4
| style="text-align:right;" | 84.2105
| 84.21
| {{Ups and downs note name|step=4}}
| {{UDnote|step=4}}
| {{Ups and downs note name|fifth=34|step=4}}
| {{UDnote|fifth=34|step=4}}
|-
|-
| style="text-align:center;" | 5
| 5
| style="text-align:right;" | 105.2632
| 105.26
| {{Ups and downs note name|step=5}}
| {{UDnote|step=5}}
| {{Ups and downs note name|fifth=34|step=5}}
| {{UDnote|fifth=34|step=5}}
|-
|-
| style="text-align:center;" | 6
| 6
| style="text-align:right;" | 126.3158
| 126.32
| {{Ups and downs note name|step=6}}
| {{UDnote|step=6}}
| {{Ups and downs note name|fifth=34|step=6}}
| {{UDnote|fifth=34|step=6}}
|-
|-
| style="text-align:center;" | 7
| 7
| style="text-align:right;" | 147.3684
| 147.37
| {{Ups and downs note name|step=7}}
| {{UDnote|step=7}}
| {{Ups and downs note name|fifth=34|step=7}}
| {{UDnote|fifth=34|step=7}}
|-
|-
| style="text-align:center;" | 8
| 8
| style="text-align:right;" | 168.42105
| 168.42
| {{Ups and downs note name|step=8}}
| {{UDnote|step=8}}
| {{Ups and downs note name|fifth=34|step=8}}
| {{UDnote|fifth=34|step=8}}
|-
|-
| style="text-align:center;" | 9
| 9
| style="text-align:right;" | 189.4737
| 189.47
| {{Ups and downs note name|step=9}}
| {{UDnote|step=9}}
| {{Ups and downs note name|fifth=34|step=9}}
| {{UDnote|fifth=34|step=9}}
|-
|-
| style="text-align:center;" | 10
| 10
| style="text-align:right;" | 210.5263
| 210.53
| {{Ups and downs note name|step=10}}
| {{UDnote|step=10}}
| {{Ups and downs note name|fifth=34|step=10}}
| {{UDnote|fifth=34|step=10}}
|-
|-
| style="text-align:center;" | 11
| 11
| style="text-align:right;" | 231.57895
| 231.58
| {{Ups and downs note name|step=11}}
| {{UDnote|step=11}}
| {{Ups and downs note name|fifth=34|step=11}}
| {{UDnote|fifth=34|step=11}}
|-
|-
| style="text-align:center;" | 12
| 12
| style="text-align:right;" | 252.6316
| 252.63
| {{Ups and downs note name|step=12}}
| {{UDnote|step=12}}
| {{Ups and downs note name|fifth=34|step=12}}
| {{UDnote|fifth=34|step=12}}
|-
|-
| style="text-align:center;" | 13
| 13
| style="text-align:right;" | 273.6842
| 273.68
| {{Ups and downs note name|step=13}}
| {{UDnote|step=13}}
| {{Ups and downs note name|fifth=34|step=13}}
| {{UDnote|fifth=34|step=13}}
|-
|-
| style="text-align:center;" | 14
| 14
| style="text-align:right;" | 294.7368
| 294.74
| {{Ups and downs note name|step=14}}
| {{UDnote|step=14}}
| {{Ups and downs note name|fifth=34|step=14}}
| {{UDnote|fifth=34|step=14}}
|-
|-
| style="text-align:center;" | 15
| 15
| style="text-align:right;" | 315.7895
| 315.79
| {{Ups and downs note name|step=15}}
| {{UDnote|step=15}}
| {{Ups and downs note name|fifth=34|step=15}}
| {{UDnote|fifth=34|step=15}}
|-
|-
| style="text-align:center;" | 16
| 16
| style="text-align:right;" | 336.8421
| 336.84
| {{Ups and downs note name|step=16}}
| {{UDnote|step=16}}
| {{Ups and downs note name|fifth=34|step=16}}
| {{UDnote|fifth=34|step=16}}
|-
|-
| style="text-align:center;" | 17
| 17
| style="text-align:right;" | 357.8947
| 357.89
| {{Ups and downs note name|step=17}}
| {{UDnote|step=17}}
| {{Ups and downs note name|fifth=34|step=17}}
| {{UDnote|fifth=34|step=17}}
|-
|-
| style="text-align:center;" | 18
| 18
| style="text-align:right;" | 378.9474
| 378.95
| {{Ups and downs note name|step=18}}
| {{UDnote|step=18}}
| {{Ups and downs note name|fifth=34|step=18}}
| {{UDnote|fifth=34|step=18}}
|-
|-
| style="text-align:center;" | 19
| 19
| style="text-align:right;" | 400
| 400.00
| {{Ups and downs note name|step=19}}
| {{UDnote|step=19}}
| {{Ups and downs note name|fifth=34|step=19}}
| {{UDnote|fifth=34|step=19}}
|-
|-
| style="text-align:center;" | 20
| 20
| style="text-align:right;" | 421.0526
| 421.05
| {{Ups and downs note name|step=20}}
| {{UDnote|step=20}}
| {{Ups and downs note name|fifth=34|step=20}}
| {{UDnote|fifth=34|step=20}}
|-
|-
| style="text-align:center;" | 21
| 21
| style="text-align:right;" | 442.1053
| 442.11
| {{Ups and downs note name|step=21}}
| {{UDnote|step=21}}
| {{Ups and downs note name|fifth=34|step=21}}
| {{UDnote|fifth=34|step=21}}
|-
|-
| style="text-align:center;" | 22
| 22
| style="text-align:right;" | 463.1579
| 463.16
| {{Ups and downs note name|step=22}}
| {{UDnote|step=22}}
| {{Ups and downs note name|fifth=34|step=22}}
| {{UDnote|fifth=34|step=22}}
|-
|-
| style="text-align:center;" | 23
| 23
| style="text-align:right;" | 484.2105
| 484.21
| {{Ups and downs note name|step=23}}
| {{UDnote|step=23}}
| {{Ups and downs note name|fifth=34|step=23}}
| {{UDnote|fifth=34|step=23}}
|-
|-
| style="text-align:center;" | 24
| 24
| style="text-align:right;" | 505.2632
| 505.26
| {{Ups and downs note name|step=24}}
| {{UDnote|step=24}}
| {{Ups and downs note name|fifth=34|step=24}}
| {{UDnote|fifth=34|step=24}}
|-
|-
| style="text-align:center;" | 25
| 25
| style="text-align:right;" | 526.3158
| 526.32
| {{Ups and downs note name|step=25}}
| {{UDnote|step=25}}
| {{Ups and downs note name|fifth=34|step=25}}
| {{UDnote|fifth=34|step=25}}
|-
|-
| style="text-align:center;" | 26
| 26
| style="text-align:right;" | 547.3684
| 547.37
| {{Ups and downs note name|step=26}}
| {{UDnote|step=26}}
| {{Ups and downs note name|fifth=34|step=26}}
| {{UDnote|fifth=34|step=26}}
|-
|-
| style="text-align:center;" | 27
| 27
| style="text-align:right;" | 568.42105
| 568.42
| {{Ups and downs note name|step=27}}
| {{UDnote|step=27}}
| {{Ups and downs note name|fifth=34|step=27}}
| {{UDnote|fifth=34|step=27}}
|-
|-
| style="text-align:center;" | 28
| 28
| style="text-align:right;" | 589.4737
| 589.47
| {{Ups and downs note name|step=28}}
| {{UDnote|step=28}}
| {{Ups and downs note name|fifth=34|step=28}}
| {{UDnote|fifth=34|step=28}}
|-
|-
| style="text-align:center;" | 29
| 29
| style="text-align:right;" | 610.5263
| 610.53
| {{Ups and downs note name|step=29}}
| {{UDnote|step=29}}
| {{Ups and downs note name|fifth=34|step=29}}
| {{UDnote|fifth=34|step=29}}
|-
|-
| style="text-align:center;" | 30
| 30
| style="text-align:right;" | 631.57895
| 631.58
| {{Ups and downs note name|step=30}}
| {{UDnote|step=30}}
| {{Ups and downs note name|fifth=34|step=30}}
| {{UDnote|fifth=34|step=30}}
|-
|-
| style="text-align:center;" | 31
| 31
| style="text-align:right;" | 652.6316
| 652.63
| {{Ups and downs note name|step=31}}
| {{UDnote|step=31}}
| {{Ups and downs note name|fifth=34|step=31}}
| {{UDnote|fifth=34|step=31}}
|-
|-
| style="text-align:center;" | 32
| 32
| style="text-align:right;" | 673.6842
| 673.68
| {{Ups and downs note name|step=32}}
| {{UDnote|step=32}}
| {{Ups and downs note name|fifth=34|step=32}}
| {{UDnote|fifth=34|step=32}}
|-
|-
| style="text-align:center;" | 33
| 33
| style="text-align:right;" | 694.7368
| 694.74
| {{Ups and downs note name|step=33}}
| {{UDnote|step=33}}
| {{Ups and downs note name|fifth=34|step=33}}
| {{UDnote|fifth=34|step=33}}
|-
|-
| style="text-align:center;" | 34
| 34
| style="text-align:right;" | 715.7895
| 715.79
| {{Ups and downs note name|step=34}}
| {{UDnote|step=34}}
| {{Ups and downs note name|fifth=34|step=34}}
| {{UDnote|fifth=34|step=34}}
|-
|-
| style="text-align:center;" | 35
| 35
| style="text-align:right;" | 736.8421
| 736.84
| {{Ups and downs note name|step=35}}
| {{UDnote|step=35}}
| {{Ups and downs note name|fifth=34|step=35}}
| {{UDnote|fifth=34|step=35}}
|-
|-
| style="text-align:center;" | 36
| 36
| style="text-align:right;" | 757.8947
| 757.89
| {{Ups and downs note name|step=36}}
| {{UDnote|step=36}}
| {{Ups and downs note name|fifth=34|step=36}}
| {{UDnote|fifth=34|step=36}}
|-
|-
| style="text-align:center;" | 37
| 37
| style="text-align:right;" | 778.9474
| 778.95
| {{Ups and downs note name|step=37}}
| {{UDnote|step=37}}
| {{Ups and downs note name|fifth=34|step=37}}
| {{UDnote|fifth=34|step=37}}
|-
|-
| style="text-align:center;" | 38
| 38
| style="text-align:right;" | 800
| 800.00
| {{Ups and downs note name|step=38}}
| {{UDnote|step=38}}
| {{Ups and downs note name|fifth=34|step=38}}
| {{UDnote|fifth=34|step=38}}
|-
|-
| style="text-align:center;" | 39
| 39
| style="text-align:right;" | 821.0526
| 821.05
| {{Ups and downs note name|step=39}}
| {{UDnote|step=39}}
| {{Ups and downs note name|fifth=34|step=39}}
| {{UDnote|fifth=34|step=39}}
|-
|-
| style="text-align:center;" | 40
| 40
| style="text-align:right;" | 842.1053
| 842.11
| {{Ups and downs note name|step=40}}
| {{UDnote|step=40}}
| {{Ups and downs note name|fifth=34|step=40}}
| {{UDnote|fifth=34|step=40}}
|-
|-
| style="text-align:center;" | 41
| 41
| style="text-align:right;" | 863.1579
| 863.16
| {{Ups and downs note name|step=41}}
| {{UDnote|step=41}}
| {{Ups and downs note name|fifth=34|step=41}}
| {{UDnote|fifth=34|step=41}}
|-
|-
| style="text-align:center;" | 42
| 42
| style="text-align:right;" | 884.2105
| 884.21
| {{Ups and downs note name|step=42}}
| {{UDnote|step=42}}
| {{Ups and downs note name|fifth=34|step=42}}
| {{UDnote|fifth=34|step=42}}
|-
|-
| style="text-align:center;" | 43
| 43
| style="text-align:right;" | 905.2632
| 905.26
| {{Ups and downs note name|step=43}}
| {{UDnote|step=43}}
| {{Ups and downs note name|fifth=34|step=43}}
| {{UDnote|fifth=34|step=43}}
|-
|-
| style="text-align:center;" | 44
| 44
| style="text-align:right;" | 926.3158
| 926.32
| {{Ups and downs note name|step=44}}
| {{UDnote|step=44}}
| {{Ups and downs note name|fifth=34|step=44}}
| {{UDnote|fifth=34|step=44}}
|-
|-
| style="text-align:center;" | 45
| 45
| style="text-align:right;" | 947.3684
| 947.37
| {{Ups and downs note name|step=45}}
| {{UDnote|step=45}}
| {{Ups and downs note name|fifth=34|step=45}}
| {{UDnote|fifth=34|step=45}}
|-
|-
| style="text-align:center;" | 46
| 46
| style="text-align:right;" | 968.42105
| 968.42
| {{Ups and downs note name|step=46}}
| {{UDnote|step=46}}
| {{Ups and downs note name|fifth=34|step=46}}
| {{UDnote|fifth=34|step=46}}
|-
|-
| style="text-align:center;" | 47
| 47
| style="text-align:right;" | 989.4737
| 989.47
| {{Ups and downs note name|step=47}}
| {{UDnote|step=47}}
| {{Ups and downs note name|fifth=34|step=47}}
| {{UDnote|fifth=34|step=47}}
|-
|-
| style="text-align:center;" | 48
| 48
| style="text-align:right;" | 1010.5263
| 1010.53
| {{Ups and downs note name|step=48}}
| {{UDnote|step=48}}
| {{Ups and downs note name|fifth=34|step=48}}
| {{UDnote|fifth=34|step=48}}
|-
|-
| style="text-align:center;" | 49
| 49
| style="text-align:right;" | 1031.57895
| 1031.58
| {{Ups and downs note name|step=49}}
| {{UDnote|step=49}}
| {{Ups and downs note name|fifth=34|step=49}}
| {{UDnote|fifth=34|step=49}}
|-
|-
| style="text-align:center;" | 50
| 50
| style="text-align:right;" | 1052.6316
| 1052.63
| {{Ups and downs note name|step=50}}
| {{UDnote|step=50}}
| {{Ups and downs note name|fifth=34|step=50}}
| {{UDnote|fifth=34|step=50}}
|-
|-
| style="text-align:center;" | 51
| 51
| style="text-align:right;" | 1073.6842
| 1073.68
| {{Ups and downs note name|step=51}}
| {{UDnote|step=51}}
| {{Ups and downs note name|fifth=34|step=51}}
| {{UDnote|fifth=34|step=51}}
|-
|-
| style="text-align:center;" | 52
| 52
| style="text-align:right;" | 1094.7368
| 1094.74
| {{Ups and downs note name|step=52}}
| {{UDnote|step=52}}
| {{Ups and downs note name|fifth=34|step=52}}
| {{UDnote|fifth=34|step=52}}
|-
|-
| style="text-align:center;" | 53
| 53
| style="text-align:right;" | 1115.7895
| 1115.79
| {{Ups and downs note name|step=53}}
| {{UDnote|step=53}}
| {{Ups and downs note name|fifth=34|step=53}}
| {{UDnote|fifth=34|step=53}}
|-
|-
| style="text-align:center;" | 54
| 54
| style="text-align:right;" | 1136.8421
| 1136.84
| {{Ups and downs note name|step=54}}
| {{UDnote|step=54}}
| {{Ups and downs note name|fifth=34|step=54}}
| {{UDnote|fifth=34|step=54}}
|-
|-
| style="text-align:center;" | 55
| 55
| style="text-align:right;" | 1157.8947
| 1157.89
| {{Ups and downs note name|step=55}}
| {{UDnote|step=55}}
| {{Ups and downs note name|fifth=34|step=55}}
| {{UDnote|fifth=34|step=55}}
|-
|-
| style="text-align:center;" | 56
| 56
| style="text-align:right;" | 1178.9474
| 1178.95
| {{Ups and downs note name|step=56}}
| {{UDnote|step=56}}
| {{Ups and downs note name|fifth=34|step=56}}
| {{UDnote|fifth=34|step=56}}
|-
|-
| style="text-align:center;" | 57
| 57
| style="text-align:right;" | 1200
| 1200.00
| {{Ups and downs note name|step=57}}
| {{UDnote|step=57}}
| {{Ups and downs note name|fifth=34|step=57}}
| {{UDnote|fifth=34|step=57}}
|}
|}


<u>'''Modes of 57edo'''</u>
== Notation ==
=== 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.
{{sharpness-sharp3a}}
 
Using [[Helmholtz–Ellis]] accidentals, 57edo can also be notated using [[Alternative symbols for ups and downs notation#Sharp-3|alternative ups and downs]]:
{{Sharpness-sharp3}}
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.
 
=== 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|19-EDO]].
 
==== 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.


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)
== 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:57edo]]
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Heinz]]
[[Category:Heinz]]
[[Category:Mothra]]
[[Category:Mothra]]
[[Category:Subgroup]]
{{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)
Retrieved from "https://en.xen.wiki/w/57edo"