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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|57}}
{{EDO intro|57}}
== Theory ==
== Theory ==
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]].
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]].


[[5-limit|5-limit]] [[comma]]s: [[81/80]], [[3125/3072]]
[[5-limit|5-limit]] [[comma]]s: [[81/80]], [[3125/3072]]
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[[11-limit|11-limit]] commas: [[99/98]], [[385/384]], [[441/440]], [[625/616]]
[[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}}


==Intervals==
== Intervals ==


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