57edo: Difference between revisions

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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 ===

{{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 ===

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== Intervals ==

{| class="wikitable center-1 right-2 center-3 center-4"

{| class="wikitable center-1 right-2 center-4 center-5"

|-

! #

! [[Cent]]s

!Approximate ratios

! Approximate ratios*

~~(in ''italics'' if~~

! [[Kite's ups and downs notation|Ups and downs notation]] (Flat fifth 11\19)

~~inconsistent)~~

! [[Kite's ups and downs notation|Ups and downs notation]] (Sharp fifth 34\57)

! [[~~Ups~~ and downs notation|Ups and downs notation]] (Flat ~~Fifth~~ 11\19)

! [[~~Ups~~ and downs notation|Ups and downs notation]] (Sharp ~~Fifth~~ 34\57)

|-

| 0

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| 6

| 126.32

|[[14/13]]

|[[14/13]], ''[[16/15]]''

| {{UDnote|step=6}}

| {{UDnote|fifth=34|step=6}}

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| 9

| 189.47

|[[19/17]], [[29/26]]

|[[19/17]], [[29/26]], [[10/9]], ''[[9/8]]''

| {{UDnote|step=9}}

| {{UDnote|fifth=34|step=9}}

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| 20

| 421.05

|''[[9/7]]'', [[14/11]]

|[[14/11]], ''[[9/7]]''

| {{UDnote|step=20}}

| {{UDnote|fifth=34|step=20}}

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| 22

| 463.16

|[[17/13]]

|[[17/13]], [[21/16]], [[38/29]]

| {{UDnote|step=22}}

| {{UDnote|fifth=34|step=22}}

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| 24

| 505.26

|

|[[4/3]]

| {{UDnote|step=24}}

| {{UDnote|fifth=34|step=24}}

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| 25

| 526.32

|[[19/14]], [[23/17]]

|[[19/14]], [[42/31]], [[23/17]]

| {{UDnote|step=25}}

| {{UDnote|fifth=34|step=25}}

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| 29

| 610.53

|

|[[44/31]]

| {{UDnote|step=29}}

| {{UDnote|fifth=34|step=29}}

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| 33

| 694.74

|

|[[3/2]]

| {{UDnote|step=33}}

| {{UDnote|fifth=34|step=33}}

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| 34

| 715.79

|

|[[50/33]]

| {{UDnote|step=34}}

| {{UDnote|fifth=34|step=34}}

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| 35

| 736.84

|[[26/17]], [[29/19]]

|[[26/17]], [[32/21]], [[29/19]]

| {{UDnote|step=35}}

| {{UDnote|fifth=34|step=35}}

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| 37

| 778.95

|''[[14/9]]'', [[11/7]]

|[[11/7]], ''[[14/9]]''

| {{UDnote|step=37}}

| {{UDnote|fifth=34|step=37}}

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| 40

| 842.11

|[[14/8]]

|[[13/8]]

| {{UDnote|step=40}}

| {{UDnote|fifth=34|step=40}}

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| 48

| 1010.53

|[[34/19]]

|[[34/19]], [[52/29]], [[9/5]], ''[[16/9]]''

| {{UDnote|step=48}}

| {{UDnote|fifth=34|step=48}}

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| 51

| 1073.68

|[[13/7]]

|[[13/7]], ''[[15/8]]''

| {{UDnote|step=51}}

| {{UDnote|fifth=34|step=51}}

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| 54

| 1136.84

|

|[[56/29]]

| {{UDnote|step=54}}

| {{UDnote|fifth=34|step=54}}

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| {{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 ==

=== ~~Ups~~ and downs notation ===

=== Stein–Zimmermann–Gould notation ===

57edo can be notated using [[Stein–Zimmermann–Gould 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.

=== 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.

~~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~~]].

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 ====

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</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 ==

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== Music ==

; [[Bryan Deister]]

* [https://www.youtube.com/watch?v=4NZvSgS3ryY ''57edo improv''] (2025)

* [https://www.youtube.com/watch?v=4NZvSgS3ryY ''57edo improv''] (2025-05-12)

* ''Prelude in 57edo'' (2025) (~~switches Lumatone mapping between~~ sections)

* ''Prelude in 57edo'' (2025) (Uses different temperaments in different sections)

** [https://www.youtube.com/shorts/FylRB5MGBz4 ''Part 1 (short clip)'']

** [https://www.youtube.com/shorts/FylRB5MGBz4 ''Part 1 (short clip)''] (Lumatone view using [[5L 2s|diatonic]] sharp fifth + [[mothra]] flat fifth mapping)

** [https://www.youtube.com/shorts/yVv0s8t--pg ''Part 2 (short clip)'']

** [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)

@@ Line 6: / Line 6: @@
 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 ===
@@ Line 14: / Line 17: @@
 == Intervals ==
-{| class="wikitable center-1 right-2 center-3 center-4"
+{| class="wikitable center-1 right-2 center-4 center-5"
 |-
 ! #
 ! [[Cent]]s
-!Approximate ratios
+! Approximate ratios*
-(in ''italics'' if
+! [[Kite's ups and downs notation|Ups and downs notation]]<br>(Flat fifth 11\19)
-inconsistent)
+! [[Kite's ups and downs notation|Ups and downs notation]]<br>(Sharp fifth 34\57)
-! [[Ups and downs notation|Ups and downs notation]]<br>(Flat Fifth 11\19)
-! [[Ups and downs notation|Ups and downs notation]]<br>(Sharp Fifth 34\57)
 |-
 | 0
@@ Line 62: / Line 63: @@
 | 6
 | 126.32
-|[[14/13]]
+|[[14/13]], ''[[16/15]]''
 | {{UDnote|step=6}}
 | {{UDnote|fifth=34|step=6}}
@@ Line 80: / Line 81: @@
 | 9
 | 189.47
-|[[19/17]], [[29/26]]
+|[[19/17]], [[29/26]],<br>[[10/9]], ''[[9/8]]''
 | {{UDnote|step=9}}
 | {{UDnote|fifth=34|step=9}}
@@ Line 146: / Line 147: @@
 | 20
 | 421.05
-|''[[9/7]]'', [[14/11]]
+|[[14/11]], ''[[9/7]]''
 | {{UDnote|step=20}}
 | {{UDnote|fifth=34|step=20}}
@@ Line 158: / Line 159: @@
 | 22
 | 463.16
-|[[17/13]]
+|[[17/13]], [[21/16]], [[38/29]]
 | {{UDnote|step=22}}
 | {{UDnote|fifth=34|step=22}}
@@ Line 170: / Line 171: @@
 | 24
 | 505.26
-|
+|[[4/3]]
 | {{UDnote|step=24}}
 | {{UDnote|fifth=34|step=24}}
@@ Line 176: / Line 177: @@
 | 25
 | 526.32
-|[[19/14]], [[23/17]]
+|[[19/14]], [[42/31]], [[23/17]]
 | {{UDnote|step=25}}
 | {{UDnote|fifth=34|step=25}}
@@ Line 200: / Line 201: @@
 | 29
 | 610.53
-|
+|[[44/31]]
 | {{UDnote|step=29}}
 | {{UDnote|fifth=34|step=29}}
@@ Line 224: / Line 225: @@
 | 33
 | 694.74
-|
+|[[3/2]]
 | {{UDnote|step=33}}
 | {{UDnote|fifth=34|step=33}}
@@ Line 230: / Line 231: @@
 | 34
 | 715.79
-|
+|[[50/33]]
 | {{UDnote|step=34}}
 | {{UDnote|fifth=34|step=34}}
@@ Line 236: / Line 237: @@
 | 35
 | 736.84
-|[[26/17]], [[29/19]]
+|[[26/17]], [[32/21]], [[29/19]]
 | {{UDnote|step=35}}
 | {{UDnote|fifth=34|step=35}}
@@ Line 248: / Line 249: @@
 | 37
 | 778.95
-|''[[14/9]]'', [[11/7]]
+|[[11/7]], ''[[14/9]]''
 | {{UDnote|step=37}}
 | {{UDnote|fifth=34|step=37}}
@@ Line 266: / Line 267: @@
 | 40
 | 842.11
-|[[14/8]]
+|[[13/8]]
 | {{UDnote|step=40}}
 | {{UDnote|fifth=34|step=40}}
@@ Line 314: / Line 315: @@
 | 48
 | 1010.53
-|[[34/19]]
+|[[34/19]], [[52/29]],<br>[[9/5]], ''[[16/9]]''
 | {{UDnote|step=48}}
 | {{UDnote|fifth=34|step=48}}
@@ Line 332: / Line 333: @@
 | 51
 | 1073.68
-|[[13/7]]
+|[[13/7]], ''[[15/8]]''
 | {{UDnote|step=51}}
 | {{UDnote|fifth=34|step=51}}
@@ Line 350: / Line 351: @@
 | 54
 | 1136.84
-|
+|[[56/29]]
 | {{UDnote|step=54}}
 | {{UDnote|fifth=34|step=54}}
@@ Line 372: / Line 373: @@
 | {{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 ==
-=== Ups and downs notation ===
+=== Stein–Zimmermann–Gould notation ===
+edo 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}}
-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]].
+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 ====
@@ Line 405: / Line 409: @@
 </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 ==
@@ Line 421: / Line 425: @@
 == Music ==
 ; [[Bryan Deister]]
-* [https://www.youtube.com/watch?v=4NZvSgS3ryY ''57edo improv''] (2025)
+* [https://www.youtube.com/watch?v=4NZvSgS3ryY ''57edo improv''] (2025-05-12)
-* ''Prelude in 57edo'' (2025) (switches Lumatone mapping between sections)
+* ''Prelude in 57edo'' (2025) (Uses different temperaments in different sections)
-** [https://www.youtube.com/shorts/FylRB5MGBz4 ''Part 1 (short clip)'']
+** [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)'']
+** [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)

57edo: Difference between revisions

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