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57edo: Difference between revisions

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→Intervals: Add some more common intervals and intervals specified by Orwell, including some for which only the octave complement was present in the automatically generated output
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=== 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 ===
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== Intervals ==
== Intervals ==
{| class="wikitable center-1 right-2 center-3 center-4"
{| class="wikitable center-1 right-2 center-4 center-5"
|-
|-
! #
! #
! [[Cent]]s
! [[Cent]]s
!Approximate<br>ratios (in ''italics''<br>if inconsistent)
! Approximate ratios*
! [[Ups and downs notation|Ups and downs notation]]<br>(Flat Fifth 11\19)
! [[Kite's 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)
! [[Kite's ups and downs notation|Ups and downs notation]]<br>(Sharp fifth 34\57)
|-
|-
| 0
| 0
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| 9
| 9
| 189.47
| 189.47
|''[[9/8]]'', [[10/9]],
|[[19/17]], [[29/26]],<br>[[10/9]], ''[[9/8]]''
[[19/17]], [[29/26]]
| {{UDnote|step=9}}
| {{UDnote|step=9}}
| {{UDnote|fifth=34|step=9}}
| {{UDnote|fifth=34|step=9}}
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| 20
| 20
| 421.05
| 421.05
|''[[9/7]]'', [[14/11]]
|[[14/11]], ''[[9/7]]''
| {{UDnote|step=20}}
| {{UDnote|step=20}}
| {{UDnote|fifth=34|step=20}}
| {{UDnote|fifth=34|step=20}}
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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]], [[23/17]]
|[[19/14]], [[42/31]], [[23/17]]
| {{UDnote|step=25}}
| {{UDnote|step=25}}
| {{UDnote|fifth=34|step=25}}
| {{UDnote|fifth=34|step=25}}
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| 29
| 29
| 610.53
| 610.53
|
|[[44/31]]
| {{UDnote|step=29}}
| {{UDnote|step=29}}
| {{UDnote|fifth=34|step=29}}
| {{UDnote|fifth=34|step=29}}
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| 33
| 33
| 694.74
| 694.74
|
|[[3/2]]
| {{UDnote|step=33}}
| {{UDnote|step=33}}
| {{UDnote|fifth=34|step=33}}
| {{UDnote|fifth=34|step=33}}
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| 34
| 34
| 715.79
| 715.79
|
|[[50/33]]
| {{UDnote|step=34}}
| {{UDnote|step=34}}
| {{UDnote|fifth=34|step=34}}
| {{UDnote|fifth=34|step=34}}
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| 35
| 35
| 736.84
| 736.84
|[[26/17]], [[32/21]] [[29/19]]
|[[26/17]], [[32/21]], [[29/19]]
| {{UDnote|step=35}}
| {{UDnote|step=35}}
| {{UDnote|fifth=34|step=35}}
| {{UDnote|fifth=34|step=35}}
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| 37
| 37
| 778.95
| 778.95
|''[[14/9]]'', [[11/7]]
|[[11/7]], ''[[14/9]]''
| {{UDnote|step=37}}
| {{UDnote|step=37}}
| {{UDnote|fifth=34|step=37}}
| {{UDnote|fifth=34|step=37}}
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| 40
| 40
| 842.11
| 842.11
|[[14/8]]
|[[13/8]]
| {{UDnote|step=40}}
| {{UDnote|step=40}}
| {{UDnote|fifth=34|step=40}}
| {{UDnote|fifth=34|step=40}}
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| 48
| 48
| 1010.53
| 1010.53
|''[[16/9]]'', [[9/5]],
|[[34/19]], [[52/29]],<br>[[9/5]], ''[[16/9]]''
[[34/19]], [[52/29]]
| {{UDnote|step=48}}
| {{UDnote|step=48}}
| {{UDnote|fifth=34|step=48}}
| {{UDnote|fifth=34|step=48}}
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| 54
| 54
| 1136.84
| 1136.84
|
|[[56/29]]
| {{UDnote|step=54}}
| {{UDnote|step=54}}
| {{UDnote|fifth=34|step=54}}
| {{UDnote|fifth=34|step=54}}
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| {{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 ==
=== Ups and downs 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.
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}}
{{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 ===
=== 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 ====
==== Evo flavor ====
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</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 ==
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