User:Eufalesio/Mappings of edos: Difference between revisions

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Different ways edos I deem important map intervals, made mostly for myself for notekeeping but may be useful to you.
Different ways edos I deem important map intervals, made mostly for myself for notekeeping but may be useful to you. Nomenclature is a mix of my [[User:Eufalesio/Holopyth and Hemipyth|Holopyth and Hemipyth]] and [[Kite's ups and downs notation]], but resumed: sub/super/hypo/hyper add -1/+1/-2/+2 mapped pythagorean commas, up/down add edosteps.
 
Before going into the tables: edos listed here are 
 
== Meantonoids* ==
Edos that temper the syntonic comma '''in the golden series'''. Up/down can be used for diesis halves.
 
* 19edo is coarse, decent 5-limit.
* 31edo has a great 11-limit, usable 13-limit, still a bit coarse.
* 50 has a worse 7-limit, but better overall 19-limit.
* 62edo greatly improves upon 31edo expanding it to the 23-limit. Finest reasonably usable meantone edo.


== Meantonoid edos ==
Edos that temper out the syntonic comma, in the golden series.
{| class="wikitable"
{| class="wikitable"
|+
!Edo
!Prime
!m2:d2
!5
!5
!7
!7
Line 15: Line 23:
!29
!29
|-
|-
|12edo
|[[19edo|19]]
|1:1
| rowspan="4" |major third
| rowspan="4" |major third
| rowspan="4" |subminor seventh
|tritone
|minor sixth
| rowspan="2" |minor second
| rowspan="2" |minor third
| rowspan="4" |supertritone
|minor seventh
|-
|'''[[31edo|31]]'''
|2:1
| rowspan="3" |superfourth
|superminor sixth
|superminor seventh
|-
|[[50edo|50]]
|3:2
| rowspan="2" |upminor sixth
| rowspan="2" |downminor second
| rowspan="2" |downminor third
| rowspan="2" |upminor seventh
|-
|[[62edo|62]]
|4:2
|}
<nowiki>*</nowiki>Treating super/sub as meantone dieses (d2) not pythagorean commas.
== Comptons ==
Edos that temper out the [[Pythagorean comma|poma]]. Not using up/down in 24edo because up/down differ too much in size from 72 and 84. The mapping of up/down is obviously fractions of 1\12.
* 72edo has an astounding 11-limit, usable in the 19-limit.
* 84edo has a great 2.3.5.7.13, worse 11.
{| class="wikitable"
!'''Edo'''
!'''n:12edo'''
!'''5'''
!'''7'''
!'''11'''
!'''13'''
!'''17'''
!'''19'''
!'''23'''
!'''29'''
|-
|[[12edo|12]]
|1
| rowspan="2" |major third
|minor seventh
|minor seventh
| rowspan="2" |tritone
| rowspan="2" |minor sixth
| rowspan="3" |minor second
| rowspan="3" |minor third
|tritone
|tritone
| rowspan="2" |minor seventh
|minor sixth
| rowspan="4" |minor second
| rowspan="4" |minor third
|tritone
|minor seventh
|-
|[[24edo|24]]
|2
|halfdimminor seventh
|halfaugfourth
|halfaugminor sixth
|halfaugtritone
|halfaugminor seventh
|-
|[[72edo|'''72''']]
|6
| rowspan="2" |downmajor third
| rowspan="2" |dudminor seventh
| rowspan="2" |trupfourth
| rowspan="2" |trupminor sixth
|uptritone
|upminor seventh
|-
|[[84edo|84]]
|7
|duptritone
|dupminor seventh
|}
 
== Superpythoids ==
Edos with sharp fifths. Up/down can be used for limma (halves).
 
* 22edo has a usable 11-limit, though quite exaggerated.
* 27edo has a usable no-11 13-limit.
* 34edo has a great 2.3.5.13.17.
 
{| class="wikitable"
!Edo
!A1:m2
!5
!7
!11
!13
!17
!19
!23
!29
|-
|-
|19edo
|[[22edo|22]]
| rowspan="3" |subminor seventh
|3:1
| rowspan="3" |supertritone
| rowspan="3" |downmajor third
| rowspan="3" |minor seventh
| rowspan="3" |upfourth
|upminor sixth
| rowspan="2" |upminor second
|minor third
|tritone
| rowspan="3" |upminor seventh
|-
|-
|31edo
|[[27edo|27]]
| rowspan="2" |superfourth
|4:1
|superminor sixth
|dupminor sixth
|superminor seventh
|upminor third
|downtritone
|-
|-
|50edo
|[[34edo|34]]
|1/2perminor sixth
|4:2
|1/2subminor second
|upminor sixth
|1/2subminor third
|trupminor second
|1/2perminor seventh
|minor third
|tritone
|}
|}
Treating super/sub as meantone dieses (d2) not pythagorean commas
== Panschismoids ==
Edos that have very accurate fifths and temper out very small or unnoticeable commas.


== Schismoid edos ==
* 41edo has a great 11-limit, usable no-17,23 29-limit
Edos that temper the schisma(s) or garischisma(c), or both(g).
* 53edo has an extremely accurate 2.3.5.13.19, decent 13-limit.
* 94edo has a well-rounded 23-limit with good accuracy.
 
=== Cassandroids ===
Have fifths close to just, supporting garibaldi. Up/down can be used for pc halves (or mercommas) in the case of 94edo.
{| class="wikitable"
{| class="wikitable"
!Prime
!Edo
!m2:pc
!5
!5
!7
!7
Line 55: Line 168:
!29
!29
|-
|-
|(g)41edo
|[[41edo|41]]
| rowspan="6" |submajor third
|3:1
| rowspan="3" |submajor third
| rowspan="3" |subminor seventh
| rowspan="3" |subminor seventh
| rowspan="3" |hyperfourth
| rowspan="3" |hyperfourth
| rowspan="3" |hyperminor sixth
| rowspan="3" |hyperminor sixth
| rowspan="2" |superminor second
| rowspan="2" |superminor second
| rowspan="6" |minor third
| rowspan="3" |minor third
|tritone
|tritone
| rowspan="2" |superminor seventh
| rowspan="2" |superminor seventh
|-
|-
|(g)53edo
|[[53edo|53]]
|4:1
|supertritone
|supertritone
|-
|-
|(g)94edo
|'''[[94edo|94]]'''
| rowspan="2" |1/2perminor second
|7:2
|1/2pertritone
|upperminor second
|1/2perminor seventh
|uppertritone
|upperminor seventh
|}
 
=== Helmholtzoids ===
Have fifths a smidge flatter than just, along the optimal range for schismic and pontiac. Up/down can be used for pc fractions. The true mappings for 171 and 224's edostep are -53 fifths (negative merccomma). 130 and 159 instead have a poma fraction as the only possible reasonable mapping.
 
* 130 has a well rounded 13-limit with very good accuracy, usable all the way to the no-29 31-limit.
* 159 has an unfathomably accurate 2.3.11, extremely accurate 2.3.5.11.17, usable in the no-17 29-limit.
* 171 has an unfathomably accurate 7-limit. Usable in the no-11 19-limit.
* 224 has an extremely accurate 13-limit. Bad for higher limits.
 
{| class="wikitable"
!Edo
!m2:pc
!5
!7
!11
!13
!17
!19
!23
!29
|-
|-
|(s)130edo
|[[130edo|130]]
|3/2subminor seventh
|10:2
| rowspan="4" |submajor third
| rowspan="4" |downsubminor seventh
|3perfourth
|3perfourth
|5/2perminor sixth
| rowspan="4" |upperminor sixth
| rowspan="4" |downperminor second
| rowspan="3" |minor third
|supertritone
|supertritone
|hyperminor seventh
|hyperminor seventh
|-
|-
|(s)159edo
|[[159edo|159]]
| rowspan="2" |4/3subminor seventh
|12:3
| rowspan="2" |7/3perfourth
| rowspan="2" |uphyperfourth
|hyperminor sixth
|downpertritone
| rowspan="2" |2/3perminor second
|upperminor seventh
|2/3pertritone
|4/3perminor seventh
|-
|-
|(s)171edo
|[[171edo|171]]
|7/3perminor sixth
|13:3
|supertritone
|supertritone
|5/3perminor seventh
| rowspan="2" |dupperminor seventh
|-
|[[224edo|224]]
|17:4
|duphyperfourth
|upperminor third
|downpertritone
|}
 
=== Non-cassandroid Ultimates ===
Have fifths a smidge sharper than just, along the optimal range for [[Cassaschismic|cassaschismic (Ultimate)]]. Up/down can be used for pc fractions.
 
{{Databox|The true mappings of the up/down are contrived.|41-comma (transsuperunison) for 217edo<br>53-comma - half poma (transsemisubunison) for 270edo<br>135-comma for 311edo<br>also -41 hemififths (sesquisubbarbaric) for 270edo and 311edo}}
* 217 has a well rounded 31-limit with great accuracy.
* 270 has an astonishingly accurate yazalathana. Usable in higher limits.
* 311 has a well rounded 41-limit with great accuracy.
 
{| class="wikitable"
!Edo
!m2:pc
!5
!7
!11
!13
!17
!19
!23
!29
|-
|-
|(c)217edo
|[[217edo|217]]
|4/5submajor third
|16:5
| rowspan="3" |upsubmajor third
| rowspan="3" |subminor seventh
| rowspan="3" |subminor seventh
| rowspan="3" |hyperfourth
| rowspan="3" |hyperfourth
|9/5perminor sixth
| rowspan="3" |downperminor sixth
|3/5perminor second
|dudperminor second
|1/5perminor third
| rowspan="3" |upperminor third
|3/5pertritone
|duppertritone
|6/5perminor seventh
|upperminor seventh
|-
|-
|(c)270edo
|'''[[270edo|270]]'''
|5/6submajor third
|20:6
|11/6perminor sixth
|downperminor second
|2/3perminor second
| rowspan="2" |truppertritone
|1/6perminor third
| rowspan="2" |dupperminor seventh
|1/2pertritone
|4/3perminor seventh
|-
|-
|(c)311edo
|'''[[311edo|311]]'''
|6/7submajor third
|23:7
|13/7perminor sixth
|trudperminor second
|
|1/7perminor third
|4/7pertritone
|9/7perminor seventh
|}
|}
WIP

Latest revision as of 10:49, 27 May 2026

Different ways edos I deem important map intervals, made mostly for myself for notekeeping but may be useful to you. Nomenclature is a mix of my Holopyth and Hemipyth and Kite's ups and downs notation, but resumed: sub/super/hypo/hyper add -1/+1/-2/+2 mapped pythagorean commas, up/down add edosteps.

Before going into the tables: edos listed here are

Meantonoids*

Edos that temper the syntonic comma in the golden series. Up/down can be used for diesis halves.

  • 19edo is coarse, decent 5-limit.
  • 31edo has a great 11-limit, usable 13-limit, still a bit coarse.
  • 50 has a worse 7-limit, but better overall 19-limit.
  • 62edo greatly improves upon 31edo expanding it to the 23-limit. Finest reasonably usable meantone edo.
Edo m2:d2 5 7 11 13 17 19 23 29
19 1:1 major third subminor seventh tritone minor sixth minor second minor third supertritone minor seventh
31 2:1 superfourth superminor sixth superminor seventh
50 3:2 upminor sixth downminor second downminor third upminor seventh
62 4:2

*Treating super/sub as meantone dieses (d2) not pythagorean commas.

Comptons

Edos that temper out the poma. Not using up/down in 24edo because up/down differ too much in size from 72 and 84. The mapping of up/down is obviously fractions of 1\12.

  • 72edo has an astounding 11-limit, usable in the 19-limit.
  • 84edo has a great 2.3.5.7.13, worse 11.
Edo n:12edo 5 7 11 13 17 19 23 29
12 1 major third minor seventh tritone minor sixth minor second minor third tritone minor seventh
24 2 halfdimminor seventh halfaugfourth halfaugminor sixth halfaugtritone halfaugminor seventh
72 6 downmajor third dudminor seventh trupfourth trupminor sixth uptritone upminor seventh
84 7 duptritone dupminor seventh

Superpythoids

Edos with sharp fifths. Up/down can be used for limma (halves).

  • 22edo has a usable 11-limit, though quite exaggerated.
  • 27edo has a usable no-11 13-limit.
  • 34edo has a great 2.3.5.13.17.
Edo A1:m2 5 7 11 13 17 19 23 29
22 3:1 downmajor third minor seventh upfourth upminor sixth upminor second minor third tritone upminor seventh
27 4:1 dupminor sixth upminor third downtritone
34 4:2 upminor sixth trupminor second minor third tritone

Panschismoids

Edos that have very accurate fifths and temper out very small or unnoticeable commas.

  • 41edo has a great 11-limit, usable no-17,23 29-limit
  • 53edo has an extremely accurate 2.3.5.13.19, decent 13-limit.
  • 94edo has a well-rounded 23-limit with good accuracy.

Cassandroids

Have fifths close to just, supporting garibaldi. Up/down can be used for pc halves (or mercommas) in the case of 94edo.

Edo m2:pc 5 7 11 13 17 19 23 29
41 3:1 submajor third subminor seventh hyperfourth hyperminor sixth superminor second minor third tritone superminor seventh
53 4:1 supertritone
94 7:2 upperminor second uppertritone upperminor seventh

Helmholtzoids

Have fifths a smidge flatter than just, along the optimal range for schismic and pontiac. Up/down can be used for pc fractions. The true mappings for 171 and 224's edostep are -53 fifths (negative merccomma). 130 and 159 instead have a poma fraction as the only possible reasonable mapping.

  • 130 has a well rounded 13-limit with very good accuracy, usable all the way to the no-29 31-limit.
  • 159 has an unfathomably accurate 2.3.11, extremely accurate 2.3.5.11.17, usable in the no-17 29-limit.
  • 171 has an unfathomably accurate 7-limit. Usable in the no-11 19-limit.
  • 224 has an extremely accurate 13-limit. Bad for higher limits.
Edo m2:pc 5 7 11 13 17 19 23 29
130 10:2 submajor third downsubminor seventh 3perfourth upperminor sixth downperminor second minor third supertritone hyperminor seventh
159 12:3 uphyperfourth downpertritone upperminor seventh
171 13:3 supertritone dupperminor seventh
224 17:4 duphyperfourth upperminor third downpertritone

Non-cassandroid Ultimates

Have fifths a smidge sharper than just, along the optimal range for cassaschismic (Ultimate). Up/down can be used for pc fractions.

The true mappings of the up/down are contrived.
41-comma (transsuperunison) for 217edo
53-comma - half poma (transsemisubunison) for 270edo
135-comma for 311edo
also -41 hemififths (sesquisubbarbaric) for 270edo and 311edo
  • 217 has a well rounded 31-limit with great accuracy.
  • 270 has an astonishingly accurate yazalathana. Usable in higher limits.
  • 311 has a well rounded 41-limit with great accuracy.
Edo m2:pc 5 7 11 13 17 19 23 29
217 16:5 upsubmajor third subminor seventh hyperfourth downperminor sixth dudperminor second upperminor third duppertritone upperminor seventh
270 20:6 downperminor second truppertritone dupperminor seventh
311 23:7 trudperminor second
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