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. See [[User:Eufalesio/Holopyth and Hemipyth|Holopyth and Hemipyth]] for mapping nomenclature.
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.  


== Meantonoid edos ==
== Meantonoid edos ==
Edos with flat fifths that temper the syntonic comma in the golden series.
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.
 
{| class="wikitable"
{| class="wikitable"
|+
!Edo
!Edo
!m2:d2
!m2:d2
Line 16: Line 21:
!29
!29
|-
|-
|19
|[[19edo|19]]
|1:1
|1:1
| rowspan="3" |major third
| rowspan="4" |major third
| rowspan="3" |subminor seventh
| rowspan="4" |subminor seventh
|tritone
|tritone
|minor sixth
|minor sixth
| rowspan="2" |minor second
| rowspan="2" |minor second
| rowspan="2" |minor third
| rowspan="2" |minor third
| rowspan="3" |supertritone
| rowspan="4" |supertritone
|minor seventh
|minor seventh
|-
|-
|'''31'''
|'''[[31edo|31]]'''
|2:1
|2:1
| rowspan="2" |superfourth
| rowspan="3" |superfourth
|superminor sixth
|superminor sixth
|superminor seventh
|superminor seventh
|-
|-
|50
|[[50edo|50]]
|3:2
|3:2
|1/2perminor sixth
| rowspan="2" |upminor sixth
|1/2subminor second
| rowspan="2" |downminor second
|1/2subminor third
| rowspan="2" |downminor third
|1/2perminor seventh
| rowspan="2" |upminor seventh
|-
|[[62edo|62]]
|4:2
|}
|}
Treating super/sub as meantone dieses (d2) not pythagorean commas.
Treating super/sub as meantone dieses (d2) not pythagorean commas.


== Compton edos ==
== Compton edos ==
Edos that temper the pythagorean comma. Not using up/down in 24edo because up/down differ too much in size from 72 and 84.
* 72edo has an astounding 11-limit, usable in the 19-limit.
* 84edo has a great 2.3.5.7.13, worse 11.
{| class="wikitable"
{| class="wikitable"
!Edo
!'''Edo'''
!n:12edo
!'''n:12edo'''
!5
!'''5'''
!7
!'''7'''
!11
!'''11'''
!13
!'''13'''
!17
!'''17'''
!19
!'''19'''
!23
!'''23'''
!29
!'''29'''
|-
|-
|12
|[[12edo|12]]
|1
|1
| rowspan="2" |major third
| rowspan="2" |major third
Line 66: Line 79:
|minor seventh
|minor seventh
|-
|-
|24
|[[24edo|24]]
|2
|2
|1/2dimminor seventh
|halfdimminor seventh
| rowspan="2" |1/2augfourth
|halfaugfourth
| rowspan="2" |1/2augminor sixth
|halfaugminor sixth
|1/2augtritone
|halfaugtritone
|1/2augminor seventh
|halfaugminor seventh
|-
|-
|'''72'''
|[[72edo|'''72''']]
|6
|6
|1/6dimmajor third
| rowspan="2" |downmajor third
|1/3dimminor seventh
| rowspan="2" |dudminor seventh
|1/3augtritone
| rowspan="2" |trupfourth
|1/3augminor seventh
| rowspan="2" |trupminor sixth
|uptritone
|upminor seventh
|-
|-
|84
|[[84edo|84]]
|7
|7
|1/7dimmajor third
|duptritone
|2/7dimminor seventh
|dupminor seventh
|3/7augfourth
|3/7augminor sixth
|2/7augtritone
|2/7augminor seventh
|}
|}
Using aug/dim fractions because the chain of fifth is heavily enfactored.


== Superpythoid edos ==
== Superpythoid edos ==
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"
{| class="wikitable"
!Edo
!Edo
Line 105: Line 121:
!29
!29
|-
|-
|22
|[[22edo|22]]
|3:1
|3:1
| rowspan="2" |limmajor third
| rowspan="3" |downmajor third
| rowspan="3" |lumminor seventh
| rowspan="3" |minor seventh
| rowspan="3" |limfourth
| rowspan="3" |upfourth
|limminor sixth
|upminor sixth
| rowspan="2" |limminor second
| rowspan="2" |upminor second
|minor third
|minor third
|tritone
|tritone
| rowspan="2" |limminor seventh
| rowspan="3" |upminor seventh
|-
|-
|27
|[[27edo|27]]
|4:1
|4:1
|2limminor sixth
|dupminor sixth
|2limminor third
|upminor third
|lumtritone
|downtritone
|-
|-
|34
|[[34edo|34]]
|2:1
|4:2
|1/2limmajor third
|upminor sixth
|limminor sixth
|trupminor second
|3/2limminor second
|minor third
|minor third
|tritone
|tritone
|1/2limminor seventh
|}
|}
Using m2 fractions because the mapped pythagorean comma is far too big. 34edo can use either m2 or pc since both are equal there.
== Panschismoid edos ==
== Panschismoid edos ==
Edos that have very accurate fifths and temper schisma-sized commas.
Edos that have very accurate fifths and temper schisma-sized 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 ===
=== Cassandroids ===
Have fifths close to just, and are marvel systems. Up/down can be used for pc halves.
{| class="wikitable"
{| class="wikitable"
!Edo
!Edo
Line 149: Line 166:
!29
!29
|-
|-
|41
|[[41edo|41]]
|3:1
|3:1
| rowspan="3" |submajor third
| rowspan="3" |submajor third
Line 160: Line 177:
| rowspan="2" |superminor seventh
| rowspan="2" |superminor seventh
|-
|-
|53
|[[53edo|53]]
|4:1
|4:1
|supertritone
|supertritone
|-
|-
|'''94'''
|'''[[94edo|94]]'''
|7:2
|7:2
|1/2perminor second
|upperminor second
|1/2pertritone
|uppertritone
|1/2perminor seventh
|upperminor seventh
|}
|}


=== Euschismoids ===
=== Helmholtzoids ===
Have fifths a smidge flatter than just, along the optimal range for schismic. Up/down can be used for pc fractions.
 
* 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"
{| class="wikitable"
!Edo
!Edo
Line 184: Line 208:
!29
!29
|-
|-
|130
|[[130edo|130]]
|10:2
|10:2
| rowspan="4" |submajor third
| rowspan="4" |submajor third
|3/2subminor seventh
| rowspan="4" |downsubminor seventh
|3perfourth
|3perfourth
|5/2perminor sixth
| rowspan="4" |upperminor sixth
|1/2perminor second
| rowspan="4" |downperminor second
| rowspan="3" |minor third
| rowspan="3" |minor third
|supertritone
|supertritone
|hyperminor seventh
|hyperminor seventh
|-
|-
|159
|[[159edo|159]]
|12:3
|12:3
| rowspan="2" |4/3subminor seventh
| rowspan="2" |uphyperfourth
| rowspan="2" |7/3perfourth
|downpertritone
|hyperminor sixth
|upperminor seventh
| rowspan="2" |2/3perminor second
|2/3pertritone
|4/3perminor seventh
|-
|-
|171
|[[171edo|171]]
|13:3
|13:3
|7/3perminor sixth
|supertritone
|supertritone
|5/3perminor seventh
| rowspan="2" |dupperminor seventh
|-
|-
|224
|[[224edo|224]]
|17:4
|17:4
|5/4subminor seventh
|duphyperfourth
|5/2perfourth
|9/4perminor sixth
|3/4perminor second
|1/4perminor third
|1/4perminor third
|3/4pertritone
|downpertritone
|3/2perminor seventh
|}
|}


=== Garischismoids ===
=== Garytonoids ===
Have fifths a smidge sharper than just, along the optimal range for cassaschismic. Up/down can be used for schisma-sized steps.
 
* 217 has a well rounded 31-limit with great accuracy.
* 270 has an astonishingly accurate 2.3.5.7.11.13.19. Usable in higher limits.
* 311 has a well rounded 41-limit with great accuracy.
 
{| class="wikitable"
{| class="wikitable"
!Edo
!Edo
!m2:pc
!5
!5
!7
!7
Line 233: Line 256:
!29
!29
|-
|-
|217
|[[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
|-
|-
|'''270'''
|'''[[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
|-
|-
|'''311'''
|'''[[311edo|311]]'''
|6/7submajor third
|23:7
|13/7perminor sixth
|trudperminor second
|4/7perminor second
|1/7perminor third
|4/7pertritone
|9/7perminor seventh
|}
|}

Revision as of 17:49, 1 March 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.

Meantonoid edos

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.

Compton edos

Edos that temper the pythagorean comma. Not using up/down in 24edo because up/down differ too much in size from 72 and 84.

  • 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

Superpythoid edos

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

Panschismoid edos

Edos that have very accurate fifths and temper schisma-sized 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, and are marvel systems. Up/down can be used for pc halves.

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. Up/down can be used for pc fractions.

  • 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 1/4perminor third downpertritone

Garytonoids

Have fifths a smidge sharper than just, along the optimal range for cassaschismic. Up/down can be used for schisma-sized steps.

  • 217 has a well rounded 31-limit with great accuracy.
  • 270 has an astonishingly accurate 2.3.5.7.11.13.19. 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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