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. 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. | 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 | ||
Edos that temper the syntonic comma in the golden series. Up/down can be used for diesis halves. | |||
== 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. | * 19edo is coarse, decent 5-limit. | ||
| Line 48: | Line 50: | ||
|4:2 | |4:2 | ||
|} | |} | ||
Treating super/sub as meantone dieses (d2) not pythagorean commas. | <nowiki>*</nowiki>Treating super/sub as meantone dieses (d2) not pythagorean commas. | ||
== | == Comptons == | ||
Edos that temper the | 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. | * 72edo has an astounding 11-limit, usable in the 19-limit. | ||
| Line 102: | Line 104: | ||
|} | |} | ||
== | == Superpythoids == | ||
Edos with sharp fifths. Up/down can be used for limma (halves). | Edos with sharp fifths. Up/down can be used for limma (halves). | ||
| Line 145: | Line 147: | ||
|tritone | |tritone | ||
|} | |} | ||
== | == Panschismoids == | ||
Edos that have very accurate fifths and temper | 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 | * 41edo has a great 11-limit, usable no-17,23 29-limit | ||
| Line 153: | Line 155: | ||
=== Cassandroids === | === Cassandroids === | ||
Have fifths close to just, | 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" | ||
!Edo | !Edo | ||
| Line 189: | Line 191: | ||
=== Helmholtzoids === | === Helmholtzoids === | ||
Have fifths a smidge flatter than just, along the optimal range for schismic. Up/down can be used for pc fractions. | 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. | * 130 has a well rounded 13-limit with very good accuracy, usable all the way to the no-29 31-limit. | ||
| Line 233: | Line 235: | ||
|17:4 | |17:4 | ||
|duphyperfourth | |duphyperfourth | ||
| | |upperminor third | ||
|downpertritone | |downpertritone | ||
|} | |} | ||
=== | === Non-cassandroid Ultimates === | ||
Have fifths a smidge sharper than just, along the optimal range for cassaschismic. Up/down can be used for | 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. | * 217 has a well rounded 31-limit with great accuracy. | ||
* 270 has an astonishingly accurate yazalathana. Usable in higher limits. | * 270 has an astonishingly accurate yazalathana. Usable in higher limits. | ||
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
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 |