179edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|179}} | {{EDO intro|179}} | ||
== Theory == | == Theory == | ||
179edo | 179edo does not approximate well any odd [[harmonic]] up to 23, best being [[21/16]] with 22% error. Nonetheless, it is [[consistent]] in the [[7-odd-limit]] and there are a number of temperaments to be considered. | ||
The equal temperament [[tempering out|tempers out]] the [[parakleisma]], {{monzo| 8 14 -13 }} in the 5-limit, and [[support]]s [[parakleismic]] and its [[extension]]s, providing the [[optimal patent val]] for 11- and 13-limit [[Ragismic microtemperaments #parkleismic|parkleismic]] temperament. In the 7-limit it tempers out [[3136/3125]], [[4375/4374]] and [[10976/10935]], in the 11-limit [[176/175]] and 1375/1372 and in the 13-limit [[169/168]], [[325/324]], [[351/350]] and [[352/351]], providing the optimal patent val for 11- and 13-limit [[ulmo]] temperament. | |||
=== Odd harmonics === | === Odd harmonics === | ||
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179edo is the 41st [[prime edo]]. | 179edo is the 41st [[prime edo]]. | ||
==Regular temperament properties== | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" |[[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" |Tuning Error | ! colspan="2" | Tuning Error | ||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3 | | 2.3 | ||
|{{monzo|284 -179}} | | {{monzo| 284 -179 }} | ||
|{{ | | {{mapping| 179 284 }} | ||
| -0.6169 | | -0.6169 | ||
| 0.6166 | | 0.6166 | ||
| 9.20 | | 9.20 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{monzo|20 -17 3}}, {{monzo|28 -3 -10}} | | {{monzo| 20 -17 3 }}, {{monzo| 28 -3 -10 }} | ||
|{{ | | {{mapping| 179 284 416 }} | ||
| -0.7718 | | -0.7718 | ||
| 0.5490 | | 0.5490 | ||
| 8.19 | | 8.19 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|3136/3125, 4375/4374, 10976/10935 | | 3136/3125, 4375/4374, 10976/10935 | ||
|{{ | | {{mapping| 179 284 416 503 }} | ||
| -0.8673 | | -0.8673 | ||
| 0.5034 | | 0.5034 | ||
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|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br> | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|35\179 | | 35\179 | ||
|234.64 | | 234.64 | ||
|8/7 | | 8/7 | ||
|[[Rodan]] | | [[Rodan]] (179d) | ||
|- | |- | ||
|1 | | 1 | ||
|47\179 | | 47\179 | ||
|315.08 | | 315.08 | ||
|6/5 | | 6/5 | ||
|[[Parakleismic]] | | [[Parakleismic]] | ||
|- | |- | ||
|1 | | 1 | ||
|79\ | | 79\179 | ||
|529.61 | | 529.61 | ||
|512/375 | | 512/375 | ||
|[[Mabila]] | | [[Mabila]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
== Music == | == Music == | ||
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* "in a hopeless situation" from ''hope in dark times'' (2024) – [https://open.spotify.com/track/787NGqkI5FHjQcfEOu4y4w Spotify] | [https://francium223.bandcamp.com/track/in-a-hopeless-situation Bandcamp] | [https://www.youtube.com/watch?v=kGdWmP81jwc YouTube] | * "in a hopeless situation" from ''hope in dark times'' (2024) – [https://open.spotify.com/track/787NGqkI5FHjQcfEOu4y4w Spotify] | [https://francium223.bandcamp.com/track/in-a-hopeless-situation Bandcamp] | [https://www.youtube.com/watch?v=kGdWmP81jwc YouTube] | ||
[[Category:Listen]] | [[Category:Listen]] | ||
Revision as of 09:33, 25 April 2024
| ← 178edo | 179edo | 180edo → |
Theory
179edo does not approximate well any odd harmonic up to 23, best being 21/16 with 22% error. Nonetheless, it is consistent in the 7-odd-limit and there are a number of temperaments to be considered.
The equal temperament tempers out the parakleisma, [8 14 -13⟩ in the 5-limit, and supports parakleismic and its extensions, providing the optimal patent val for 11- and 13-limit parkleismic temperament. In the 7-limit it tempers out 3136/3125, 4375/4374 and 10976/10935, in the 11-limit 176/175 and 1375/1372 and in the 13-limit 169/168, 325/324, 351/350 and 352/351, providing the optimal patent val for 11- and 13-limit ulmo temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.96 | +2.51 | +3.24 | -2.79 | -1.60 | -2.54 | -2.24 | +2.31 | -2.54 | -1.51 | +1.89 |
| Relative (%) | +29.2 | +37.5 | +48.3 | -41.7 | -23.8 | -37.9 | -33.3 | +34.4 | -37.9 | -22.5 | +28.2 | |
| Steps (reduced) |
284 (105) |
416 (58) |
503 (145) |
567 (30) |
619 (82) |
662 (125) |
699 (162) |
732 (16) |
760 (44) |
786 (70) |
810 (94) | |
Subsets and supersets
179edo is the 41st prime edo.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [284 -179⟩ | [⟨179 284]] | -0.6169 | 0.6166 | 9.20 |
| 2.3.5 | [20 -17 3⟩, [28 -3 -10⟩ | [⟨179 284 416]] | -0.7718 | 0.5490 | 8.19 |
| 2.3.5.7 | 3136/3125, 4375/4374, 10976/10935 | [⟨179 284 416 503]] | -0.8673 | 0.5034 | 7.51 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 35\179 | 234.64 | 8/7 | Rodan (179d) |
| 1 | 47\179 | 315.08 | 6/5 | Parakleismic |
| 1 | 79\179 | 529.61 | 512/375 | Mabila |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct