179edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|179}} | {{EDO intro|179}} | ||
== Theory == | |||
179edo doesn't approximate well any odd harmonic up to 23, best being [[21/16]] with 22% error. Nonetheless, it is consistent in the 7-limit and there are anumber of temperaments to be considered. | 179edo doesn't approximate well any odd harmonic up to 23, best being [[21/16]] with 22% error. Nonetheless, it is consistent in the 7-limit and there are anumber of temperaments to be considered. | ||
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=== Odd harmonics === | === Odd harmonics === | ||
{{Harmonics in equal|179}} | {{Harmonics in equal|179}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
179edo is the 41st [[prime edo]]. | |||
==Regular temperament properties== | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" |[[Subgroup]] | |||
! rowspan="2" |[[Comma list|Comma List]] | |||
! rowspan="2" |[[Mapping]] | |||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | |||
! colspan="2" |Tuning Error | |||
|- | |||
![[TE error|Absolute]] (¢) | |||
![[TE simple badness|Relative]] (%) | |||
|- | |||
|2.3 | |||
|{{monzo|284 -179}} | |||
|{{val|179 284}} | |||
| -0.6169 | |||
| 0.6166 | |||
| 9.20 | |||
|- | |||
|2.3.5 | |||
|{{monzo|20 -17 3}}, {{monzo|28 -3 -10}} | |||
|{{val|179 284 416}} | |||
| -0.7718 | |||
| 0.5490 | |||
| 8.19 | |||
|- | |||
|2.3.5.7 | |||
|3136/3125, 4375/4374, 10976/10935 | |||
|{{val|179 284 416 503}} | |||
| -0.8673 | |||
| 0.5034 | |||
| 7.51 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per 8ve | |||
! Generator<br>(reduced) | |||
! Cents<br>(reduced) | |||
! Associated<br>ratio | |||
! Temperaments | |||
|- | |||
|1 | |||
|35\179 | |||
|234.64 | |||
|8/7 | |||
|[[Rodan]] | |||
|- | |||
|1 | |||
|47\179 | |||
|315.08 | |||
|6/5 | |||
|[[Parakleismic]] | |||
|- | |||
|1 | |||
|79\167 | |||
|529.61 | |||
|512/375 | |||
|[[Mabila]] | |||
|} | |||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
[[Category:Prime EDO]] | [[Category:Prime EDO]] | ||
Revision as of 19:50, 3 November 2023
| ← 178edo | 179edo | 180edo → |
Theory
179edo doesn't approximate well any odd harmonic up to 23, best being 21/16 with 22% error. Nonetheless, it is consistent in the 7-limit and there are anumber of temperaments to be considered.
179edo tempers out the parakleisma, 1224440064/1220703125 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 (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 35\179 | 234.64 | 8/7 | Rodan |
| 1 | 47\179 | 315.08 | 6/5 | Parakleismic |
| 1 | 79\167 | 529.61 | 512/375 | Mabila |