330edo: Difference between revisions
Cleanup and expansion |
Added regular temperament properties |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|330}} | {{EDO intro|330}} | ||
==Theory== | |||
330edo has a flat tendency, with its 3rd, 5th, and 7th harmonics tuned progressively flatter. In the 11-limit, the 330e [[val]] {{val| 330 523 766 926 '''1141''' }} scores significantly better in [[TE error]] than its [[patent val]] {{val| 330 523 766 926 '''1142''' }} and allows an extension to the 13-limit. | 330edo has a flat tendency, with its 3rd, 5th, and 7th harmonics tuned progressively flatter. In the 11-limit, the 330e [[val]] {{val| 330 523 766 926 '''1141''' }} scores significantly better in [[TE error]] than its [[patent val]] {{val| 330 523 766 926 '''1142''' }} and allows an extension to the 13-limit. | ||
| Line 13: | Line 15: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 330 factors into 2 × 3 × 5 × 11, it has subset edos {{EDOs| 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, and 165 }}. | Since 330 factors into 2 × 3 × 5 × 11, it has subset edos {{EDOs| 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, and 165 }}. | ||
==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 | |||
|- | |||
|2.3 | |||
|{{monzo|-523 330}} | |||
|{{val|330 523}} | |||
| 0.0432 | |||
| 0.0432 | |||
| 1.19 | |||
|- | |||
|2.3.5 | |||
|32805/32768, {{monzo|-2 -50 35}} | |||
|{{val|330 523 766}} | |||
| 0.1521 | |||
| 0.1581 | |||
| 4.35 | |||
|- | |||
|2.3.5.7 | |||
|32805/32768, 250047/250000, 703125/702464 | |||
|{{val|330 523 766 926}} | |||
| 0.2524 | |||
| 0.2212 | |||
| 6.08 | |||
|} | |||
=== 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 | |||
|137\330 | |||
|498.182 | |||
|4/3 | |||
|[[Helmholtz]] | |||
|- | |||
|3 | |||
|137\330<br>(27\330) | |||
|498.182<br>(98.182) | |||
|4/3<br>(200/189) | |||
|[[Term]] | |||
|} | |||
[[Category:Canou]] | [[Category:Canou]] | ||
[[Category:Semicanousmic]] | [[Category:Semicanousmic]] | ||
[[Category:Keenanismic]] | [[Category:Keenanismic]] | ||
Revision as of 04:47, 12 October 2023
| ← 329edo | 330edo | 331edo → |
Theory
330edo has a flat tendency, with its 3rd, 5th, and 7th harmonics tuned progressively flatter. In the 11-limit, the 330e val ⟨330 523 766 926 1141] scores significantly better in TE error than its patent val ⟨330 523 766 926 1142] and allows an extension to the 13-limit.
It tempers out 32805/32768 (schisma) in the 5-limit; 250047/250000 (landscape comma), 703125/702464 (meter) and 4802000/4782969 (canousma) in the 7-limit. Using the 330e val, it tempers out 385/384 (keenanisma), 9801/9800 (kalisma), and 14641/14580 (semicanousma) in the 11-limit; 847/845 (cuthbert) and 1001/1000 (sinbadma) in the 13-limit.
It provides a nice tuning for keenanismic, the rank-4 temperament that tempers out 385/384 (even better than its optimal patent val 284edo), and actually a close-to-optimal tuning for 11-limit semicanou, the rank-3 temperament that tempers out 9801/9800 and 14641/14580.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.14 | -0.86 | -1.55 | +1.41 | -0.53 | +0.50 | +0.67 | +0.82 | -0.49 | +0.42 |
| Relative (%) | +0.0 | -3.8 | -23.6 | -42.7 | +38.8 | -14.5 | +13.7 | +18.4 | +22.5 | -13.4 | +11.5 | |
| Steps (reduced) |
330 (0) |
523 (193) |
766 (106) |
926 (266) |
1142 (152) |
1221 (231) |
1349 (29) |
1402 (82) |
1493 (173) |
1603 (283) |
1635 (315) | |
Subsets and supersets
Since 330 factors into 2 × 3 × 5 × 11, it has subset edos 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, and 165.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| 2.3 | [-523 330⟩ | ⟨330 523] | 0.0432 | 0.0432 | 1.19 |
| 2.3.5 | 32805/32768, [-2 -50 35⟩ | ⟨330 523 766] | 0.1521 | 0.1581 | 4.35 |
| 2.3.5.7 | 32805/32768, 250047/250000, 703125/702464 | ⟨330 523 766 926] | 0.2524 | 0.2212 | 6.08 |
Rank-2 temperaments
| Periods per 8ve |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 137\330 | 498.182 | 4/3 | Helmholtz |
| 3 | 137\330 (27\330) |
498.182 (98.182) |
4/3 (200/189) |
Term |