330edo: Difference between revisions
Cleanup; review RTT tables; clarify the title row of the rank-2 temp table |
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {{comma basis begin}} | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| Line 46: | Line 38: | ||
| 0.2212 | | 0.2212 | ||
| 6.08 | | 6.08 | ||
{{comma basis end}} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {{rank-2 begin}} | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 64: | Line 50: | ||
|- | |- | ||
| 3 | | 3 | ||
| 137\330<br>(27\330) | | 137\330<br />(27\330) | ||
| 498.182<br>(98.182) | | 498.182<br />(98.182) | ||
| 4/3<br>(200/189) | | 4/3<br />(200/189) | ||
| [[Term (temperament)|Term]] | | [[Term (temperament)|Term]] | ||
|- | |- | ||
| 6 | | 6 | ||
| 137\330<br>(27\330) | | 137\330<br />(27\330) | ||
| 498.182<br>(98.182) | | 498.182<br />(98.182) | ||
| 4/3<br>(200/189) | | 4/3<br />(200/189) | ||
| [[Semiterm]] (330eff) | | [[Semiterm]] (330eff) | ||
|- | |- | ||
| 22 | | 22 | ||
| 137\330<br>(2\330) | | 137\330<br />(2\330) | ||
| 498.182<br>(7.273) | | 498.182<br />(7.273) | ||
| 4/3<br>({{monzo| 16 -13 2 }}) | | 4/3<br />({{monzo| 16 -13 2 }}) | ||
| [[Major arcana]] (5-limit) | | [[Major arcana]] (5-limit) | ||
{{rank-2 end}} | |||
{{orf}} | |||
[[Category:Canou]] | [[Category:Canou]] | ||
[[Category:Semicanousmic]] | [[Category:Semicanousmic]] | ||
[[Category:Keenanismic]] | [[Category:Keenanismic]] | ||
Revision as of 04:13, 16 November 2024
| ← 329edo | 330edo | 331edo → |
Theory
330edo has a flat tendency, with its harmonics 3, 5, and 7 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.
The equal temperament 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, 330edo has subset edos 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, and 165.
Regular temperament properties
Template:Comma basis begin |- | 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, 823543/819200 | [⟨330 523 766 926]] | +0.2524 | 0.2212 | 6.08 Template:Comma basis end
Rank-2 temperaments
Template:Rank-2 begin
|-
| 1
| 137\330
| 498.182
| 4/3
| Helmholtz
|-
| 3
| 137\330
(27\330)
| 498.182
(98.182)
| 4/3
(200/189)
| Term
|-
| 6
| 137\330
(27\330)
| 498.182
(98.182)
| 4/3
(200/189)
| Semiterm (330eff)
|-
| 22
| 137\330
(2\330)
| 498.182
(7.273)
| 4/3
([16 -13 2⟩)
| Major arcana (5-limit)
Template:Rank-2 end
Template:Orf