378edo: Difference between revisions
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[[ | The equal temperament [[tempering out|tempers out]] 32805/32768 ([[schisma]]) in the 5-limit and [[3136/3125]] in the 7-limit, so that it [[support]]s [[bischismic]], and in fact provides the [[optimal patent val]]. It tempers out [[441/440]] and [[8019/8000]] in the 11-limit and [[729/728]] and [[1001/1000]] in the 13-limit so that it supports 11- and 13-limit bischismatic, and it also gives the optimal patent val for 13-limit bischismic. | ||
=== Prime harmonics === | |||
{{Harmonics in equal|378}} | |||
=== Subsets and supersets === | |||
Since 378 factors into {{factorization|378}}, 378edo has subset edos {{EDOs| 2, 3, 6, 7, 9, 14, 18, 21, 27, 42, 54, 63, 126, and 189 }}. | |||
[[Category:Bischismic]] | |||
Latest revision as of 14:48, 20 February 2025
| ← 377edo | 378edo | 379edo → |
378 equal divisions of the octave (abbreviated 378edo or 378ed2), also called 378-tone equal temperament (378tet) or 378 equal temperament (378et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 378 equal parts of about 3.17 ¢ each. Each step represents a frequency ratio of 21/378, or the 378th root of 2.
The equal temperament tempers out 32805/32768 (schisma) in the 5-limit and 3136/3125 in the 7-limit, so that it supports bischismic, and in fact provides the optimal patent val. It tempers out 441/440 and 8019/8000 in the 11-limit and 729/728 and 1001/1000 in the 13-limit so that it supports 11- and 13-limit bischismatic, and it also gives the optimal patent val for 13-limit bischismic.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.37 | +0.99 | -0.57 | +1.06 | +0.74 | -0.19 | +0.90 | +0.30 | -1.01 | +1.00 |
| Relative (%) | +0.0 | -11.6 | +31.1 | -18.0 | +33.5 | +23.4 | -6.1 | +28.3 | +9.4 | -31.7 | +31.4 | |
| Steps (reduced) |
378 (0) |
599 (221) |
878 (122) |
1061 (305) |
1308 (174) |
1399 (265) |
1545 (33) |
1606 (94) |
1710 (198) |
1836 (324) |
1873 (361) | |
Subsets and supersets
Since 378 factors into 2 × 33 × 7, 378edo has subset edos 2, 3, 6, 7, 9, 14, 18, 21, 27, 42, 54, 63, 126, and 189.