144edo: Difference between revisions
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=== Subsets and supersets === | === Subsets and supersets === | ||
144edo is the square of world-dominant [[12edo]]. One step of it is called ''[[farab]]''. | |||
Since 144 factors into 2<sup>4</sup> × 3<sup>2</sup>, 144edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, and 72 }}. | Since 144 factors into 2<sup>4</sup> × 3<sup>2</sup>, 144edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, and 72 }}. | ||
=== Approximation to φ === | === Approximation to φ === | ||
144edo is the 12th Fibonacci edo | 144edo is the 12th Fibonacci edo. As a consequence of being a Fibonacci EDO, it can produce extremely precise approximation of the [[Logarithmic phi|logarithmic golden ratio]] at 89 steps. In addition, it ''also'' excellently represents the [[Acoustic phi|acoustic golden ratio]] by 100 steps. | ||
== Music == | == Music == | ||
* [https://www.youtube.com/watch?v=p71oF-A9gF8 ''Like an endless uphill''] by [[5 hideya]] | * [https://www.youtube.com/watch?v=p71oF-A9gF8 ''Like an endless uphill''] by [[5 hideya]] | ||
Revision as of 19:04, 2 September 2023
| ← 143edo | 144edo | 145edo → |
Theory
144edo is closely related to 72edo, but the patent vals differ on the mapping for 13 and 17. It is contorted in the 11-limit, tempering out 225/224, 243/242, 385/384, 441/440, and 4000/3993. Using the patent val, it tempers out 847/845, 1188/1183, 1701/1690, 1875/1859, and 4225/4224 in the 13-limit; 273/272, 715/714, 833/832, 875/867, 891/884, and 1275/1274 in the 17-limit; 210/209, 325/323, 343/342, 363/361, 400/399, 513/512, and 665/663 in the 19-limit.
The 144gh val is a tuning for semihemisecordite temperament, while 144eff val is a tuning for hemimiracle. 144ee is a tuning for oracle. 144cf val supports necromanteion.
Prime harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.96 | -2.98 | -2.16 | -3.91 | -1.32 | +1.14 | +3.40 | +3.38 | +2.49 | -4.11 | -3.27 |
| Relative (%) | -23.5 | -35.8 | -25.9 | -46.9 | -15.8 | +13.7 | +40.8 | +40.5 | +29.8 | -49.4 | -39.3 | |
| Steps (reduced) |
228 (84) |
334 (46) |
404 (116) |
456 (24) |
498 (66) |
533 (101) |
563 (131) |
589 (13) |
612 (36) |
632 (56) |
651 (75) | |
Subsets and supersets
144edo is the square of world-dominant 12edo. One step of it is called farab.
Since 144 factors into 24 × 32, 144edo has subset edos 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, and 72.
Approximation to φ
144edo is the 12th Fibonacci edo. As a consequence of being a Fibonacci EDO, it can produce extremely precise approximation of the logarithmic golden ratio at 89 steps. In addition, it also excellently represents the acoustic golden ratio by 100 steps.