388edo: Difference between revisions
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
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{{Harmonics in equal|388|columns=12}} | {{Harmonics in equal|388|columns=12}} | ||
{{Harmonics in equal|388|start=13|columns=12|collapsed=1}} | {{Harmonics in equal|388|start=13|columns=12|collapsed=1}} | ||
=== Approximation to JI | === Subsets and supersets === | ||
This | Since 388 factors into primes as {{nowrap| 2<sup>2</sup> × 97 }}, 388edo has subset edos {{EDOs| 2, 4, 97, and 194 }}. | ||
== Approximation to JI == | |||
This edo has a high consistency limit, although due to [[311edo]] having a higher consistency limit, among other things, it is mostly unexplored. However, it still makes for an interesting comparison. | |||
388edo has a consistency limit of 37 compared to 311edo's consistency limit of 41, so most people wishing to approximate higher-limit JI choose the latter. 311edo also approximates harmonics up to 42 with under 25% error, while 388edo fails as early as harmonic [[7/1|7]]. One thing to note is that 388edo's distinct consistency limit is 27 compared to 311edo's 23, so those who want distinct consistency in the 27-odd-limit may choose 388. | 388edo has a consistency limit of 37 compared to 311edo's consistency limit of 41, so most people wishing to approximate higher-limit JI choose the latter. 311edo also approximates harmonics up to 42 with under 25% error, while 388edo fails as early as harmonic [[7/1|7]]. One thing to note is that 388edo's distinct consistency limit is 27 compared to 311edo's 23, so those who want distinct consistency in the 27-odd-limit may choose 388. | ||
311edo also deals better with composite harmonics than | 311edo also deals better with composite harmonics than 388edo. 311edo is consistent to the 41-limit 77-odd-limit, while 388edo has inconsistencies involving composite harmonics as low as 39, and harmonic 49 itself is inconsistent. The 7th and 11th harmonics both being flat by just over 25% of a step is less than ideal. However, it approximates some higher primes better than 311 does. The only inconsistencies in the 41-odd-limit in 388edo are 39/28, 39/22 ,39/37, 41/28, 41/22, 41/37 and their octave complements. This is due to the fact that harmonics 39 and 41 are quite sharp, both just over 1/4 of a step. 311edo misses most primes after 41, though it hits 73, 89, (101,) 109, and 113. 388, on the other hand, hits primes 47, 61, 71, 79, 97, 109, and 113. Still, 311 does much better at composite harmonics due to having much lower error in the 13-limit, which is also important to note by itself, though if one wants to approximate the 13-limit specifically they may prefer [[270edo]] or [[494edo]]. Note that 311edo has generally higher absolute errors than 388edo due to its smaller size, but having a smaller size also means the system is easier to handle. | ||
Another system notable in high limits around this size is [[422edo]]. | Another system notable in high limits around this size is [[422edo]], using the sharp-tending 422[[wart|l]] [[val]] for prime 37. | ||
=== 37-odd-limit intervals === | |||
{{Q-odd-limit intervals|388|limit=37}} | {{Q-odd-limit intervals|388|limit=37}} | ||
== Regular temperament properties == | == Regular temperament properties == | ||
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| [[Berkelium]] | | [[Berkelium]] | ||
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<nowiki/>* [[Normal | <nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | ||
[[Category:Cuthbert]] | [[Category:Cuthbert]] | ||