388edo: Difference between revisions

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It provides the [[optimal patent val]] for the rank-5 cuthbert temperament, which tempers out [[847/845]], the cuthbert comma, and for a number of other temperaments tempering it out, e.g. [[neusec]], the {{nowrap| 190 & 198 }} temperament. By tempering out cuthbert it [[support]]s [[cuthbert chords]], in addition to [[sinbadmic chords]].

~~{{Todo|inline=1| expand | comment=Explain its higher limit properties.}}~~

=== Prime harmonics ===

=== 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.

311edo also deals better with composite harmonics than 388. 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, 41/28, 41/22, and their octave complements. 311edo misses most primes after 41, though it hits 73, 89, (101,) 109, 113, 139, 149, and 151. 388, on the other hand, hits primes 47, 61, 71, 79, 97, 109, 113, 131, 137, 139, and 149. 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]]. Note that 311 has generally higher absolute errors than 388 due to its smaller size, but smaller size also means the system is easier to handle.

Another system notable in high limits around this size is [[422edo]].

=== Subsets and supersets ===

Since 388 factors into {{nowrap| 2<sup>2</sup> × 97 }}, 388edo has subset edos {{EDOs| 2, 4, 97, and 194 }}.

== Regular temperament properties ==

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 It provides the [[optimal patent val]] for the rank-5 cuthbert temperament, which tempers out [[847/845]], the cuthbert comma, and for a number of other temperaments tempering it out, e.g. [[neusec]], the {{nowrap| 190 & 198 }} temperament. By tempering out cuthbert it [[support]]s [[cuthbert chords]], in addition to [[sinbadmic chords]].
-{{Todo|inline=1| expand | comment=Explain its higher limit properties.}}
 === Prime harmonics ===
 {{Harmonics in equal|388|columns=12}}
+{{Harmonics in equal|388|start=13|columns=12|collapsed=1}}
+{{Harmonics in equal|388|start=25|columns=12|collapsed=1}}
+=== 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.
+edo 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.
+edo also deals better with composite harmonics than 388. 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, 41/28, 41/22, and their octave complements. 311edo misses most primes after 41, though it hits 73, 89, (101,) 109, 113, 139, 149, and 151. 388, on the other hand, hits primes 47, 61, 71, 79, 97, 109, 113, 131, 137, 139, and 149. 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]]. Note that 311 has generally higher absolute errors than 388 due to its smaller size, but smaller size also means the system is easier to handle.
+Another system notable in high limits around this size is [[422edo]].
 === Subsets and supersets ===
 Since 388 factors into {{nowrap| 2<sup>2</sup> × 97 }}, 388edo has subset edos {{EDOs| 2, 4, 97, and 194 }}.
 == Regular temperament properties ==