248edo: Difference between revisions
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{{Infobox ET}} | |||
{{ED intro}} | |||
[[ | == Theory == | ||
[[Category: | 248edo shares the mapping of [[harmonic]]s [[5/1|5]] and [[7/1|7]] with [[31edo]]. It has a decent 13-limit interpretation despite not being [[consistent]]. The equal temperament [[tempering out|tempers out]] [[32805/32768]] in the 5-limit; [[3136/3125]] and [[420175/419904]] in the 7-limit; [[441/440]], [[8019/8000]] in the 11-limit; [[729/728]], [[847/845]], [[1001/1000]], [[1575/1573]] and [[2200/2197]] in the 13-limit. It also notably tempers out the [[quartisma]]. 248edo, additionally, has the interesting property of its mapping for all prime harmonics 3 to 23 being a multiple of 3, and therefore derived from [[131edt]]. Similarly, using the lower-error 248[[Wart notation|h]] val, the mappings of all its [[2.5.7_subgroup|no-3]] harmonics up to [[23-limit|28]] are multiples of 2 and derived from [[124edo]]. | ||
It [[support]]s the [[bischismic]] temperament, providing the [[optimal patent val]] for 11-limit bischismic, and excellent tunings in the 7- and 13-limits. It also provides the optimal patent val for the [[essence]] temperament. It is notable for its combination of precise intonation with an abundance of essentially tempered chords. | |||
=== Prime harmonics === | |||
{{Harmonics in equal|248|}} | |||
=== Subsets and supersets === | |||
Since 248 factors into {{factorization|248}}, 248edo has subset edos {{EDOs| 2, 4, 8, 31, 62, and 124 }}. | |||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| 287 -181 }} | |||
| {{mapping| 248 393 }} | |||
| +0.108 | |||
| 0.108 | |||
| 2.23 | |||
|- | |||
| 2.3.5 | |||
| 32805/32768, {{monzo| 12 32 -27 }} | |||
| {{mapping| 248 393 576 }} | |||
| -0.041 | |||
| 0.228 | |||
| 4.70 | |||
|- | |||
| 2.3.5.7 | |||
| 3136/3125, 32805/32768, 420175/419904 | |||
| {{mapping| 248 393 576 696 }} | |||
| +0.066 | |||
| 0.270 | |||
| 5.58 | |||
|- | |||
| 2.3.5.7.11 | |||
| 441/440, 3136/3125, 8019/8000, 41503/41472 | |||
| {{mapping| 248 393 576 696 858 }} | |||
| +0.036 | |||
| 0.249 | |||
| 5.15 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 441/440, 729/728, 847/845, 1001/1000, 3136/3125 | |||
| {{mapping| 248 393 576 696 858 918 }} | |||
| +0.079 | |||
| 0.275 | |||
| 5.69 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 5\248 | |||
| 24.19 | |||
| 686/675 | |||
| [[Sengagen]] | |||
|- | |||
| 1 | |||
| 103\248 | |||
| 498.39 | |||
| 4/3 | |||
| [[Helmholtz (temperament)|Helmholtz]] | |||
|- | |||
| 2 | |||
| 77\248<br />(47\248) | |||
| 372.58<br />(227.42) | |||
| 26/21<br />(154/135) | |||
| [[Essence]] | |||
|- | |||
| 2 | |||
| 103\248 | |||
| 498.39 | |||
| 4/3 | |||
| [[Bischismic]] | |||
|- | |||
| 8 | |||
| 117\248<br />(7\248) | |||
| 566.13<br />(33.87) | |||
| 104/75<br />(49/48) | |||
| [[Octowerck]] | |||
|- | |||
| 31 | |||
| 103\248<br />(1\248) | |||
| 498.39<br />(4.84) | |||
| 4/3<br />(385/384) | |||
| [[Birds]] | |||
|} | |||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
[[Category:Bischismic]] | |||
[[Category:Essence]] | |||