381edo: Difference between revisions
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{{ED intro}} | {{ED intro}} | ||
381edo is [[consistent]] to the [[13-odd-limit]] | == Theory == | ||
381edo is [[consistent]] to the [[13-odd-limit]] and almost the [[15-odd-limit]]; the only inconsistently mapped intervals in the 15-odd-limit are [[15/11]] and its [[octave complement]]. It has a sharp tendency, with odd [[harmonic]]s 3 through 15 all tuned sharp except for 11, which is very slightly flat. | |||
As an equal temperament, it [[tempering out|tempers out]] the [[vulture comma]], {{monzo| 24 -21 4 }}, in the 5-limit and 6144/6125 ([[porwell comma]]) and 250047/250000 ([[landscape comma]]) in the 7-limit. It provides the [[optimal patent val]] for the porwell planar temperament tempering out 6144/6125, and [[nessafof]], the {{nowrap| 99 & 282 }} temperament tempering out it and the landscape comma 250047/250000. | |||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 8: | Line 11: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 381 factors into {{ | Since 381 factors into primes as {{nowrap| 3 × 127 }}, 381edo contains [[3edo]] and [[127edo]] as subsets. | ||
== 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| 604 -381 }} | |||
| {{Mapping| 381 604 }} | |||
| −0.1285 | |||
| 0.1284 | |||
| 4.08 | |||
|- | |||
| 2.3.5 | |||
| {{Monzo| 24 -21 4 }}, {{monzo| 25 15 -21 }} | |||
| {{Mapping| 381 604 885 }} | |||
| −0.2418 | |||
| 0.1915 | |||
| 6.08 | |||
|- | |||
| 2.3.5.7 | |||
| 6144/6125, 250047/250000, 43046721/43025920 | |||
| {{Mapping| 381 604 885 1070 }} | |||
| −0.2929 | |||
| 0.1880 | |||
| 5.97 | |||
|- | |||
| 2.3.5.7.11 | |||
| 3025/3024, 6144/6125, 19712/19683, 160083/160000 | |||
| {{Mapping| 381 604 885 1070 1318 }} | |||
| −0.2264 | |||
| 0.2144 | |||
| 6.81 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 676/675, 1001/1000, 3025/3024, 6144/6125, 10985/10976 | |||
| {{Mapping| 381 604 885 1070 1318 1410 }} | |||
| −0.2075 | |||
| 0.2002 | |||
| 6.36 | |||
|} | |||
=== 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 | |||
| 151\381 | |||
| 475.59 | |||
| 320/243 | |||
| [[Vulture]] (5-limit) | |||
|- | |||
| 3 | |||
| 24\381 | |||
| 475.59<br>(75.59) | |||
| 320/243<br>(117/112) | |||
| [[Terture]] | |||
|- | |||
| 3 | |||
| 50\381 | |||
| 157.48 | |||
| 35/32 | |||
| [[Nessafof]] | |||
|} | |||
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[Normal forms|minimal form]] in parentheses if distinct | |||
[[Category:Porwell]] | [[Category:Porwell]] | ||
[[Category:Nessafof]] | [[Category:Nessafof]] | ||