414edo: Difference between revisions
Cleanup; +prime error table |
+RTT table and rank-2 temperaments |
||
| Line 1: | Line 1: | ||
'''414edo''' is the [[EDO|equal division of the octave]] into 414 parts of 2.89855 [[cent]]s each. | '''414edo''' is the [[EDO|equal division of the octave]] into 414 parts of 2.89855 [[cent]]s each. | ||
414edo is closely related to [[207edo]], but the [[patent val]]s differ on the mapping for 5. It is [[consistent]] to the [[17-odd-limit]], tempering out {{monzo| -36 11 8 }} (submajor comma) and {{monzo|1 -27 18}} ([[ennealimma]]) in the 5-limit; [[2401/2400]], [[4375/4374]], and | 414edo is closely related to [[207edo]], but the [[patent val]]s differ on the mapping for 5. It is [[consistent]] to the [[17-odd-limit]], tempering out {{monzo| -36 11 8 }} (submajor comma) and {{monzo|1 -27 18}} ([[ennealimma]]) in the 5-limit; [[2401/2400]], [[4375/4374]], and {{monzo| -37 4 12 1 }} in the 7-limit; [[3025/3024]], [[9801/9800]], [[41503/41472]], and 1265625/1261568 in the 11-limit; [[625/624]], [[729/728]], [[1575/1573]], [[2200/2197]], and 26411/26364 in the 13-limit; [[833/832]], [[1089/1088]], [[1225/1224]], 1275/1274, and [[1701/1700]] in the 17-limit. It [[support]]s the 11-limit [[Ragismic microtemperaments|hemiennealimmal]] and the 13-limit [[Ragismic microtemperaments|quatracot]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|414}} | {{Harmonics in equal|414}} | ||
== 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]] (%) | |||
|- | |||
| 2.3.5 | |||
| {{monzo| -36 11 8 }}, {{monzo| 1 -27 18 }} | |||
| [{{val| 414 656 961 }}] | |||
| +0.2222 | |||
| 0.1575 | |||
| 5.43 | |||
|- | |||
| 2.3.5.7 | |||
| 2401/2400, 4375/4374, {{monzo| -36 11 8 }} | |||
| [{{val| 414 656 961 1162 }}] | |||
| +0.2299 | |||
| 0.1371 | |||
| 4.73 | |||
|- | |||
| 2.3.5.7.11 | |||
| 2401/2400, 3025/3024, 4375/4374, 1366875/1362944 | |||
| [{{val| 414 656 961 1162 1432 }}] | |||
| +0.2182 | |||
| 0.1248 | |||
| 4.30 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 625/624, 729/728, 1575/1573, 2200/2197, 2401/2400 | |||
| [{{val| 414 656 961 1162 1432 1532 }}] | |||
| +0.1795 | |||
| 0.1431 | |||
| 4.94 | |||
|- | |||
| 2.3.5.7.11.13.17 | |||
| 625/624, 729/728, 833/832, 1089/1088, 1225/1224, 2200/2197 | |||
| [{{val| 414 656 961 1162 1432 1532 1692 }}] | |||
| +0.1751 | |||
| 0.1329 | |||
| 4.58 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per octave | |||
! Generator<br>(reduced) | |||
! Cents<br>(reduced) | |||
! Associated<br>ratio | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 125\414 | |||
| 362.31 | |||
| 10125/8192 | |||
| [[Submajor]] (5-limit) | |||
|- | |||
| 2 | |||
| 61\414 | |||
| 176.81 | |||
| 195/176 | |||
| [[Quatracot]] | |||
|- | |||
| 9 | |||
| 109\414<br>(17\414) | |||
| 315.94<br>(49.28) | |||
| 6/5<br>(36/35) | |||
| [[Ennealimmal]] | |||
|- | |||
| 18 | |||
| 86\414<br>(6\414) | |||
| 249.28<br>(17.39) | |||
| 231/200<br>(99/98) | |||
| [[Hemiennealimmal]] | |||
|- | |||
| 18 | |||
| 164\414<br>(3\414) | |||
| 475.36<br>(8.70) | |||
| 1053/800<br>(1287/1280) | |||
| [[Semihemiennealimmal]] | |||
|} | |||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||