414edo: Difference between revisions
m Moving from Category:Edo to Category:Equal divisions of the octave using Cat-a-lot |
ArrowHead294 (talk | contribs) mNo edit summary |
||
| (16 intermediate revisions by 7 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | |||
{{ED intro}} | |||
[[ | == Theory == | ||
414edo is [[consistent]] to the [[17-odd-limit]] with a flat tendency for most of the [[harmonic]]s, making for a good full [[17-limit]] system. It is closely related to [[207edo]], but the [[patent val]]s differ on the mapping for [[harmonic]] [[5/1|5]]. It [[tempering out|tempers 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 [[hemiennealimmal]] and the 13-limit [[quatracot]]. | |||
=== Prime harmonics === | |||
{{Harmonics in equal|414}} | |||
=== Subsets and supersets === | |||
Since 414 factors into 2 × 3<sup>2</sup> × 23, 414edo has subset edos {{EDOs| 2, 3, 6, 9, 18, 23, 46, 69, 138, and 207 }}. | |||
== 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 }} | |||
| {{mapping| 414 656 961 }} | |||
| +0.2222 | |||
| 0.1575 | |||
| 5.43 | |||
|- | |||
| 2.3.5.7 | |||
| 2401/2400, 4375/4374, {{monzo| -36 11 8 }} | |||
| {{mapping| 414 656 961 1162 }} | |||
| +0.2299 | |||
| 0.1371 | |||
| 4.73 | |||
|- | |||
| 2.3.5.7.11 | |||
| 2401/2400, 3025/3024, 4375/4374, 1366875/1362944 | |||
| {{mapping| 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 | |||
| {{mapping| 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 | |||
| {{mapping| 414 656 961 1162 1432 1532 1692 }} | |||
| +0.1751 | |||
| 0.1329 | |||
| 4.58 | |||
|} | |||
=== 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 | |||
| 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]] | |||
|} | |||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
== Music == | |||
; [[No Clue Music]] | |||
* [https://www.youtube.com/watch?v=j6KPW-Hr1sI ''DISconnectioN''] (2024) | |||
[[Category:Listen]] | |||