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'''414EDO''' is the [[EDO|equal division of the octave]] into 414 parts of 2.89855 [[cent]]s each. It is closely related to [[207edo]], but the patent vals differ on the mapping for 5. It is consistent to the 17-limit, tempering out 69198046875/68719476736 (submajor comma) and 7629394531250/7625597484987 (ennealimmal comma) in the 5-limit; 2401/2400, 4375/4374, and 138427734375/137438953472 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 supports the 11-limit [[Ragismic microtemperaments|hemiennealimmal]] and the 13-limit [[Ragismic microtemperaments|quatracot]].
{{Infobox ET}}
{{ED intro}}


[[Category:Edo]]
== 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]]
Retrieved from "https://en.xen.wiki/w/414edo"