445edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
445edo is [[enfactoring|enfactored]] in the [[3-limit]] with the same tuning as [[89edo]], but the approximation to some of the higher harmonics are improved. It is [[consistent]] to the [[7-odd-limit]] with [[harmonic]]s [[3/1|3]], [[5/1|5]], [[7/1|7]] all tuned flat, and it allows an extension to the [[11-limit]]. The equal temperament [[tempering out|tempers out]] [[2401/2400]], 7381125/7340032, 33756345/33554432, 43046721/42875000, and 48828125/48771072 in the 7-limit; [[3025/3024]], [[8019/8000]], 24057/24010, 35937/35840, [[41503/41472]], 137781/137500, 151263/151250, and 234375/234256 in the 11-limit. It notably [[support]]s [[neptune]]. | 445edo is [[enfactoring|enfactored]] in the [[3-limit]] with the same tuning as [[89edo]], but the approximation to some of the higher harmonics are improved. It is [[consistent]] to the [[7-odd-limit]] with [[harmonic]]s [[3/1|3]], [[5/1|5]], [[7/1|7]] all tuned flat, and it allows an extension to the [[11-limit]]. The equal temperament [[tempering out|tempers out]] [[2401/2400]], 7381125/7340032, 33756345/33554432, 43046721/42875000, and 48828125/48771072 in the 7-limit; [[3025/3024]], [[8019/8000]], 24057/24010, 35937/35840, [[41503/41472]], 137781/137500, 151263/151250, and 234375/234256 in the 11-limit. It notably [[support]]s [[Neptune (temperament)|neptune]]. | ||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 9: | Line 9: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 445 factors into | Since 445 factors into {{factorisation|445}}, 445edo has [[5edo]] and 89edo as its subsets. | ||
== Regular temperament properties == | == 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 | | 2.3.5 | ||
| {{ | | {{Monzo| -28 25 -5 }}, {{monzo| -29 -11 20 }} | ||
| {{ | | {{Mapping| 445 705 1033 }} | ||
| +0.2748 | | +0.2748 | ||
| 0.2149 | | 0.2149 | ||
| Line 23: | Line 32: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 2401/2400, 7381125/7340032, 43046721/42875000 | | 2401/2400, 7381125/7340032, 43046721/42875000 | ||
| {{ | | {{Mapping| 445 705 1033 1249 }} | ||
| +0.2716 | | +0.2716 | ||
| 0.1862 | | 0.1862 | ||
| Line 30: | Line 39: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 2401/2400, 3025/3024, 8019/8000, 234375/234256 | | 2401/2400, 3025/3024, 8019/8000, 234375/234256 | ||
| {{ | | {{Mapping| 445 705 1033 1249 1539 }} | ||
| +0.2870 | | +0.2870 | ||
| 0.1694 | | 0.1694 | ||
| 6.28 | | 6.28 | ||
|} | |||
=== Rank-2 temperaments === | === 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 | | 1 | ||
| Line 55: | Line 71: | ||
| 582.47 | | 582.47 | ||
| 7/5 | | 7/5 | ||
| [[Neptune]] (7-limit) | | [[Neptune (temperament)|Neptune]] (7-limit) | ||
|- | |- | ||
| 5 | | 5 | ||
| | | 7\445 | ||
| | | 18.88 | ||
| | | 81/80 | ||
| [[ | | [[Quintile]] | ||
|} | |||
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]] | |||