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 (temperament)|neptune]]. | |||
=== Odd harmonics === | === Odd harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
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" | {| 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 | ||
| | |||
|- | |- | ||
|2.3.5 | ! [[TE error|Absolute]] (¢) | ||
|{{monzo|-28 25 -5}}, {{monzo|-29 -11 20}} | ! [[TE simple badness|Relative]] (%) | ||
|{{mapping|445 705 1033}} | |- | ||
| 0.2748 | | 2.3.5 | ||
| {{monzo| -28 25 -5 }}, {{monzo| -29 -11 20 }} | |||
| {{mapping| 445 705 1033 }} | |||
| +0.2748 | |||
| 0.2149 | | 0.2149 | ||
| 7.97 | | 7.97 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|2401/2400, | | 2401/2400, 7381125/7340032, 43046721/42875000 | ||
|{{mapping|445 705 1033 1249}} | | {{mapping| 445 705 1033 1249 }} | ||
| 0.2716 | | +0.2716 | ||
| 0.1862 | | 0.1862 | ||
| 6.90 | | 6.90 | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|3025/3024 | | 2401/2400, 3025/3024, 8019/8000, 234375/234256 | ||
|{{mapping|445 705 1033 1249 1539}} | | {{mapping| 445 705 1033 1249 1539 }} | ||
| 0.2870 | | +0.2870 | ||
| 0.1694 | | 0.1694 | ||
| 6.28 | | 6.28 | ||
| Line 53: | Line 47: | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | |- | ||
! Periods<br />per 8ve | |||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|13\445 | | 13\445 | ||
|35.06 | | 35.06 | ||
|1990656/1953125 | | 1990656/1953125 | ||
|[[Gammic]] | | [[Gammic]] (5-limit) | ||
|- | |- | ||
|1 | | 1 | ||
|42\445 | | 42\445 | ||
|113.26 | | 113.26 | ||
|16/15 | | 16/15 | ||
|[[Misneb]] | | [[Misneb]] | ||
|- | |- | ||
|1 | | 1 | ||
|216\445 | | 216\445 | ||
|582.47 | | 582.47 | ||
|7/5 | | 7/5 | ||
|[[Neptune]] | | [[Neptune (temperament)|Neptune]] (7-limit) | ||
|- | |- | ||
|5 | | 5 | ||
|185\445<br>(7\445) | | 185\445<br>(7\445) | ||
|498.88<br>(18.88) | | 498.88<br>(18.88) | ||
|4/3<br>(81/80) | | 4/3<br>(81/80) | ||
|[[Pental]] | | [[Pental (temperament)|Pental]] (5-limit) | ||
|} | |} | ||
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | |||
<nowiki>* | |||