441edo: Difference between revisions
→Music: if there's music it must be in Listen, especially for large edos where almost no one composes |
Cleanup; clarify the title row of the rank-2 temp table; +music |
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== Theory == | == Theory == | ||
441edo is a very strong [[7-limit]] system; strong enough to qualify as a [[ | 441edo is a very strong [[7-limit]] system; strong enough to qualify as a [[zeta peak edo]]. It is also very strong simply considered as a 5-limit system; it is the first division past [[118edo|118]] with a lower [[5-limit]] [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. In the 5-limit It [[tempering out|tempers out]] the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, the [[ennealimma]], {{monzo| 1 -27 18 }}, whoosh, {{monzo| 37 25 -33 }}, and egads, {{monzo| -36 -52 51 }}. In the 7-limit it tempers out [[2401/2400]], [[4375/4374]], [[420175/419904]] and [[250047/250000]], so that it [[support]]s [[ennealimmal]]. In the [[11-limit]] it tempers out [[4000/3993]], and in the 13-limit, [[1575/1573]], [[2080/2079]] and [[4096/4095]]. It provides the [[optimal patent val]] for 11- and [[13-limit]] [[Ragismic microtemperaments #Ennealimmal|semiennealimmal]], the 72 & 369f temperament, and for the 7-limit 41 & 400 temperament. Since it tempers out 1575/1573, the nicola, it allows the [[nicolic chords]] in the [[15-odd-limit]]. | ||
The steps of 441 are only 1/30 of a cent sharp of 1/8 syntonic comma. Lowering the fifth, which is only 1/12 of a cent sharp, by two steps gives a generator, 256\441, close to 1/4 comma meantone. Like [[205edo]] but even more accurately, 441 can be used as a basis for a Vicentino style "adaptive JI" system. | The steps of 441 are only 1/30 of a cent sharp of 1/8 syntonic comma. Lowering the fifth, which is only 1/12 of a cent sharp, by two steps gives a generator, 256\441, close to 1/4 comma meantone. Like [[205edo]] but even more accurately, 441 can be used as a basis for a Vicentino style "adaptive JI" system. | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
441 factors into primes as 3<sup>2</sup> × 7<sup>2</sup>, and has divisors {{EDOs| 3, 7, 9, 21, 49, 63 and 147 }}. | 441 factors into primes as 3<sup>2</sup> × 7<sup>2</sup>, and 441edo has divisors {{EDOs| 3, 7, 9, 21, 49, 63 and 147 }}. | ||
[[1323edo]], which divides the edostep into three, is the smallest distinctly consistent edo in the 29-odd-limit and thus provides good correction for prime harmonics from 11 to 29. | [[1323edo]], which divides the edostep into three, is the smallest distinctly consistent edo in the 29-odd-limit and thus provides good correction for prime harmonics from 11 to 29. | ||
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| 2.3.5 | | 2.3.5 | ||
| {{monzo| 38 -2 -15 }}, {{monzo| 1 -27 18 }} | | {{monzo| 38 -2 -15 }}, {{monzo| 1 -27 18 }} | ||
| | | {{mapping| 441 699 1024 }} | ||
| -0.0297 | | -0.0297 | ||
| 0.0224 | | 0.0224 | ||
| Line 93: | Line 93: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 2401/2400, 4375/4374, {{monzo| 38 -2 -15 }} | | 2401/2400, 4375/4374, {{monzo| 38 -2 -15 }} | ||
| | | {{mapping| 441 699 1024 1238 }} | ||
| -0.0117 | | -0.0117 | ||
| 0.0367 | | 0.0367 | ||
| Line 100: | Line 100: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 2401/2400, 4000/3993, 4375/4374, 131072/130977 | | 2401/2400, 4000/3993, 4375/4374, 131072/130977 | ||
| | | {{mapping| 441 699 1024 1238 1526 }} | ||
| -0.0708 | | -0.0708 | ||
| 0.1227 | | 0.1227 | ||
| Line 107: | Line 107: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 1575/1573, 2080/2079, 2401/2400, 4096/4095, 4375/4374 | | 1575/1573, 2080/2079, 2401/2400, 4096/4095, 4375/4374 | ||
| | | {{mapping| 441 699 1024 1238 1526 1632 }} | ||
| -0.0720 | | -0.0720 | ||
| 0.1120 | | 0.1120 | ||
| Line 114: | Line 114: | ||
| 2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
| 936/935, 1225/1224, 1575/1573, 1701/1700, 2025/2023, 4096/4095 | | 936/935, 1225/1224, 1575/1573, 1701/1700, 2025/2023, 4096/4095 | ||
| | | {{mapping| 441 699 1024 1238 1526 1632 1803 }} | ||
| -0.1025 | | -0.1025 | ||
| 0.1278 | | 0.1278 | ||
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|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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| [[Akjayland]] | | [[Akjayland]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
== Scales == | == Scales == | ||
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== Music == | == Music == | ||
; [[Eliora]] | ; [[Eliora]] | ||
* [https://www.youtube.com/watch?v=j3sq5jkFjUE ''Etude in G Akjayland for Piano and Tribal Pan'' | * [https://www.youtube.com/watch?v=j3sq5jkFjUE ''Etude in G Akjayland for Piano and Tribal Pan''] (2022) | ||
; [[Gene Ward Smith]] | |||
* ''Bodacious Breed'' (archived 2010) – [http://www.archive.org/details/BodaciousBreed details] | [http://www.archive.org/download/BodaciousBreed/Genewardsmith-BodaciousBreed.mp3 play] – breed in 441edo tuning | |||
[[Category:Ennealimmal]] | [[Category:Ennealimmal]] | ||