441edo: Difference between revisions
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== Theory == | == Theory == | ||
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 [[ | 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 [[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 {{nowrap|72 & 369f}} temperament, and for the 7-limit {{nowrap|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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| {{monzo| 38 -2 -15 }}, {{monzo| 1 -27 18 }} | | {{monzo| 38 -2 -15 }}, {{monzo| 1 -27 18 }} | ||
| {{mapping| 441 699 1024 }} | | {{mapping| 441 699 1024 }} | ||
| | | −0.0297 | ||
| 0.0224 | | 0.0224 | ||
| 0.82 | | 0.82 | ||
| Line 96: | Line 96: | ||
| 2401/2400, 4375/4374, {{monzo| 38 -2 -15 }} | | 2401/2400, 4375/4374, {{monzo| 38 -2 -15 }} | ||
| {{mapping| 441 699 1024 1238 }} | | {{mapping| 441 699 1024 1238 }} | ||
| | | −0.0117 | ||
| 0.0367 | | 0.0367 | ||
| 1.35 | | 1.35 | ||
| Line 103: | Line 103: | ||
| 2401/2400, 4000/3993, 4375/4374, 131072/130977 | | 2401/2400, 4000/3993, 4375/4374, 131072/130977 | ||
| {{mapping| 441 699 1024 1238 1526 }} | | {{mapping| 441 699 1024 1238 1526 }} | ||
| | | −0.0708 | ||
| 0.1227 | | 0.1227 | ||
| 4.51 | | 4.51 | ||
| Line 110: | Line 110: | ||
| 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 }} | | {{mapping| 441 699 1024 1238 1526 1632 }} | ||
| | | −0.0720 | ||
| 0.1120 | | 0.1120 | ||
| 4.12 | | 4.12 | ||
| Line 117: | Line 117: | ||
| 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 }} | | {{mapping| 441 699 1024 1238 1526 1632 1803 }} | ||
| | | −0.1025 | ||
| 0.1278 | | 0.1278 | ||
| 4.70 | | 4.70 | ||
| Line 201: | Line 201: | ||
* Balzano-200[9]: 77 41 41 41 77 41 41 41 41 ([[2L 7s]], generator = 200\441) | * Balzano-200[9]: 77 41 41 41 77 41 41 41 41 ([[2L 7s]], generator = 200\441) | ||
* OEIS-A163205[11]: 9 12 24 4 44 12 84 28 8 156 60 (rank 10) | * OEIS-A163205[11]: 9 12 24 4 44 12 84 28 8 156 60 (rank 10) | ||
* Lafa[14]: 34 34 27 34 34 27 34 34 27 34 34 27 34 27 | * Lafa[14]: 34 34 27 34 34 27 34 34 27 34 34 27 34 27 – [[9L 5s]] (m-chro semiquartal) | ||
* Ennealimmal[27]: 18 18 13 18 18 13 18 18 13 18 18 13 18 18 13 18 18 13 18 18 13 18 18 13 18 18 13 (18L 9s) | * Ennealimmal[27]: 18 18 13 18 18 13 18 18 13 18 18 13 18 18 13 18 18 13 18 18 13 18 18 13 18 18 13 (18L 9s) | ||
* Akjayland[84]: 6 5 5 5, repeated 21 times | * Akjayland[84]: 6 5 5 5, repeated 21 times | ||
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; [[Gene Ward Smith]] | ; [[Gene Ward Smith]] | ||
* ''Bodacious Breed'' (archived 2010) | * ''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:Akjayland]] | [[Category:Akjayland]] | ||