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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|441}}
{{ED intro}}


== 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 [[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]].
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 {{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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=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|441|prec=3|columns=11}}
{{Harmonics in equal|441|prec=3}}


=== Subsets and supersets ===
=== Subsets and supersets ===
441 factors into primes as {{factorization|441}}, and 441edo has divisors {{EDOs| 3, 7, 9, 21, 49, 63 and 147 }}.  
441 factors into primes as {{nowrap| 3<sup>2</sup> × 7<sup>2</sup> }}, and 441edo has subset edos {{EDOs| 3, 7, 9, 21, 49, 63 and 147 }}.  


[[882edo]], which doubles it, gives an alternative mapping for harmonics 11 and 17. [[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.
[[882edo]], which doubles it, gives an alternative mapping for harmonics 11 and 17. [[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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|-
|-
! Step
! Step
! Eliora's Naming System
! Eliora's naming system
! Asosociated Ratio
! Asosociated ratio
|-
|-
| 0
| 0
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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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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 }}
| {{Mapping| 441 699 1024 }}
| −0.0297
| −0.0297
| 0.0224
| 0.0224
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| 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 }}
| {{Mapping| 441 699 1024 1238 }}
| −0.0117
| −0.0117
| 0.0367
| 0.0367
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| 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 }}
| {{Mapping| 441 699 1024 1238 1526 }}
| −0.0708
| −0.0708
| 0.1227
| 0.1227
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| 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 }}
| {{Mapping| 441 699 1024 1238 1526 1632 }}
| −0.0720
| −0.0720
| 0.1120
| 0.1120
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| 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 }}
| {{Mapping| 441 699 1024 1238 1526 1632 1803 }}
| −0.1025
| −0.1025
| 0.1278
| 0.1278
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|+ style="font-size: 105%;" | 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 />ratio*
! Associated<br>ratio*
! Temperament
! Temperament
|-
|-
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| 193.20
| 193.20
| 262144/234375
| 262144/234375
| [[Luna]] / [[lunatic]]
| [[Lunatic]]
|-
|-
| 1
| 1
| 95\441
| 95\441
| 258.50
| 258.50
| {{monzo| -32 13 5 }}
| {{Monzo| -32 13 5 }}
| [[Lafa]]
| [[Lafa]]
|-
|-
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| 565.99
| 565.99
| 104/75
| 104/75
| [[Tricot]] / [[trillium]]
| [[Alphatrillium]]
|-
|-
| 7
| 7
| 191\441<br />(2\441)
| 191\441<br>(2\441)
| 519.73<br />(5.44)
| 519.73<br>(5.44)
| 27/20<br />(325/324)
| 27/20<br>(325/324)
| [[Brahmagupta]]
| [[Brahmagupta]]
|-
|-
| 9
| 9
| 92\441<br />(6\441)
| 92\441<br>(6\441)
| 250.34<br />(16.33)
| 250.34<br>(16.33)
| 140/121<br />(100/99)
| 140/121<br>(100/99)
| [[Semiennealimmal]]
| [[Semiennealimmal]]
|-
|-
| 9
| 9
| 116\441<br />(18\441)
| 116\441<br>(18\441)
| 315.65<br />(48.98)
| 315.65<br>(48.98)
| 6/5<br />(36/35)
| 6/5<br>(36/35)
| [[Ennealimmal]] / [[ennealimmia]]
| [[Ennealimmal]] / [[ennealimmia]]
|-
|-
| 21
| 21
| 215\441<br />(5\441)
| 215\441<br>(5\441)
| 585.03<br />(13.61)
| 585.03<br>(13.61)
| 91875/65536<br />(126/125)
| 91875/65536<br>(126/125)
| [[Akjayland]]
| [[Akjayland]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== Scales ==
== Scales ==
Scales used in ''Etude in G Akjayland'', in order of size:
Scales used in ''Etude in G Akjayland'', in order of size:
* 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)
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