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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 3<sup>2</sup> × 7<sup>2</sup>
{{ED intro}}
| Step size = 2.72109¢
| Fifth = 258\441 (7.0204¢)
| Semitones = 42:33 (114.286¢ : 89.796¢)
| Consistency = 15
}}
The '''441 equal divisions of the octave''' ('''441edo'''), or the '''441(-tone) equal temperament''' ('''441tet''', '''441et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 441 parts of about 2.721 [[cent]]s each, a size close to [[625/624]], the tunbarsma.


== Theory ==
== Theory ==
{{Primes in edo|441|prec=3|columns=10}}
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]].
It is a very strong [[7-limit]] system; strong enough to qualify as a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|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 [[Ragismic microtemperaments #Ennealimmal|ennealimmal temperament]]. 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 temperament]], and the 7-limit 41&amp;359 temperament. Since it tempers out 1575/1573, the nicola, it allows the [[nicolic tetrad]].


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.  


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}}.
One step of 441edo is also of a size close to [[625/624]], the tunbarsma.


== Table of intervals ==
=== Prime harmonics ===
{{Harmonics in equal|441|prec=3}}
 
=== Subsets and supersets ===
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.
 
== Selected intervals ==
{| class="wikitable"
{| class="wikitable"
|+Selected intervals
|+ style="font-size; 105%;" | Selected intervals
!Step
|-
!Name
! Step
!Asosociated ratio
! Eliora's naming system
!Comments
! Asosociated ratio
|-
| 0
| Prime
| 1/1
|-
| 8
| Syntonic comma
| 81/80
|-
| 9
| Pythagorean comma
| 531441/524288
|-
| 10
| Septimal comma
| 64/63
|-
| 75
| Whole tone
| 9/8
|-
| 85
| Septimal supermajor second
| 8/7
|-
| 98
| Septimal subminor third
| 7/6
|-
| 142
| Classical major 3rd
| 5/4
|-
| 150
| Pythagorean major 3rd
| 81/64
|-
| 258
| Perfect 5th
| 3/2
|-
| 356
| Harmonic 7th
| 7/4
|-
| 441
| Octave
| 2/1
|}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
|-
|0
! rowspan="2" | [[Subgroup]]
|Prime
! rowspan="2" | [[Comma list]]
|1/1
! rowspan="2" | [[Mapping]]
|Exact
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
|8
! [[TE error|Absolute]] (¢)
|Syntonic comma
! [[TE simple badness|Relative]] (%)
|81/80
|
|-
|-
|9
| 2.3.5
|Pythagorean comma
| {{Monzo| 38 -2 -15 }}, {{monzo| 1 -27 18 }}
|531441/524288
| {{Mapping| 441 699 1024 }}
|
| −0.0297
| 0.0224
| 0.82
|-
|-
|85
| 2.3.5.7
|Septimal supermajor second
| 2401/2400, 4375/4374, {{monzo| 38 -2 -15 }}
|8/7
| {{Mapping| 441 699 1024 1238 }}
|
| −0.0117
| 0.0367
| 1.35
|-
|-
|98
| 2.3.5.7.11
|Septimal subminor third
| 2401/2400, 4000/3993, 4375/4374, 131072/130977
|7/6
| {{Mapping| 441 699 1024 1238 1526 }}
|
| −0.0708
| 0.1227
| 4.51
|-
|-
|142
| 2.3.5.7.11.13
|Classical major 3rd
| 1575/1573, 2080/2079, 2401/2400, 4096/4095, 4375/4374
|5/4
| {{Mapping| 441 699 1024 1238 1526 1632 }}
|
| −0.0720
| 0.1120
| 4.12
|-
|-
|150
| 2.3.5.7.11.13.17
|Pythagorean major 3rd
| 936/935, 1225/1224, 1575/1573, 1701/1700, 2025/2023, 4096/4095
|81/64
| {{Mapping| 441 699 1024 1238 1526 1632 1803 }}
|
| −0.1025
| 0.1278
| 4.70
|}
* 441et has a lower relative error than any previous equal temperaments in the 5-limit, past [[118edo|118]] and before [[559edo|559]].
* 441et is also notable in the 7-limit, where it has a lower absolute error than any previous equal temperaments, past [[171edo|171]] and before [[612edo|612]].
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
|258
! Periods<br>per 8ve
|Perfect 5th
! Generator*
|3/2
! Cents*
|
! Associated<br>ratio*
! Temperament
|-
|-
|356
| 1
|Harmonic 7th
| 71\441
|7/4
| 193.20
|
| 262144/234375
| [[Lunatic]]
|-
|-
|441
| 1
|Octave
| 95\441
|2/1
| 258.50
|Exact
| {{Monzo| -32 13 5 }}
| [[Lafa]]
|-
| 1
| 116\441
| 315.65
| 6/5
| [[Egads]]
|-
| 1
| 128\441
| 348.30
| 57344/46875
| [[Subneutral]]
|-
| 1
| 206\441
| 560.54
| 864/625
| [[Whoosh]]
|-
| 1
| 208\441
| 565.99
| 104/75
| [[Alphatrillium]]
|-
| 7
| 191\441<br>(2\441)
| 519.73<br>(5.44)
| 27/20<br>(325/324)
| [[Brahmagupta]]
|-
| 9
| 92\441<br>(6\441)
| 250.34<br>(16.33)
| 140/121<br>(100/99)
| [[Semiennealimmal]]
|-
| 9
| 116\441<br>(18\441)
| 315.65<br>(48.98)
| 6/5<br>(36/35)
| [[Ennealimmal]] / [[ennealimmia]]
|-
| 21
| 215\441<br>(5\441)
| 585.03<br>(13.61)
| 91875/65536<br>(126/125)
| [[Akjayland]]
|}
|}
[[Category:441edo]]
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
[[Category:Equal divisions of the octave]]
 
== Scales ==
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)
* 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 – [[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)
* Akjayland[84]: 6 5 5 5, repeated 21 times
 
== Music ==
; [[Eliora]]
* [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:Akjayland]]
[[Category:Ennealimmal]]
[[Category:Ennealimmal]]
[[Category:Semienealimmal]]
[[Category:Listen]]
[[Category:Luna]]
[[Category:Luna]]
[[Category:Nicolic]]
[[Category:Nicolic]]
[[Category:Zeta]]
[[Category:Semienealimmal]]
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