441edo: Difference between revisions

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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 2.72 [[cent]]s each.

== 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 [[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]].

441edo 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.

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

441 factors into primes as 32 × 72, and has ~~divisors~~ {{EDOs|3, 7, 9, 21, 49, 63 and 147}}.

=== Prime harmonics ===

=== Subsets and supersets ===

441 factors into primes as {{nowrap| 32 × 72 }}, and 441edo has subset edos {{EDOs| 3, 7, 9, 21, 49, 63 and 147 }}.

~~=== Prime~~ harmonics ~~===~~

[[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.

~~{{Harmonics~~ in ~~equal|441|prec=3|columns=~~11}}

== Selected intervals ==

{| class="wikitable"

|+Selected intervals

|+ style="font-size; 105%;" | Selected intervals

|-

! Step

! Eliora's ~~Naming System~~

! Eliora's naming system

! Asosociated ~~Ratio~~

! Asosociated ratio

|-

| 0

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== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" | Subgroup

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

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|-

| 2.3.5

| {{~~monzo~~| 38 -2 -15 }}, {{monzo| 1 -27 18 }}

| {{Monzo| 38 -2 -15 }}, {{monzo| 1 -27 18 }}

| [{{~~val~~| 441 699 1024 }}]

| {{Mapping| 441 699 1024 }}

| -0.0297

| −0.0297

| 0.0224

| 0.82

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| 2.3.5.7

| 2401/2400, 4375/4374, {{monzo| 38 -2 -15 }}

| [{{~~val~~| 441 699 1024 1238 }}]

| {{Mapping| 441 699 1024 1238 }}

| -0.0117

| −0.0117

| 0.0367

| 1.35

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| 2.3.5.7.11

| 2401/2400, 4000/3993, 4375/4374, 131072/130977

| [{{~~val~~| 441 699 1024 1238 1526 }}]

| {{Mapping| 441 699 1024 1238 1526 }}

| -0.0708

| −0.0708

| 0.1227

| 4.51

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| 2.3.5.7.11.13

| 1575/1573, 2080/2079, 2401/2400, 4096/4095, 4375/4374

| [{{~~val~~| 441 699 1024 1238 1526 1632 }}]

| {{Mapping| 441 699 1024 1238 1526 1632 }}

| -0.0720

| −0.0720

| 0.1120

| 4.12

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| 2.3.5.7.11.13.17

| 936/935, 1225/1224, 1575/1573, 1701/1700, 2025/2023, 4096/4095

| [{{~~val~~| 441 699 1024 1238 1526 1632 1803 }}]

| {{Mapping| 441 699 1024 1238 1526 1632 1803 }}

| -0.1025

| −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"

|+Table of rank-2 temperaments by generator

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Generator~~ (reduced)~~

! Generator*

! Cents~~ (reduced)~~

! Cents*

! Associated ratio

! Associated ratio*

! ~~Temperaments~~

! Temperament

|-

| 1

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| 193.20

| 262144/234375

| [[~~Luna]] / [[lunatic~~]]

| [[Lunatic]]

|-

| 1

| 95\441

| 258.50

| {{~~monzo~~| -32 13 5 }}

| {{Monzo| -32 13 5 }}

| [[Lafa]]

|-

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| 565.99

| 104/75

| [[~~Tricot]] / [[trillium~~]]

| [[Alphatrillium]]

|-

| 7

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| [[Akjayland]]

|}

<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== 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)

* 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)

* 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 ==

* [https://www.youtube.com/watch?v=j3sq5jkFjUE Etude in G Akjayland for Piano and Tribal Pan~~, Op. 1, No. 3] by [[Eliora]~~]

; [[Eliora]]

* [https://www.youtube.com/watch?v=j3sq5jkFjUE ''Etude in G Akjayland for Piano and Tribal Pan''] (2022)

[[~~Category~~:~~441edo~~]]

; [[Gene Ward Smith]]

[[Category:~~Equal divisions of the octave|###~~]] ~~~~

* ''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:~~Semienealimmal~~]]

[[Category:Listen]]

[[Category:Luna]]

[[Category:Nicolic]]

[[Category:~~Zeta]]~~

[[Category:Semienealimmal]]

~~[[Category:Akjayland~~]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-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 2.72 [[cent]]s each.
+{{ED intro}}
 == Theory ==
+edo 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]].
-edo 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.
@@ Line 10: / Line 9: @@
 One step of 441edo is also of a size close to [[625/624]], the tunbarsma.
-factors into primes as 3<sup>2</sup> × 7<sup>2</sup>, and has divisors {{EDOs|3, 7, 9, 21, 49, 63 and 147}}.
+=== Prime harmonics ===
+{{Harmonics in equal|441|prec=3}}
+=== Subsets and supersets ===
+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 }}.
-=== Prime harmonics ===
+[[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.
-{{Harmonics in equal|441|prec=3|columns=11}}
 == Selected intervals ==
 {| class="wikitable"
-|+Selected intervals
+|+ style="font-size; 105%;" | Selected intervals
+|-
 ! Step
-! Eliora's Naming System
+! Eliora's naming system
-! Asosociated Ratio
+! Asosociated ratio
 |-
 | 0
@@ Line 73: / Line 76: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" | Subgroup
+|-
+! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
@@ Line 83: / Line 87: @@
 |-
 | 2.3.5
-| {{monzo| 38 -2 -15 }}, {{monzo| 1 -27 18 }}
+| {{Monzo| 38 -2 -15 }}, {{monzo| 1 -27 18 }}
-| [{{val| 441 699 1024 }}]
+| {{Mapping| 441 699 1024 }}
-| -0.0297
+| −0.0297
 | 0.0224
 | 0.82
@@ Line 91: / Line 95: @@
 | 2.3.5.7
 | 2401/2400, 4375/4374, {{monzo| 38 -2 -15 }}
-| [{{val| 441 699 1024 1238 }}]
+| {{Mapping| 441 699 1024 1238 }}
-| -0.0117
+| −0.0117
 | 0.0367
 | 1.35
@@ Line 98: / Line 102: @@
 | 2.3.5.7.11
 | 2401/2400, 4000/3993, 4375/4374, 131072/130977
-| [{{val| 441 699 1024 1238 1526 }}]
+| {{Mapping| 441 699 1024 1238 1526 }}
-| -0.0708
+| −0.0708
 | 0.1227
 | 4.51
@@ Line 105: / Line 109: @@
 | 2.3.5.7.11.13
 | 1575/1573, 2080/2079, 2401/2400, 4096/4095, 4375/4374
-| [{{val| 441 699 1024 1238 1526 1632 }}]
+| {{Mapping| 441 699 1024 1238 1526 1632 }}
-| -0.0720
+| −0.0720
 | 0.1120
 | 4.12
@@ Line 112: / Line 116: @@
 | 2.3.5.7.11.13.17
 | 936/935, 1225/1224, 1575/1573, 1701/1700, 2025/2023, 4096/4095
-| [{{val| 441 699 1024 1238 1526 1632 1803 }}]
+| {{Mapping| 441 699 1024 1238 1526 1632 1803 }}
-| -0.1025
+| −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"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
 ! Periods<br>per 8ve
-! Generator<br>(reduced)
+! Generator*
-! Cents<br>(reduced)
+! Cents*
-! Associated<br>ratio
+! Associated<br>ratio*
-! Temperaments
+! Temperament
 |-
 | 1
@@ Line 131: / Line 138: @@
 | 193.20
 | 262144/234375
-| [[Luna]] / [[lunatic]]
+| [[Lunatic]]
 |-
 | 1
 | 95\441
 | 258.50
-| {{monzo| -32 13 5 }}
+| {{Monzo| -32 13 5 }}
 | [[Lafa]]
 |-
@@ Line 161: / Line 168: @@
 | 565.99
 | 104/75
-| [[Tricot]] / [[trillium]]
+| [[Alphatrillium]]
 |-
 | 7
@@ Line 187: / Line 194: @@
 | [[Akjayland]]
 |}
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == 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)
 * 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)
+* 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 ==
-* [https://www.youtube.com/watch?v=j3sq5jkFjUE Etude in G Akjayland for Piano and Tribal Pan, Op. 1, No. 3] by [[Eliora]]
+; [[Eliora]]
+* [https://www.youtube.com/watch?v=j3sq5jkFjUE ''Etude in G Akjayland for Piano and Tribal Pan''] (2022)
-[[Category:441edo]]
+; [[Gene Ward Smith]]
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+* ''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:Semienealimmal]]
+[[Category:Listen]]
 [[Category:Luna]]
 [[Category:Nicolic]]
-[[Category:Zeta]]
+[[Category:Semienealimmal]]
-[[Category:Akjayland]]