441edo: Difference between revisions

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== 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 an extremely strong [[7-limit]] system; strong enough to qualify as a [[zeta peak edo]]. It is also extraordinarily 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 shines in the 2.3.5.7.13 subgroup, with a maximum error of 0.408 cents per interval (in [[14/13]] and [[13/7]]), and in the 2.3.5.7.13.23 subgroup, with a maximum error of 0.416 cents per interval (in [[28/23]] and [[23/14]]).

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. Similar to [[205edo]] but with greater accuracy, 441 can be used as a basis for a Vicentino style "adaptive JI" system.

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

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441 factors into primes as {{nowrap| 32 × 72 }}, 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.

== Selected intervals ==

Line 180:

| 250.34 (16.33)

| 140/121 (100/99)

| [[~~Semiennealimmal~~]]

| [[Ennealimmapine]]

|-

| 9

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| 315.65 (48.98)

| 6/5 (36/35)

| [[Ennealimmal]] / ~~[[ennealimmia]]~~

| [[Ennealimmal]] / ennealympic

|-

| 21

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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

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Scales ==

@@ Line 3: / Line 3: @@
 == 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 an extremely strong [[7-limit]] system; strong enough to qualify as a [[zeta peak edo]]. It is also extraordinarily 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 shines in the 2.3.5.7.13 subgroup, with a maximum error of 0.408 cents per interval (in [[14/13]] and [[13/7]]), and in the 2.3.5.7.13.23 subgroup, with a maximum error of 0.416 cents per interval (in [[28/23]] and [[23/14]]).
-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. Similar to [[205edo]] but with greater accuracy, 441 can be used as a basis for a Vicentino style "adaptive JI" system.
 One step of 441edo is also of a size close to [[625/624]], the tunbarsma.
@@ Line 15: / Line 15: @@
 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.
 == Selected intervals ==
@@ Line 180: / Line 180: @@
 | 250.34<br>(16.33)
 | 140/121<br>(100/99)
-| [[Semiennealimmal]]
+| [[Ennealimmapine]]
 |-
 | 9
@@ Line 186: / Line 186: @@
 | 315.65<br>(48.98)
 | 6/5<br>(36/35)
-| [[Ennealimmal]] / [[ennealimmia]]
+| [[Ennealimmal]] / ennealympic
 |-
 | 21
@@ Line 194: / Line 194: @@
 | [[Akjayland]]
 |}
-<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 == Scales ==