441edo - Revision history

FloraC: /* Regular temperament properties */ update

2026-05-19T09:21:00Z

Regular temperament properties: update

← Older revision		Revision as of 09:21, 19 May 2026
Line 180:		Line 180:
	\| 250.34<br>(16.33)		\| 250.34<br>(16.33)
	\| 140/121<br>(100/99)		\| 140/121<br>(100/99)
	\| [[~~Semiennealimmal~~]]		\| [[Ennealimmapine]]
	\|-		\|-
	\| 9		\| 9
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	\| 315.65<br>(48.98)		\| 315.65<br>(48.98)
	\| 6/5<br>(36/35)		\| 6/5<br>(36/35)
	\| [[Ennealimmal]] / ~~[[ennealimmia]]~~		\| [[Ennealimmal]] / ennealympic
	\|-		\|-
	\| 21		\| 21

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:31:47Z

Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

← Older revision		Revision as of 13:31, 13 March 2026
Line 194:		Line 194:
	\| [[Akjayland]]		\| [[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 ==		== Scales ==

Overthink: /* Theory */ fix typo

2026-01-10T23:31:15Z

Theory: fix typo

← Older revision		Revision as of 23:31, 10 January 2026
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	== Theory ==		== Theory ==
	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]])		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. Similar to [[205edo]] but with greater accuracy, 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.

Eufalesio: Added some things

2025-10-27T12:48:43Z

Added some things

← Older revision		Revision as of 12:48, 27 October 2025
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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 [[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.		One step of 441edo is also of a size close to [[625/624]], the tunbarsma.

FloraC: - factoid that has nothing to do with edos but only numbers

2025-09-10T13:51:01Z

- factoid that has nothing to do with edos but only numbers

← Older revision		Revision as of 13:51, 10 September 2025
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	{{Infobox ET~~\|Zeta=yes~~}}		{{Infobox ET}}
	{{ED intro}}		{{ED intro}}

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

	Coincidentally, it's a product of two Mersenne numbers, which could be an interesting fact to some. [[217edo]], which is closely related to 441 through the [[brahmagupta]] temperament (and by extension to [[224edo\|224]]), is also a product of two Mersenne numbers, which are also primes.

	=== Prime harmonics ===		=== Prime harmonics ===

Eufalesio: added some extra info

2025-09-09T16:16:01Z

added some extra info

← Older revision		Revision as of 16:16, 9 September 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET\|Zeta=yes}}
	{{ED intro}}		{{ED intro}}

Line 8:		Line 8:

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

			Coincidentally, it's a product of two Mersenne numbers, which could be an interesting fact to some. [[217edo]], which is closely related to 441 through the [[brahmagupta]] temperament (and by extension to [[224edo\|224]]), is also a product of two Mersenne numbers, which are also primes.

	=== Prime harmonics ===		=== Prime harmonics ===
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	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 }}.		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.

	== Selected intervals ==		== Selected intervals ==

FloraC: Cleanup and update

2025-03-16T13:45:27Z

Cleanup and update

← Older revision		Revision as of 13:45, 16 March 2025
Line 3:		Line 3:

	== 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 [[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 &~~amp;~~ 369f}} temperament, and for the 7-limit {{nowrap\|41 &~~amp;~~ 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.
Line 13:		Line 13:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	441 factors into primes as 3<sup>2</sup> × 7<sup>2</sup>, and 441edo has subset edos {{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 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)

ArrowHead294 at 14:56, 20 February 2025

2025-02-20T14:56:46Z

← Older revision		Revision as of 14:56, 20 February 2025
Line 1:		Line 1:
	{{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 [[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 [[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.

FloraC: /* Theory */ cleanup

2025-01-17T16:18:07Z

Theory: cleanup

← Older revision		Revision as of 16:18, 17 January 2025
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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 [[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 &~~amp;~~ 369f}} temperament, and for the 7-limit {{nowrap\|41 &~~amp;~~ 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 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.

ArrowHead294 at 15:34, 16 January 2025

2025-01-16T15:34:59Z

← Older revision		Revision as of 15:34, 16 January 2025
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	\| [[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 ==