441edo: Difference between revisions

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| Step size = 2.72109¢
| Step size = 2.72109¢
| Fifth = 258\441 (702.04¢) (→ [[147edo|86\147]])
| Fifth = 258\441 (702.04¢) (→ [[147edo|86\147]])
| Semitones = 42:33 (114.286¢ : 89.796¢)
| Semitones = 42:33 (114.29¢ : 89.80¢)
| Consistency = 17
| Consistency = 17
}}
}}
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.
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, a size close to [[625/624]], the tunbarsma.


== Theory ==
== Theory ==
{{Primes in edo|441|prec=3|columns=10}}
 
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&359 temperament. Since it tempers out 1575/1573, the nicola, it allows the [[nicolic tetrad]].
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&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}}.
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}}.


== Table of intervals ==
=== Prime harmonics ===
{{Harmonics in equal|441|prec=3|columns=11}}
 
== Selected intervals ==
{| class="wikitable"
{| class="wikitable"
|+Selected intervals
|+Selected intervals
!Step
! Step
!Name
! Eliora's Naming System
!Asosociated ratio
! Asosociated Ratio
!Comments
|-
|-
|0
| 0
|Prime
| Prime
|1/1
| 1/1
|Exact
|-
|-
|8
| 8
|Syntonic comma
| Syntonic comma
|81/80
| 81/80
|
|-
|-
|9
| 9
|Pythagorean comma
| Pythagorean comma
|531441/524288
| 531441/524288
|
|-
|-
|10
| 10
|Septimal comma
| Septimal comma
|64/63
| 64/63
|
|-
|-
|75
| 75
|Whole tone
| Whole tone
|9/8
| 9/8
|
|-
|-
|85
| 85
|Septimal supermajor second
| Septimal supermajor second
|8/7
| 8/7
|
|-
|-
|98
| 98
|Septimal subminor third
| Septimal subminor third
|7/6
| 7/6
|
|-
|-
|142
| 142
|Classical major 3rd
| Classical major 3rd
|5/4
| 5/4
|
|-
|-
|150
| 150
|Pythagorean major 3rd
| Pythagorean major 3rd
|81/64
| 81/64
|
|-
|-
|258
| 258
|Perfect 5th
| Perfect 5th
|3/2
| 3/2
|
|-
|-
|356
| 356
|Harmonic 7th
| Harmonic 7th
|7/4
| 7/4
|
|-
|-
|441
| 441
|Octave
| Octave
|2/1
| 2/1
|Exact
|}
|}
[[Category:441edo]]
[[Category:441edo]]
[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]

Revision as of 00:23, 28 January 2022

← 440edo 441edo 442edo →
Prime factorization 32 × 72
Step size 2.72109 ¢ 
Fifth 258\441 (702.041 ¢) (→ 86\147)
Semitones (A1:m2) 42:33 (114.3 ¢ : 89.8 ¢)
Consistency limit 17
Distinct consistency limit 17

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 equal division of the octave into 441 parts of about 2.72 cents each, a size close to 625/624, the tunbarsma.

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 118 with a lower 5-limit relative error. In the 5-limit It tempers out the hemithirds comma, [38 -2 -15⟩, the ennealimma, [1 -27 18⟩, whoosh, [37 25 -33⟩, and egads, [-36 -52 51⟩. In the 7-limit it tempers out 2401/2400, 4375/4374, 420175/419904 and 250047/250000, so that it supports 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 semiennealimmal temperament, and the 7-limit 41&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.

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

Prime harmonics

Approximation of prime harmonics in 441edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.086 +0.081 -0.118 +1.063 +0.289 +1.167 -0.914 +0.297 -1.006 +0.543
Relative (%) +0.0 +3.2 +3.0 -4.4 +39.1 +10.6 +42.9 -33.6 +10.9 -37.0 +19.9
Steps
(reduced) 		441
(0) 		699
(258) 		1024
(142) 		1238
(356) 		1526
(203) 		1632
(309) 		1803
(39) 		1873
(109) 		1995
(231) 		2142
(378) 		2185
(421)

Selected intervals

Selected intervals
Step Eliora's Naming System 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
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