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'''441edo''' is the [[equal division of the octave]] into 441 parts of 2.721 [[cent]]s each. 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 supports [[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 the [[equal division of the octave]] into 441 parts of 2.721 [[cent]]s each.  
 
== 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 supports [[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.
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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}}.


{{Primes in edo|441|prec=3}}
== Table of intervals ==
 
{| class="wikitable"
|+Selected intervals
!Step
!Name
!Asosociated ratio
!Comments
|-
|0
|Prime
|1/1
|Exact
|-
|8
|Syntonic comma
|81/80
|
|-
|9
|Pythagorean comma
|531441/524288
|
|-
|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
|Exact
|}
[[Category:441edo]]
[[Category:441edo]]
[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]

Revision as of 14:53, 18 January 2022

441edo is the equal division of the octave into 441 parts of 2.721 cents each.

Theory

Script error: No such module "primes_in_edo". It 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.

Table of intervals

Selected intervals
Step Name Asosociated ratio Comments
0 Prime 1/1 Exact
8 Syntonic comma 81/80
9 Pythagorean comma 531441/524288
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 Exact
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