Xenharmonic Wiki

441edo: Difference between revisions

← Older editNewer edit →
VisualWikitext
Gud2228 (talk | contribs)
autopatrolled, emailconfirmed
227 edits
No edit summary
Tags: Mobile edit Mobile web edit
Cleanup
Line 3: Line 3:
| 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]]
Retrieved from "https://en.xen.wiki/w/441edo"