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426edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|426}} == Theory == 426et is consistent to the 9-odd-limit. Using the patent val, it tempers out 283115520/282475249, 48828125/48771072, 65625..."
 
m 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"
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|426}}
{{ED intro}}


== Theory ==
== Theory ==
426et is consistent to the [[9-odd-limit]]. Using the patent val, it tempers out 283115520/282475249, 48828125/48771072, [[65625/65536]], [[250047/250000]] and 5250987/5242880 in the 7-limit; [[117440512/117406179]], 806736/805255, 25165824/25109315, 2097152/2096325, [[4000/3993]], 2734375/2725888, 166698/166375, 151263/151250, 104857600/104825259, 2359296/2358125, [[540/539]], 1265625/1261568, 107495424/107421875, 137781/137500, 5767168/5764801, 825000/823543, 24057/24010, 17537553/17500000, [[9801/9800]] and 3294225/3294172 in the 11-limit. It [[support]]s [[untriton]].
426edo is [[consistent]] to the [[9-odd-limit]]. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[65625/65536]], 118098/117649, [[250047/250000]] in the 7-limit; [[540/539]], [[4000/3993]], [[9801/9800]], 24057/24010, 137781/137500, and 151263/151250 in the 11-limit. It [[support]]s the 5-limit version of [[untriton]].


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
426 factors into 2 × 3 × 71, with subset edos {{EDOs|2, 3, 6, 71, 142, and 213}}.
Since 426 factors into {{factorisation|426}}, 426edo has subset edos {{EDOs| 2, 3, 6, 71, 142, and 213 }}.


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|-
|2.3
! rowspan="2" | [[Subgroup]]
|{{monzo|-225 142}}
! rowspan="2" | [[Comma list]]
|{{mapping|426 675}}
! rowspan="2" | [[Mapping]]
| 0.1724
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -225 142 }}
| {{mapping| 426 675 }}
| +0.1724
| 0.1724
| 0.1724
| 6.12
| 6.12
|-
|-
|2.3.5
| 2.3.5
|{{monzo|-7 22 -12}}, {{monzo|-44 -3 21}}
| {{monzo| -7 22 -12 }}, {{monzo| -44 -3 21 }}
|{{mapping|426 675 989}}
| {{mapping| 426 675 989 }}
| 0.1721
| +0.1721
| 0.1408
| 0.1408
| 5.00
| 5.00
|-
|-
|2.3.5.7
| 2.3.5.7
|250047/250000, 118098/117649, 65625/65536
| 65625/65536, 118098/117649, 250047/250000
|{{mapping|426 675 989 1196}}
| {{mapping| 426 675 989 1196 }}
| 0.1123
| +0.1123
| 0.1600
| 0.1600
| 5.68
| 5.68
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+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*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|199\426
| 199\426
|560.56
| 560.56
|864/625
| 864/625
|[[Whoosh]]
| [[Whoosh]]
|-
|-
|1
| 1
|209\426
| 209\426
|588.73
| 588.73
|7/5
| 45/32
|[[Untriton]]
| [[Untriton]] (5-limit)
|-
|-
|3
| 3
|137\426<br>(5\426)
| 137\426<br />(5\426)
|385.92<br>(14.08)
| 385.92<br />(14.08)
|5/4<br>(8393216/390625)
| 5/4<br />(126/125)
|[[Mutt]]
| [[Mutt]] (7-limit)
|}
|}
 
<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
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
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