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..."
 
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== 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 2 × 3 × 71, 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" | [[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Comma list|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
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3
| 2.3
|{{monzo|-225 142}}
| {{monzo| -225 142 }}
|{{mapping|426 675}}
| {{mapping| 426 675 }}
| 0.1724
| +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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! 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>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is 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

Revision as of 10:01, 21 January 2024

← 425edo 426edo 427edo →
Prime factorization 2 × 3 × 71
Step size 2.8169 ¢ 
Fifth 249\426 (701.408 ¢) (→ 83\142)
Semitones (A1:m2) 39:33 (109.9 ¢ : 92.96 ¢)
Consistency limit 9
Distinct consistency limit 9

Template:EDO intro

Theory

426edo is consistent to the 9-odd-limit. Using the patent val, the equal temperament 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 supports the 5-limit version of untriton.

Prime harmonics

Approximation of prime harmonics in 426edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.55 -0.40 +0.19 +0.79 -1.09 -0.73 +1.08 -0.11 -1.41 -1.37
Relative (%) +0.0 -19.4 -14.1 +6.7 +28.2 -38.7 -25.9 +38.3 -3.7 -50.0 -48.8
Steps
(reduced) 		426
(0) 		675
(249) 		989
(137) 		1196
(344) 		1474
(196) 		1576
(298) 		1741
(37) 		1810
(106) 		1927
(223) 		2069
(365) 		2110
(406)

Subsets and supersets

Since 426 factors into 2 × 3 × 71, 426edo has subset edos 2, 3, 6, 71, 142, and 213.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-225 142⟩ [⟨426 675]] +0.1724 0.1724 6.12
2.3.5 [-7 22 -12⟩, [-44 -3 21⟩ [⟨426 675 989]] +0.1721 0.1408 5.00
2.3.5.7 65625/65536, 118098/117649, 250047/250000 [⟨426 675 989 1196]] +0.1123 0.1600 5.68

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 199\426 560.56 864/625 Whoosh
1 209\426 588.73 45/32 Untriton (5-limit)
3 137\426
(5\426) 		385.92
(14.08) 		5/4
(126/125) 		Mutt (7-limit)

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

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