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" Tags: Mobile edit Mobile web edit |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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 === | ||
| Line 9: | Line 9: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
426 factors into | 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" | ||
|- | |- | ||
|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 | ||
| | | 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 | ||
| Line 46: | Line 47: | ||
=== 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> | ! 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 | ||
| | | 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>( | | 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>* | |||
Latest revision as of 13:31, 13 March 2026
| ← 425edo | 426edo | 427edo → |
426 equal divisions of the octave (abbreviated 426edo or 426ed2), also called 426-tone equal temperament (426tet) or 426 equal temperament (426et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 426 equal parts of about 2.82 ¢ each. Each step represents a frequency ratio of 21/426, or the 426th root of 2.
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
| 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
| 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 distinct