436edo: Difference between revisions
Rework; cleanup; clarify the title row of the rank-2 temp table |
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}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
436edo is [[consistent]] to the [[23-odd-limit]]. The [[patent val]] of 436edo has a distinct flat tendency, in the sense that if the [[octave]] is pure, [[harmonic]]s from 3 to 37 are all flat. | 436edo is [[consistent]] to the [[23-odd-limit]]. The [[patent val]] of 436edo has a distinct flat tendency, in the sense that if the [[octave]] is pure, [[harmonic]]s from 3 to 37 are all flat. | ||
It [[tempering out|tempers out]] [[32805/32768]] and {{monzo| 1 -68 46 }} in the 5-limit; [[390625/388962]], 420175/419904, and [[2100875/2097152]] in the 7-limit; 1375/1372, [[6250/6237]], [[41503/41472]], and 322102/321489 in the 11-limit; [[625/624]], [[1716/1715]], [[2080/2079]], [[10648/10647]], and 15379/15360 in the 13-limit; [[715/714]], [[1089/1088]], [[1225/1224]], [[1275/1274]], [[2025/2023]], and 11271/11264 in the 17-limit; 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 4394/4389, and 4875/4864 in the 19-limit; 875/874, 897/896, 1105/1104, 1863/1862, 2024/2023, 2185/2184, 2300/2299, and 2530/2527 in the 23-limit. It [[support]]s and gives a good tuning to [[quadrant]]. It also supports [[tsaharuk]], but [[171edo]] is better suited for that purpose. | |||
436edo is accurate for some intervals including [[3/2]], [[7/4]], [[11/10]], [[13/10]], [[18/17]], and [[19/18]], so it is especially suitable for the 2.3.7.11/5.13/5.17.19 [[subgroup]]. | 436edo is accurate for some intervals including [[3/2]], [[7/4]], [[11/10]], [[13/10]], [[18/17]], and [[19/18]], so it is especially suitable for the 2.3.7.11/5.13/5.17.19 [[subgroup]]. | ||
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== 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 | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
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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> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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| 498.17 | | 498.17 | ||
| 4/3 | | 4/3 | ||
| [[Helmholtz]] | | [[Helmholtz (temperament)|Helmholtz]] | ||
|- | |- | ||
| 4 | | 4 | ||
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| [[Quadrant]] | | [[Quadrant]] | ||
|} | |} | ||
<nowiki>* | <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 | ||