436edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ED intro}} | |||
== Theory == | == Theory == | ||
The [[patent val]] of 436edo has a distinct flat tendency, in the sense that if the [[octave]] is pure, | 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 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. | 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]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 11: | Line 13: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 436 factors into {{factorization|436}}, 436edo has subset edos {{EDOs| 2, 4, 109, and 218 }}. | |||
[[1308edo]], which divides edostep into three, is a [[zeta gap edo]] and is consistent in the 21-odd-limit. | [[1308edo]], which divides its edostep into three, is a [[zeta gap edo]] and is consistent in the 21-odd-limit. | ||
== 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 stretch (¢) | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
| Line 28: | Line 31: | ||
| 2.3 | | 2.3 | ||
| {{monzo| -691 436 }} | | {{monzo| -691 436 }} | ||
| | | {{mapping| 436 691 }} | ||
| +0.0379 | | +0.0379 | ||
| 0.0379 | | 0.0379 | ||
| Line 35: | Line 38: | ||
| 2.3.5 | | 2.3.5 | ||
| 32805/32768, {{monzo| 1 -68 46 }} | | 32805/32768, {{monzo| 1 -68 46 }} | ||
| | | {{mapping| 436 691 1012 }} | ||
| +0.1678 | | +0.1678 | ||
| 0.1863 | | 0.1863 | ||
| Line 42: | Line 45: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 32805/32768, 390625/388962, 420175/419904 | | 32805/32768, 390625/388962, 420175/419904 | ||
| | | {{mapping| 436 691 1012 1224 }} | ||
| +0.1275 | | +0.1275 | ||
| 0.1758 | | 0.1758 | ||
| Line 49: | Line 52: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 1375/1372, 6250/6237, 32805/32768, 41503/41472 | | 1375/1372, 6250/6237, 32805/32768, 41503/41472 | ||
| | | {{mapping| 436 691 1012 1224 1508 }} | ||
| +0.1517 | | +0.1517 | ||
| 0.1645 | | 0.1645 | ||
| Line 56: | Line 59: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 625/624, 1375/1372, 2080/2079, 10648/10647, 15379/15360 | | 625/624, 1375/1372, 2080/2079, 10648/10647, 15379/15360 | ||
| | | {{mapping| 436 691 1012 1224 1508 1613 }} | ||
| +0.1749 | | +0.1749 | ||
| 0.1589 | | 0.1589 | ||
| Line 63: | Line 66: | ||
| 2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
| 625/624, 715/714, 1089/1088, 1225/1224, 2431/2430, 10648/10647 | | 625/624, 715/714, 1089/1088, 1225/1224, 2431/2430, 10648/10647 | ||
| | | {{mapping| 436 691 1012 1224 1508 1613 1782 }} | ||
| +0.1628 | | +0.1628 | ||
| 0.1501 | | 0.1501 | ||
| Line 70: | Line 73: | ||
| 2.3.5.7.11.13.17.19 | | 2.3.5.7.11.13.17.19 | ||
| 625/624, 715/714, 1089/1088, 1225/1224, 1331/1330, 1445/1444, 1729/1728 | | 625/624, 715/714, 1089/1088, 1225/1224, 1331/1330, 1445/1444, 1729/1728 | ||
| | | {{mapping| 436 691 1012 1224 1508 1613 1782 1852 }} | ||
| +0.1503 | | +0.1503 | ||
| 0.1443 | | 0.1443 | ||
| Line 78: | Line 81: | ||
=== 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 | |- | ||
! Generator | ! Periods<br />per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br>ratio | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 95: | Line 99: | ||
| 498.17 | | 498.17 | ||
| 4/3 | | 4/3 | ||
| [[Helmholtz]] | | [[Helmholtz (temperament)|Helmholtz]] | ||
|- | |- | ||
| 4 | | 4 | ||
| Line 103: | Line 107: | ||
| [[Quadrant]] | | [[Quadrant]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
[[ | |||