436edo: Difference between revisions
m Moving from Category:Edo to Category:Equal divisions of the octave using Cat-a-lot |
+prime error table, +RTT table |
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'''436edo''' is the [[EDO|equal division of the octave]] into 436 parts of 2.7522935780 [[cent]]s each. The patent val has a distinct flat tendency, in the sense that if the [[octave]] is pure, 3 | '''436edo''' is the [[EDO|equal division of the octave]] into 436 parts of 2.7522935780 [[cent]]s each. | ||
== Theory == | |||
The [[patent val]] of 436edo has a distinct flat tendency, in the sense that if the [[octave]] is pure, harmonics from 3 to 37 are all flat. It is consistent to the [[23-odd-limit]], tempering out 32805/32768 and {{monzo| 1 -68 4 }} 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. | |||
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. | ||
=== Prime harmonics === | |||
{{Harmonics in equal|436}} | |||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" | Subgroup | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br>8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| -691 436 }} | |||
| [{{val| 436 691 }}] | |||
| +0.0379 | |||
| 0.0379 | |||
| 1.38 | |||
|- | |||
| 2.3.5 | |||
| 32805/32768, {{monzo| 1 -68 46 }} | |||
| [{{val| 436 691 1012 }}] | |||
| +0.1678 | |||
| 0.1863 | |||
| 6.77 | |||
|- | |||
| 2.3.5.7 | |||
| 32805/32768, 390625/388962, 420175/419904 | |||
| [{{val| 436 691 1012 1224 }}] | |||
| +0.1275 | |||
| 0.1758 | |||
| 6.39 | |||
|- | |||
| 2.3.5.7.11 | |||
| 1375/1372, 6250/6237, 32805/32768, 41503/41472 | |||
| [{{val| 436 691 1012 1224 1508 }}] | |||
| +0.1517 | |||
| 0.1645 | |||
| 5.98 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 625/624, 1375/1372, 2080/2079, 10648/10647, 15379/15360 | |||
| [{{val| 436 691 1012 1224 1508 1613 }}] | |||
| +0.1749 | |||
| 0.1589 | |||
| 5.77 | |||
|- | |||
| 2.3.5.7.11.13.17 | |||
| 625/624, 715/714, 1089/1088, 1225/1224, 2431/2430, 10648/10647 | |||
| [{{val| 422 669 980 1185 1460 1562 1725 }}] | |||
| +0.1628 | |||
| 0.1501 | |||
| 5.45 | |||
|- | |||
| 2.3.5.7.11.13.17.19 | |||
| 625/624, 715/714, 1089/1088, 1225/1224, 1331/1330, 1445/1444, 1729/1728 | |||
| [{{val| 422 669 980 1185 1460 1562 1725 1793 1852 }}] | |||
| +0.1503 | |||
| 0.1443 | |||
| 5.24 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per octave | |||
! Generator<br>(reduced) | |||
! Cents<br>(reduced) | |||
! Associated<br>ratio | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 51\436 | |||
| 140.37 | |||
| 243/224 | |||
| [[Tsaharuk]] | |||
|- | |||
| 1 | |||
| 181\436 | |||
| 498.17 | |||
| 4/3 | |||
| [[Helmholtz]] | |||
|- | |||
| 4 | |||
| 181\436<br>(37\436) | |||
| 498.17<br>(101.83) | |||
| 4/3<br>(35/33) | |||
| [[Quadrant]] | |||
|} | |||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||