436edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|436}} | |||
== Theory == | == 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 | 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, harmonics from 3 to 37 are all flat. It 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. | ||
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 === | === Prime harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 436 factors into 2<sup>2</sup> × 109, 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 the 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|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]] (¢) | ||
| Line 79: | Line 79: | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per | ! Periods<br>per 8ve | ||
! Generator<br>( | ! Generator<br>(Reduced) | ||
! Cents<br>( | ! Cents<br>(Reduced) | ||
! Associated<br> | ! Associated<br>Ratio | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 103: | Line 103: | ||
| [[Quadrant]] | | [[Quadrant]] | ||
|} | |} | ||
Revision as of 07:15, 31 July 2023
| ← 435edo | 436edo | 437edo → |
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, harmonics from 3 to 37 are all flat. It tempers out 32805/32768 and [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.
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
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.12 | -0.99 | -0.02 | -0.86 | -1.08 | -0.37 | -0.27 | -0.75 | -0.22 | -0.08 |
| Relative (%) | +0.0 | -4.4 | -36.1 | -0.7 | -31.2 | -39.2 | -13.4 | -9.6 | -27.3 | -8.0 | -3.0 | |
| Steps (reduced) |
436 (0) |
691 (255) |
1012 (140) |
1224 (352) |
1508 (200) |
1613 (305) |
1782 (38) |
1852 (108) |
1972 (228) |
2118 (374) |
2160 (416) | |
Subsets and supersets
Since 436 factors into 22 × 109, 436edo has subset edos 2, 4, 109, and 218.
1308edo, which divides the edostep into three, is a zeta gap edo and is consistent in the 21-odd-limit.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-691 436⟩ | [⟨436 691]] | +0.0379 | 0.0379 | 1.38 |
| 2.3.5 | 32805/32768, [1 -68 46⟩ | [⟨436 691 1012]] | +0.1678 | 0.1863 | 6.77 |
| 2.3.5.7 | 32805/32768, 390625/388962, 420175/419904 | [⟨436 691 1012 1224]] | +0.1275 | 0.1758 | 6.39 |
| 2.3.5.7.11 | 1375/1372, 6250/6237, 32805/32768, 41503/41472 | [⟨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 | [⟨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 | [⟨436 691 1012 1224 1508 1613 1782]] | +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 | [⟨436 691 1012 1224 1508 1613 1782 1852]] | +0.1503 | 0.1443 | 5.24 |
Rank-2 temperaments
| Periods per 8ve |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 51\436 | 140.37 | 243/224 | Tsaharuk |
| 1 | 181\436 | 498.17 | 4/3 | Helmholtz |
| 4 | 181\436 (37\436) |
498.17 (101.83) |
4/3 (35/33) |
Quadrant |