436edo
| ← 435edo | 436edo | 437edo → |
436 equal divisions of the octave (abbreviated 436edo or 436ed2), also called 436-tone equal temperament (436tet) or 436 equal temperament (436et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 436 equal parts of about 2.75 ¢ each. Each step represents a frequency ratio of 21/436, or the 436th root of 2.
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.
The equal temperament 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. It supports 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
| 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 | -4 | -36 | -1 | -31 | -39 | -13 | -10 | -27 | -8 | -3 | |
| 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 its 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* | Cents* | 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 |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct