1637edo
| ← 1636edo | 1637edo | 1638edo → |
1637 equal divisions of the octave (abbreviated 1637edo or 1637ed2), also called 1637-tone equal temperament (1637tet) or 1637 equal temperament (1637et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1637 equal parts of about 0.733 ¢ each. Each step represents a frequency ratio of 21/1637, or the 1637th root of 2.
Theory
1637edo is consistent to the 7-odd-limit, but the error of its harmonic 3 is quite large. Using the 2.9.5.7.11.13.17.19.23 subgroup, it tempers out 4096/4095, 67392/67375, 14400/14399, 6175/6174, 11016/11011, 1863/1862, 3060/3059 and 152361/152320. In the 2.5.11.17.23.43 subgroup it tempers out 10880/10879.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.305 | +0.003 | +0.264 | -0.123 | -0.066 | +0.279 | +0.308 | -0.130 | +0.105 | -0.164 | -0.052 |
| Relative (%) | +41.6 | +0.4 | +36.0 | -16.7 | -9.0 | +38.0 | +42.0 | -17.7 | +14.3 | -22.4 | -7.1 | |
| Steps (reduced) |
2595 (958) |
3801 (527) |
4596 (1322) |
5189 (278) |
5663 (752) |
6058 (1147) |
6396 (1485) |
6691 (143) |
6954 (406) |
7190 (642) |
7405 (857) | |
Subsets and supersets
1637edo is the 259th prime edo. 3274edo, which doubles it, gives a good correction to the harmonic 3.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [-5189 1637⟩ | [⟨1637 5189]] | +0.0193 | 0.0193 | 2.63 |
| 2.9.5 | [-53 5 16⟩, [-56 77 -81⟩ | [⟨1637 5189 3801]] | +0.0125 | 0.0185 | 2.52 |
| 2.9.5.7 | [-7 -2 13 -6⟩, [-24 12 0 -5⟩, [22 5 -3 -11⟩ | [⟨1637 5189 3801 4596]] | -0.0141 | 0.0488 | 6.66 |
| 2.9.5.7.11 | 2359296/2358125, 820125/819896, 50014503/50000000, 275653125/275365888 | [⟨1637 5189 3801 4596 5663]] | -0.0075 | 0.0456 | 6.22 |
| 2.9.5.7.11.13 | 4096/4095, 67392/67375, 3720087/3718000, 225000/224939, 6125625/6117748 | [⟨1637 5189 3801 4596 5663 6058]] | -0.0188 | 0.0487 | 6.64 |