1749edo
1749 equal divisions of the octave (abbreviated 1749edo or 1749ed2), also called 1749-tone equal temperament (1749tet) or 1749 equal temperament (1749et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1749 equal parts of about 0.686 ¢ each. Each step represents a frequency ratio of 21/1749, or the 1749th root of 2.
Theory
1749edo is consistent in the 9-odd-limit with strong approximations; however the approximation to the 11th harmonic is poor. The most straightforward extension is the 2.3.5.7.13.17.31 subgroup.
It is part of the optimal ET sequence for the aemilic temperament in the 7-limit, as a multiple of 159edo.
1749e val is the most straightforward extension into the 11-limit, as it is better tuned than the patent val and strongly supports 11-limit aemilic extension.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.068 | -0.036 | -0.044 | +0.312 | -0.047 | +0.019 | +0.257 | +0.199 | +0.268 | +0.076 |
| Relative (%) | +0.0 | -9.9 | -5.2 | -6.4 | +45.4 | -6.9 | +2.7 | +37.5 | +29.0 | +39.1 | +11.1 | |
| Steps (reduced) |
1749 (0) |
2772 (1023) |
4061 (563) |
4910 (1412) |
6051 (804) |
6472 (1225) |
7149 (153) |
7430 (434) |
7912 (916) |
8497 (1501) |
8665 (1669) | |