1342edt
| ← 1341edt | 1342edt | 1343edt → |
1342 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 1342edt or 1342ed3), is a nonoctave tuning system that divides the interval of 3/1 into 1342 equal parts of about 1.42 ¢ each. Each step represents a frequency ratio of 31/1342, or the 1342nd root of 3.
1342edt provides a very good approximation to the no-twos 7-limit, with the 5th harmonic tuned 0.55% sharp (approximately 1/181 of a step), and the 7th harmonic tuned 1.29% flat (approximately 1/110 of a step). Even with this level of accuracy, it still supports the (relatively) simple temperament Izar, equating six intervals of 49/45 to 5/3. Despite the very good tuning of prime harmonics 3, 5 and 7, 1342edt misses the octave, 2/1, by 29% (larger than 1/4 of a step), making it incomparable with its related edos, 847edo and 848edo.
1342edt is a no-twos zeta peak and integer peak edt, and can represent prime harmonics from 3 up to 41 with less than 30% of error. It is consistent in the (no-twos) 27-odd-limit (with 29/19 being the first interval 1342edt fails to represent consistently) and in the (no-twos) no-29s 39-odd-limit (failing at 41/19). 1342edt can thus be seen as the edt counterpart to 1578edo, a zeta edo with similar size, approximation to primes, and consistency limit.
1342 factors as 2 × 11 × 61, so 1342edt has subset edts 2, 11, 22, 61, 122, and 671.
Harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.414 | +0.000 | +0.008 | -0.013 | -0.181 | -0.271 | +0.161 | +0.350 | -0.191 |
| Relative (%) | +29.2 | +0.0 | +0.6 | -0.9 | -12.7 | -19.1 | +11.4 | +24.7 | -13.5 | |
| Steps (reduced) |
847 (847) |
1342 (0) |
1966 (624) |
2377 (1035) |
2929 (245) |
3133 (449) |
3461 (777) |
3597 (913) |
3830 (1146) | |
| Harmonic | 25 | 27 | 29 | 31 | 33 | 35 | 37 | 39 | 41 | 43 | 45 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.016 | +0.000 | -0.411 | +0.345 | -0.181 | -0.005 | +0.164 | -0.271 | -0.398 | -0.652 | +0.008 |
| Relative (%) | +1.1 | +0.0 | -29.0 | +24.4 | -12.7 | -0.4 | +11.6 | -19.1 | -28.1 | -46.0 | +0.6 | |
| Steps (reduced) |
3932 (1248) |
4026 (0) |
4113 (87) |
4195 (169) |
4271 (245) |
4343 (317) |
4411 (385) |
4475 (449) |
4536 (510) |
4594 (568) |
4650 (624) | |