33616edo
| ← 33615edo | 33616edo | 33617edo → |
33616 equal divisions of the octave (abbreviated 33616edo or 33616ed2), also called 33616-tone equal temperament (33616tet) or 33616 equal temperament (33616et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 33616 equal parts of about 0.0357 ¢ each. Each step represents a frequency ratio of 21/33616, or the 33616th root of 2.
33616edo is distinctly consistent to the 39-odd-limit. Doubling the monstrous 16808edo, it corrects the approximated 37th harmonic to near-just quality while maintaining the same tuning in the lower limits. Unfortunately, harmonic 41 is now tuned too off to be consistent, but on the plus side, 43 and 47 are tuned way better. In fact, it is excellent in the no-41 51-odd-limit, and the full 47-limit interpretation using the patent val is also obvious, as 41/40, 41/25, 41/29 and their octave complements exhaust all the inconsistently mapped intervals in the full 51-odd-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00000 | -0.00355 | +0.00233 | -0.00154 | -0.00904 | +0.00066 | -0.00539 | -0.01183 | -0.00210 |
| Relative (%) | +0.0 | -9.9 | +6.5 | -4.3 | -25.3 | +1.8 | -15.1 | -33.1 | -5.9 | |
| Steps (reduced) |
33616 (0) |
53280 (19664) |
78054 (10822) |
94372 (27140) |
116292 (15444) |
124394 (23546) |
137404 (2940) |
142798 (8334) |
152064 (17600) | |
| Harmonic | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00396 | -0.00939 | +0.00056 | -0.01671 | -0.01128 | -0.00210 | +0.00807 | +0.00254 | +0.01191 |
| Relative (%) | +11.1 | -26.3 | +1.6 | -46.8 | -31.6 | -5.9 | +22.6 | +7.1 | +33.4 | |
| Steps (reduced) |
163306 (28842) |
166540 (32076) |
175121 (7041) |
180099 (12019) |
182409 (14329) |
186723 (18643) |
192550 (24470) |
197751 (29671) |
199368 (31288) | |
Subsets and supersets
Since 33616 factors into primes as 24 × 11 × 191, 33616edo has subset edos 2, 4, 8, 11, 16, 22, 44, 88, 176, 191, 382, 764, 1528, 2101, 3056, 4202, 8404, and 16808.