333edo
| ← 332edo | 333edo | 334edo → |
333 equal divisions of the octave (abbreviated 333edo or 333ed2), also called 333-tone equal temperament (333tet) or 333 equal temperament (333et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 333 equal parts of about 3.6 ¢ each. Each step represents a frequency ratio of 21/333, or the 333rd root of 2.
The equal temperament tempers out 15625/15552 in the 5-limit and 5120/5103 in the 7-limit, so it supports countercata. In the 11-limit it tempers out 1375/1372 and 4000/3993, and in the 13-limit 325/324, 364/363, 625/624 and 676/675, and provides the optimal patent val for the rank-2 temperament novemkleismic, for the rank-3 temperament tempering out 325/324, 625/624 and 676/675, the rank-4 temperament tempering out 325/324 and 1375/1372, and the rank-5 temperament tempering out 325/324.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00 | +0.75 | -0.73 | +0.54 | +0.03 | -0.89 | -0.45 | +1.59 | -1.25 | +1.05 | +0.91 |
| relative (%) | +0 | +21 | -20 | +15 | +1 | -25 | -13 | +44 | -35 | +29 | +25 | |
| Steps (reduced) |
333 (0) |
528 (195) |
773 (107) |
935 (269) |
1152 (153) |
1232 (233) |
1361 (29) |
1415 (83) |
1506 (174) |
1618 (286) |
1650 (318) | |
Subsets and supersets
Since 333 factors into 32 × 37, 333edo has subset edos 3, 9, 37, and 111.