3323edo
| ← 3322edo | 3323edo | 3324edo → |
3323 equal divisions of the octave (abbreviated 3323edo or 3323ed2), also called 3323-tone equal temperament (3323tet) or 3323 equal temperament (3323et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3323 equal parts of about 0.361 ¢ each. Each step represents a frequency ratio of 21/3323, or the 3323rd root of 2.
Theory
3323edo is consistent to the 25-odd-limit, tempering out the following commas:
13-limit: 4096/4095, 6656/6655, 105644/105625, 1574640/1574573, 21437500/21434787
17-limit: 4096/4095, 12376/12375, 37180/37179, 194481/194480 25510625/25509168
19-limit: 4096/4095, 12376/12375, 14400/14399, 104272/104247, 104976/104975, 37180/37179, 203148/203125
23-limit: 4096/4095, 12376/12375, 14400/14399, 21736/21735, 10626/10625, 104272/104247, 104976/104975, 203148/203125
Using the 2.3.5.7.11.23.31 subgroup, it tempers out 19251/19250. As an equal temperament, 3323edo supports the septendecimal jacobin temperament.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.061 | +0.084 | +0.058 | +0.111 | +0.158 | +0.130 | +0.049 | +0.074 | -0.026 | +0.074 |
| Relative (%) | +0.0 | +17.0 | +23.3 | +16.0 | +30.9 | +43.9 | +36.1 | +13.7 | +20.4 | -7.1 | +20.6 | |
| Steps (reduced) |
3323 (0) |
5267 (1944) |
7716 (1070) |
9329 (2683) |
11496 (1527) |
12297 (2328) |
13583 (291) |
14116 (824) |
15032 (1740) |
16143 (2851) |
16463 (3171) | |
Subsets and supersets
3323edo is the 468th prime edo.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [5267 -3323⟩ | [⟨3323 5267]] | -0.0193 | 0.0193 | 0.70 |
| 2.3.5 | [38 -2 -15⟩, [151 -211 79⟩ | [⟨3323 5267 7716]] | -0.0250 | 0.0177 | 0.64 |
| 2.3.5.7 | [3 -13 10 -2⟩, [41 -15 -5 -2⟩, [-1 -18 -3 13⟩ | [⟨3323 5267 7716 9329]] | -0.0239 | 0.0154 | 0.56 |
| 2.3.5.7.11 | 151263/151250, 21437500/21434787, 2097152/2096325, 1395654656/1395032625 | [⟨3323 5267 7716 9329 11496]] | -0.0255 | 0.0142 | 0.51 |
| 2.3.5.7.11.13 | 4096/4095, 6656/6655, 105644/105625, 1574640/1574573, 21437500/21434787 | [⟨3323 5267 7716 9329 11496 12297]] | -0.0284 | 0.0145 | 0.52 |
| 2.3.5.7.11.13.17 | 4096/4095, 12376/12375, 14400/14399, 37180/37179, 194481/194480, 25510625/25509168 | [⟨3323 5267 7716 9329 11496 12297 13583]] | -0.0289 | 0.0135 | 0.49 |