User:Francium/3217edo
| ← 3216edo | 3217edo | 3218edo → |
3217 equal divisions of the octave (abbreviated 3217edo or 3217ed2), also called 3217-tone equal temperament (3217tet) or 3217 equal temperament (3217et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3217 equal parts of about 0.373 ¢ each. Each step represents a frequency ratio of 21/3217, or the 3217th root of 2.
Theory
3217edo is only consistent to the 5-limit. The errors of the lower harmonics are high, except for its 11th harmonic with a relative error of only 0.8 percent.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.066 | +0.133 | -0.097 | +0.131 | +0.003 | -0.117 | -0.174 | -0.137 | +0.156 | -0.032 | -0.111 |
| Relative (%) | +17.6 | +35.7 | -26.1 | +35.1 | +0.8 | -31.5 | -46.7 | -36.8 | +41.7 | -8.5 | -29.9 | |
| Steps (reduced) |
5099 (1882) |
7470 (1036) |
9031 (2597) |
10198 (547) |
11129 (1478) |
11904 (2253) |
12568 (2917) |
13149 (281) |
13666 (798) |
14130 (1262) |
14552 (1684) | |
Subsets and supersets
3217edo is the 455th prime edo. 16085edo, which quintuples it, gives a good correction to its harmonic 7.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [5099 -3217⟩ | [⟨3217 5099]] | −0.0207 | 0.0207 | 5.55 |
| 2.3.5 | [77 -31 -12⟩, [5 95 -67⟩ | [⟨3217 5099 7470]] | −0.0329 | 0.0242 | 6.49 |