User:Francium/2039edo
Jump to navigation
Jump to search
Prime factorization
2039 (prime)
Step size
0.588524 ¢
Fifth
1193\2039 (702.109 ¢)
Semitones (A1:m2)
195:152 (114.8 ¢ : 89.46 ¢)
Consistency limit
3
Distinct consistency limit
3
| ← 2038edo | 2039edo | 2040edo → |
2039 equal divisions of the octave (abbreviated 2039edo or 2039ed2), also called 2039-tone equal temperament (2039tet) or 2039 equal temperament (2039et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2039 equal parts of about 0.589 ¢ each. Each step represents a frequency ratio of 21/2039, or the 2039th root of 2.
Theory
2039edo is only consistent to the 3-limit because of the high error of its harmonic 5. It is strong in the 2.15.21.13 subgroup, tempering out 2453371218821120/2451942503795547, 36000000000000000/35974211341046251 and 1546322941845703125/1544734672188080128.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.154 | -0.242 | -0.116 | -0.281 | +0.129 | -0.116 | -0.088 | -0.198 | +0.280 | +0.038 | +0.269 |
| Relative (%) | +26.1 | -41.1 | -19.7 | -47.7 | +21.9 | -19.7 | -15.0 | -33.7 | +47.6 | +6.5 | +45.7 | |
| Steps (reduced) |
3232 (1193) |
4734 (656) |
5724 (1646) |
6463 (346) |
7054 (937) |
7545 (1428) |
7966 (1849) |
8334 (178) |
8662 (506) |
8956 (800) |
9224 (1068) | |
Subsets and supersets
2039edo is the 309th prime edo. 6117edo, which triples it, gives a good correction to its harmonic 5.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [3232 -2039⟩ | [⟨2039 3232]] | −0.0485 | 0.0485 | 8.24 |