User:Francium/3919edo
| ← 3918edo | 3919edo | 3920edo → |
3919 equal divisions of the octave (abbreviated 3919edo or 3919ed2), also called 3919-tone equal temperament (3919tet) or 3919 equal temperament (3919et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3919 equal parts of about 0.306 ¢ each. Each step represents a frequency ratio of 21/3919, or the 3919th root of 2.
Theory
3919edo is only consistent to the 3-limit and its harmonic 3 is about halfway its steps. It is a strong 2.9.15.7.13.23 subgroup tuning, tempering out 76545/76544, 287500/287469, 66560000/66548223, 2951690625/2951578112 and 743115202375/743008370688.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.143 | +0.111 | -0.007 | +0.020 | +0.149 | -0.007 | -0.032 | +0.071 | +0.114 | -0.151 | +0.049 |
| Relative (%) | -46.8 | +36.4 | -2.4 | +6.4 | +48.7 | -2.3 | -10.4 | +23.3 | +37.2 | -49.2 | +16.1 | |
| Steps (reduced) |
6211 (2292) |
9100 (1262) |
11002 (3164) |
12423 (666) |
13558 (1801) |
14502 (2745) |
15311 (3554) |
16019 (343) |
16648 (972) |
17213 (1537) |
17728 (2052) | |
Subsets and supersets
3919edo is the 543rd prime edo. 7838edo, which doubles it, gives a good correction to its harmonic 3.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [12423 -3919⟩ | [⟨3919 12423]] | −0.0031 | 0.0031 | 1.01 |
| 2.9.5 | [146 -38 -11⟩, [105 35 -93⟩ | [⟨3919 12423 9100]] | −0.0181 | 0.0213 | 6.96 |
| 2.9.5.7 | 4398046511104/4395357421875, 96889010407/96855122250, 40041913260899432169/40000000000000000000 | [⟨3919 12423 9100 11002]] | −0.0129 | 0.0205 | 6.69 |