User:Francium/4243edo
| ← 4242edo | 4243edo | 4244edo → |
4243 equal divisions of the octave (abbreviated 4243edo or 4243ed2), also called 4243-tone equal temperament (4243tet) or 4243 equal temperament (4243et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 4243 equal parts of about 0.283 ¢ each. Each step represents a frequency ratio of 21/4243, or the 4243rd root of 2.
Theory
4243edo is consistent to the 9-odd-limit, tempering out 78125000/78121827, [46 -14 -3 -6⟩ and [-14 26 4 -13⟩ in the 7-limit. Its harmonic 3 has a remarkably low relative error of 0.4 percent. As an equal temperament, 4243edo is strong in the 2.3.5.13.17.19 subgroup, tempering out 104976/104975, 28900/28899, 19922944/19921875, 8450649/8450000 and 18051780123/18050000000. It also tempers out the lakisma in the 2.3.5.17.19 subgroup and 7956/7955 in the 2.3.5.13.17.37.43 subgroup. 4243edo supports whoosh.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.001 | +0.017 | +0.111 | -0.104 | +0.010 | -0.030 | +0.012 | -0.134 | -0.117 | +0.098 |
| Relative (%) | +0.0 | +0.4 | +5.9 | +39.3 | -36.8 | +3.4 | -10.5 | +4.4 | -47.3 | -41.3 | +34.5 | |
| Steps (reduced) |
4243 (0) |
6725 (2482) |
9852 (1366) |
11912 (3426) |
14678 (1949) |
15701 (2972) |
17343 (371) |
18024 (1052) |
19193 (2221) |
20612 (3640) |
21021 (4049) | |
Subsets and supersets
4243edo is the 582nd prime edo. 8486edo, which doubles it, gives a good correction to the harmonic 11.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [6725 -4243⟩ | [⟨4243 6725]] | −0.0004 | 0.0004 | 0.14 |
| 2.3.5 | [37 25 -33⟩, [215 -121 -10⟩ | [⟨4243 6725 9852]] | −0.0026 | 0.0032 | 1.31 |
| 2.3.5.7 | 78125000/78121827, [46 -14 -3 -6⟩, [-14 26 4 -13⟩ | [⟨4243 6725 9852 11912]] | −0.0119 | 0.0162 | 5.73 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 1982\4243 | 560.5468 | 864/625 | Whoosh |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct