User:Francium/2423edo
| ← 2422edo | 2423edo | 2424edo → |
2423 equal divisions of the octave (abbreviated 2423edo or 2423ed2), also called 2423-tone equal temperament (2423tet) or 2423 equal temperament (2423et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2423 equal parts of about 0.495 ¢ each. Each step represents a frequency ratio of 21/2423, or the 2423rd root of 2.
Theory
2423edo is consistent to the 7-limit, although the error of its harmonic 3 is quite high. As an equal temperament, it tempers out 160083/160000, 2460375/2458624, 3750705/3748096 and 107495424/107421875 in the 11-limit and 67392/67375, 160083/160000, 50193/50176, 91125/91091 and 3750705/3748096 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.180 | -0.016 | -0.109 | +0.135 | -0.100 | -0.082 | -0.196 | +0.038 | +0.135 | +0.205 | +0.203 |
| Relative (%) | -36.4 | -3.2 | -22.1 | +27.2 | -20.3 | -16.5 | -39.6 | +7.8 | +27.2 | +41.5 | +40.9 | |
| Steps (reduced) |
3840 (1417) |
5626 (780) |
6802 (1956) |
7681 (412) |
8382 (1113) |
8966 (1697) |
9466 (2197) |
9904 (212) |
10293 (601) |
10643 (951) |
10961 (1269) | |
Subsets and supersets
2423edo is the 360th prime edo. 4846edo, which doubles it, gives a good correction to its harmonic 3, but is only consistent to the 5-limit. To ensure the consistency of 2423edo, 7269edo, which triples it, is a better choice.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [7681 -2423⟩ | [⟨2423 7681]] | −0.0212 | 0.0212 | 4.28 |
| 2.9.5 | [-13 -23 37⟩, [222 -40 -41⟩ | [⟨2423 7681 5626]] | −0.0119 | 0.0218 | 4.40 |
| 2.9.5.7 | 420175/419904, [44 -14 5 -4⟩, [39 -3 -26 11⟩ | [⟨2423 7681 5626 6802]] | +0.0008 | 0.0290 | 5.86 |
| 2.9.5.7.11 | 420175/419904, 151263/151250, 2791309312/2790703125, 4194304000000/4193866120059 | [⟨2423 7681 5626 6802 8382]] | +0.0065 | 0.0283 | 5.71 |
| 2.9.5.7.11.13 | 10648/10647, 1146880/1146717, 105644/105625, 1890625/1889568, 167518208/167403915 | [⟨2423 7681 5626 6802 8382 8966]] | +0.0091 | 0.0265 | 5.35 |
| 2.9.5.7.11.13.17 | 12376/12375, 131648/131625, 194481/194480, 1146880/1146717, 4685824/4685625, 2127125/2125764 | [⟨2423 7681 5626 6802 8382 8966 9904]] | +0.0064 | 0.0254 | 5.13 |