4380edo
| ← 4379edo | 4380edo | 4381edo → |
4380 equal divisions of the octave (abbreviated 4380edo or 4380ed2), also called 4380-tone equal temperament (4380tet) or 4380 equal temperament (4380et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 4380 equal parts of about 0.274 ¢ each. Each step represents a frequency ratio of 21/4380, or the 4380th root of 2.
4380edo is consistent in the 31-odd-limit and holds the record of lowest relative error in the 47-limit, being only beaten by 7361o-edo. It is closely related to 2190edo, inheriting the same excellent tuning in the 2.3.5.7.11.13.19.29 subgroup while improving the mapping for many other primes.
Some of the simpler commas tempered out include 31213/31212 in the 17-limit, 23409/23408 in the 19-limit, 10625/10625 in the 23-limit, 7425/7424 in the 29-limit, and 6138/6137 in the 31-limit.
In light of having 60 as a divisor, 4380edo is a tuning for the neodymium temperament in the 17-limit. It is worth noting that 4380edo tempers out the magnetisma on its 43-limit patent val, and therefore tunes the extension neodymium magnet.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.037 | -0.012 | -0.059 | -0.085 | +0.020 | -0.024 | +0.021 | -0.055 |
| Relative (%) | +0.0 | -13.6 | -4.5 | -21.5 | -31.0 | +7.4 | -8.7 | +7.7 | -20.1 | |
| Steps (reduced) |
4380 (0) |
6942 (2562) |
10170 (1410) |
12296 (3536) |
15152 (2012) |
16208 (3068) |
17903 (383) |
18606 (1086) |
19813 (2293) | |
| Harmonic | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.012 | -0.104 | -0.111 | -0.021 | -0.011 | -0.027 | -0.080 | +0.006 | +0.101 |
| Relative (%) | +4.3 | -38.0 | -40.6 | -7.8 | -4.0 | -9.9 | -29.2 | +2.3 | +37.0 | |
| Steps (reduced) |
21278 (3758) |
21699 (4179) |
22817 (917) |
23466 (1566) |
23767 (1867) |
24329 (2429) |
25088 (3188) |
25766 (3866) |
25977 (4077) | |
Subsets and supersets
4380edo has subset edos 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60, 73, 146, 219, 292, 365, 438, 730, 876, 1095, 1460, 2190. One step of 4380edo is one sixth of a Woolhouse unit (1\730).