432edo
| ← 431edo | 432edo | 433edo → |
432 equal divisions of the octave (abbreviated 432edo or 432ed2), also called 432-tone equal temperament (432tet) or 432 equal temperament (432et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 432 equal parts of about 2.78 ¢ each. Each step represents a frequency ratio of 21/432, or the 432nd root of 2.
Theory
432edo has a reasonable approximation to the 7-limit, where the equal temperament tempers out 5120/5103, 703125/702464, 40353607/40310784, 102760448/102515625, and 283115520/282475249. It supports and provides a good tuning for the 5-limit maja temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.82 | -0.20 | +0.62 | -1.13 | -1.32 | +1.14 | +0.62 | +0.60 | -0.29 | -1.34 | -0.50 |
| Relative (%) | +29.6 | -7.3 | +22.3 | -40.8 | -47.4 | +41.0 | +22.3 | +21.6 | -10.5 | -48.1 | -17.9 | |
| Steps (reduced) |
685 (253) |
1003 (139) |
1213 (349) |
1369 (73) |
1494 (198) |
1599 (303) |
1688 (392) |
1766 (38) |
1835 (107) |
1897 (169) |
1954 (226) | |
Subsets and supersets
432 is a highly factorable number, factoring into 24 × 33, so 432edo has subset edos 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72, 108, 144, and 216.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [685 -432⟩ | [⟨432 685]] | −0.2596 | 0.2595 | 9.34 |
| 2.3.5 | [41 -20 -4⟩, [-3 -23 17⟩ | [⟨432 685 1003]] | −0.1440 | 0.2676 | 9.63 |
| 2.3.5.7 | 5120/5103, 703125/702464, 6565234375/6530347008 | [⟨432 685 1003 1213]] | −0.1631 | 0.2341 | 8.43 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 163\432 | 452.78 | 125/96 | Maja |
| 4 | 179\432 (37\432) |
497.22 (102.78) |
4/3 | Undim |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct