412edo
| ← 411edo | 412edo | 413edo → |
412 equal divisions of the octave (abbreviated 412edo or 412ed2), also called 412-tone equal temperament (412tet) or 412 equal temperament (412et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 412 equal parts of about 2.91 ¢ each. Each step represents a frequency ratio of 21/412, or the 412th root of 2.
Theory
412edo has a very accurate perfect fifth, but it is not quite accurate beyond that. The equal temperament tempers out [32 -7 -9⟩ (escapade comma) and [-69 45 -1⟩ (counterschisma) in the 5-limit; 6144/6125, 118098/117649, 2460375/2458624, 49009212/48828125, and notably the nanisma in the 7-limit. It supports nanic and counterschismic.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00 | -0.01 | +1.06 | +1.08 | -0.83 | +1.22 | -0.10 | -0.43 | +0.85 | -1.42 | -0.38 |
| relative (%) | +0 | -0 | +37 | +37 | -29 | +42 | -3 | -15 | +29 | -49 | -13 | |
| Steps (reduced) |
412 (0) |
653 (241) |
957 (133) |
1157 (333) |
1425 (189) |
1525 (289) |
1684 (36) |
1750 (102) |
1864 (216) |
2001 (353) |
2041 (393) | |
Subsets and supersets
412 factors into 22 × 103, with subset edos 2, 4, 103, and 206. 1236edo, which triples it, gives a good correction to harmonics 5, 7, and 11.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-653 412⟩ | [⟨412 653]] | +0.0042 | 0.0042 | 0.14 |
| 2.3.5 | [32 -7 -9⟩, [-5 31 -19⟩ | [⟨412 653 957]] | -0.1501 | 0.2182 | 7.49 |
| 2.3.5.7 | 6144/6125, 2460375/2458624, 49009212/48828125 | [⟨412 653 957 1157]] | -0.2085 | 0.2143 | 7.36 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 9\412 | 26.21 | 49/48 | Sfourth (5-limit) |
| 1 | 19\412 | 55.34 | 16875/16384 | Escapade (5-limit) |
| 1 | 171\412 | 498.06 | 4/3 | Counterschismic Nanic |
| 2 | 19\412 | 55.34 | 16875/16384 | Septisuperfourth (7-limit) |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct