412edo
| ← 411edo | 412edo | 413edo → |
Theory
412et tempers out 2460375/2458624, 6144/6125, 102760448/102515625, 1640558367/1638400000 and 200120949/200000000 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 | -0.5 | +36.6 | +37.0 | -28.6 | +41.9 | -3.5 | -14.6 | +29.2 | -48.8 | -12.9 | |
| 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 the harmonic 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, 100442349/100000000 | [⟨412 653 957 1157]] | -0.2085 | 0.2143 | 7.36 |
Rank-2 temperaments
| Periods per 8ve |
Generator (reduced)* |
Cents (reduced)* |
Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 9\412 | 26.21 | 49/48 | Sfourth |
| 1 | 19\412 | 55.34 | 16875/16384 | Escapade |
| 1 | 171\412 | 498.06 | 4/3 | Counterschismic |
| 2 | 19\412 | 55.34 | 16875/16384 | Semisuperfourth |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct