402edo
Theory
402et is only consistent to the 5-limit, tempering out the semicomma. There are three possible mappings in the 7-limit:
- ⟨402 637 933 1129] (patent val)
- ⟨402 637 934 1129] (402c)
- ⟨402 637 933 1128] (402d)
Using the patent val, it tempers out 4375/4374, 7381125/7340032 and 3200000/3176523 in the 7-limit. It supports ragismic and abigail.
Using the 402c val, it tempers out 3136/3125, 321489/320000 and 13060694016/12867859375 in the 7-limit. It supports bischismic and parahemwuer.
Using the 402d val, it tempers out 250047/250000, 1500625/1492992 and 2460375/2458624 in the 7-limit. It supports the landscape temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.46 | -1.24 | +1.32 | -0.92 | +0.92 | +1.26 | +1.28 | -0.48 | +0.99 | +0.86 | -1.41 |
| Relative (%) | -15.5 | -41.5 | +44.3 | -31.0 | +30.8 | +42.3 | +43.0 | -16.0 | +33.3 | +28.8 | -47.2 | |
| Steps (reduced) |
637 (235) |
933 (129) |
1129 (325) |
1274 (68) |
1391 (185) |
1488 (282) |
1571 (365) |
1643 (35) |
1708 (100) |
1766 (158) |
1818 (210) | |
Subsets and supersets
402 factors into 2 × 3 × 67, with subset edos 2, 3, 6, 67, 134, and 201. 804edo, which doubles it, gives a good correction to the harmonic 7.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-637 402⟩ | [⟨402 637]] | 0.1459 | 0.1459 | 4.89 |
| 2.3.5 | 2109375/2097152, [25 -48 22⟩ | [⟨402 637 933]] | 0.2752 | 0.2182 | 7.31 |
Rank-2 temperaments
| Periods per 8ve |
Generator (reduced)* |
Cents (reduced)* |
Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 91\402 | 271.64 | 75/64 | Orson |
| 1 | 115\402 | 343.28 | 8000/6561 | Raider |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct