402edo
| ← 401edo | 402edo | 403edo → |
402 equal divisions of the octave (abbreviated 402edo or 402ed2), also called 402-tone equal temperament (402tet) or 402 equal temperament (402et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 402 equal parts of about 2.99 ¢ each. Each step represents a frequency ratio of 21/402, or the 402nd root of 2.
Theory
402edo is only consistent to the 5-odd-limit. There are three possible mappings in the 7-limit:
- ⟨402 637 933 1129] (patent val)
- ⟨402 637 933 1128] (402d)
- ⟨402 637 934 1129] (402c)
Using the patent val, it tempers out the semicomma in the 5-limit; 4375/4374, 7381125/7340032 and 3200000/3176523 in the 7-limit. It supports abigail.
Using the 402d val, it tempers out 250047/250000, 1500625/1492992 and 2460375/2458624 in the 7-limit.
Using the 402c val, it tempers out the schisma in the 5-limit; and 3136/3125, 321489/320000 and 13060694016/12867859375 in the 7-limit. It supports bischismic.
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 | -42 | +44 | -31 | +31 | +42 | +43 | -16 | +33 | +29 | -47 | |
| 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
Since 402 factors into 2 × 3 × 67, 402edo has subset edos 2, 3, 6, 67, 134, and 201. 804edo, which doubles it, gives a good correction to the harmonics 5 and 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* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 91\402 | 271.64 | 75/64 | Orson (402) |
| 1 | 115\402 | 343.28 | 8000/6561 | Raider (402) |
| 2 | 70\402 | 208.96 | 44/39 | Abigail (402) |
| 2 | 167\402 (34\402) |
498.51 (101.49) |
4/3 (200/189) |
Bischismic (402c, 7-limit) |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct