351edo
| ← 350edo | 351edo | 352edo → |
351 equal divisions of the octave (abbreviated 351edo or 351ed2), also called 351-tone equal temperament (351tet) or 351 equal temperament (351et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 351 equal parts of about 3.42 ¢ each. Each step represents a frequency ratio of 21/351, or the 351st root of 2.
Theory
351et is consistent to the 7-odd-limit with a reasonable approximation to the 11-limit. The equal temperament tempers out 19683/19600, 65625/65536, and 235298/234375 in the 7-limit; 441/440, 24057/24010, 35937/35840, 41503/41472, 43923/43904, and 46656/46585 in the 11-limit. It supports snape.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.10 | +0.01 | -1.30 | +1.22 | -0.89 | +0.50 | -1.09 | +1.03 | -0.08 | +1.01 | +0.79 |
| Relative (%) | -32.2 | +0.3 | -38.2 | +35.6 | -26.0 | +14.6 | -31.9 | +30.1 | -2.3 | +29.7 | +23.0 | |
| Steps (reduced) |
556 (205) |
815 (113) |
985 (283) |
1113 (60) |
1214 (161) |
1299 (246) |
1371 (318) |
1435 (31) |
1491 (87) |
1542 (138) |
1588 (184) | |
Subsets and supersets
351 factors into 33 × 13 with subset edos 3, 9, 13, 27, 39, and 117.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-556 351⟩ | [⟨351 556]] | 0.3471 | 0.3472 | 10.16 |
| 2.3.5 | [-36 11 8⟩, [-11 26 -13⟩ | [⟨351 556 815]] | 0.2298 | 0.3284 | 9.61 |
| 2.3.5.7 | 19683/19600, 65625/65536, 235298/234375 | [⟨351 556 815 985]] | 0.2885 | 0.3021 | 8.84 |
| 2.3.5.7.11 | 441/440, 19683/19600, 35937/35840, 65625/65536 | [⟨351 556 815 985 1214]] | 0.2823 | 0.2705 | 7.91 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 116\351 | 396.58 | 98304/78125 | Squarschmidt |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct