342edo
The 342 equal divisions of the octave (342edo), or the 342(-tone) equal temperament (342tet, 342et) when viewed from a regular temperament perspective, is the equal division of the octave into 342 parts of 3.50877 cents each.
Theory
342edo is a very strong 11-limit system. It is, as one would expect, distinctly consistent through the 11-odd-limit, but goes no higher; nonetheless, it is a zeta peak edo. A basis for the 11-limit commas is 2401/2400, 3025/3024, 4375/4374 and 32805/32768. It is the optimal patent val for 11-limit hemitert temperament, and supports hemiennealimmal.
342 factors as 2 × 32 × 19, with subset edos 2, 3, 6, 9, 18, 19, 38, 57, 114, and 171.
Prime harmonics
Script error: No such module "primes_in_edo".
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5.7.11 | 2401/2400, 3025/3024, 4375/4374, 32805/32768 | [⟨342 542 794 960 1183]] | +0.110 | 0.0556 | 1.59 |
| 2.3.5.7.11.13 | 676/675, 1001/1000, 1716/1715, 3025/3024, 32805/32768 | [⟨342 542 794 960 1183 1265]] (342f) | +0.178 | 0.1618 | 4.61 |
| 2.3.5.7.11.13 | 625/624, 729/728, 847/845, 1575/1573, 4096/4095 | [⟨342 542 794 960 1183 1266]] (342) | +0.020 | 0.2061 | 5.87 |
- 342et is lower in relative error than any previous ETs in the 11-limit. Not until 612 do we find a better ET in terms of absolute error, and not until 1848 do we find one in terms of relative error.