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 about 3.51 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.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.20 | -0.35 | -0.40 | -0.44 | +1.58 | +0.31 | +0.73 | -0.20 | -1.51 | -1.18 |
| Relative (%) | +0.0 | -5.7 | -9.9 | -11.5 | -12.6 | +45.0 | +8.8 | +20.9 | -5.8 | -43.0 | -33.5 | |
| Steps (reduced) |
342 (0) |
542 (200) |
794 (110) |
960 (276) |
1183 (157) |
1266 (240) |
1398 (30) |
1453 (85) |
1547 (179) |
1661 (293) |
1694 (326) | |
Miscellany
342 factors as 2 × 32 × 19, with subset edos 2, 3, 6, 9, 18, 19, 38, 57, 114, and 171.
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, 19773/19712 | [⟨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 equal temperaments in the 11-limit. Not until 612 do we find a better equal temperament in terms of absolute error, and not until 1848 do we find one in terms of relative error.
Rank-2 temperaments
| Periods per 8ve |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 11\342 | 38.60 | 45/44 | Hemitert |
| 2 | 5\342 | 17.54 | 99/98 | Poseidon |
| 2 | 50\342 | 175.44 | 448/405 | Bisesqui |
| 2 | 124\342 (47\342) |
435.09 (164.91) |
9/7 (11/10) |
Semisupermajor |
| 2 | 142\342 (29\342) |
498.25 (101.75) |
4/3 (35/33) |
Bipont |
| 3 | 71\342 (43\342) |
249.12 (150.88) |
15/13 (12/11) |
Hemiterm |
| 6 | 142\342 (28\342) |
498.25 (98.25) |
4/3 (200/189) |
Semiterm |
| 9 | 63\342 (13\342) |
221.05 (45.61) |
25/22 (77/75) |
Quadraennealimmal |
| 18 | 71\342 (5\342) |
249.12 (17.54) |
15/13 (99/98) |
Hemiennealimmal |
| 38 | 142\342 (2\342) |
498.25 (7.02) |
4/3 (225/224) |
Hemienneadecal |