# 342edo

 ← 341edo 342edo 343edo →
Prime factorization 2 × 32 × 19
Step size 3.50877¢
Fifth 200\342 (701.754¢) (→100\171)
Semitones (A1:m2) 32:26 (112.3¢ : 91.23¢)
Consistency limit 11
Distinct consistency limit 11
Special properties

342 equal divisions of the octave (abbreviated 342edo or 342ed2), also called 342-tone equal temperament (342tet) or 342 equal temperament (342et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 342 equal parts of about 3.51 ¢ each. Each step represents a frequency ratio of 21/342, or the 342nd root of 2.

## 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 consists of 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

Approximation of prime harmonics in 342edo
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)

### Subset and supersets

342 factors as 2 × 32 × 19, with subset edos 2, 3, 6, 9, 18, 19, 38, 57, 114, and 171.

684edo, which doubles 342edo, provides an approximation of harmonic 13 that works well with the flat tendency of its 11-limit mapping.

## 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, being the first to beat 270. 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

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* 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 97\342
(17\342)
340.35
(59.65)
162/133
(88/85)
Semiseptichrome
6 142\342
(28\342)
498.25
(98.25)
4/3
(18/17)
Semiterm
9 63\342
(13\342)
221.05
(45.61)
25/22
(77/75)
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)