# 343edo

 ← 342edo 343edo 344edo →
Prime factorization 73
Step size 3.49854¢
Fifth 201\343 (703.207¢)
Semitones (A1:m2) 35:24 (122.4¢ : 83.97¢)
Dual sharp fifth 201\343 (703.207¢)
Dual flat fifth 200\343 (699.708¢)
Dual major 2nd 58\343 (202.915¢)
Consistency limit 3
Distinct consistency limit 3

343 equal divisions of the octave (abbreviated 343edo or 343ed2), also called 343-tone equal temperament (343tet) or 343 equal temperament (343et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 343 equal parts of about 3.5 ¢ each. Each step represents a frequency ratio of 21/343, or the 343rd root of 2.

## Theory

343edo is only consistent to the 3-odd-limit since its errors of harmonics 3 and 5 are quite large. To start with, consider the 2.9.15.7 subgroup, where it tempers out 5250987/5242880. In the 2.5.7 subgroup it tempers out 2100875/2097152 and in the 2.3.7 subgroup it tempers out 118098/117649.

For the full 7-limit, the 343c val tempers out 4375/4374 and 5120/5103, supporting amity. The 343cdd val tempers out 16875/16807 and 65536/64827. The patent val tempers out 10976/10935 and 390625/387072.

### Odd harmonics

Approximation of odd harmonics in 343edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +1.25 -1.47 +0.27 -0.99 +1.45 -0.88 -0.22 +0.00 -0.14 +1.52 +1.46
Relative (%) +35.8 -42.1 +7.7 -28.4 +41.5 -25.1 -6.3 +0.0 -3.9 +43.5 +41.8
Steps
(reduced)
544
(201)
796
(110)
963
(277)
1087
(58)
1187
(158)
1269
(240)
1340
(311)
1402
(30)
1457
(85)
1507
(135)
1552
(180)

### Subsets and supersets

Since 343 factors into 73, 343edo has 7edo and 49edo as its subsets. 686edo, which doubles it, gives a good correction to the harmonics 3 and 5.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.9 [-1087 343 [343 1087]] 0.1569 0.1569 4.48
2.9.5 [-27 -1 13, [40 -28 21 [343 1087 796]] 0.3162 0.2592 7.41
2.9.5.7 118098/117649, 7381125/7340032, 9765625/9680832 [343 1087 796 963]] 0.2130 0.2869 8.20