343edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 342edo343edo344edo →
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 343-tone equal temperament (343tet), 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 of 343edo 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 (%) +36 -42 +8 -28 +41 -25 -6 +0 -4 +44 +42
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