# 348edo

 ← 347edo 348edo 349edo →
Prime factorization 22 × 3 × 29
Step size 3.44828¢
Fifth 204\348 (703.448¢) (→17\29)
Semitones (A1:m2) 36:24 (124.1¢ : 82.76¢)
Dual sharp fifth 204\348 (703.448¢) (→17\29)
Dual flat fifth 203\348 (700¢) (→7\12)
Dual major 2nd 59\348 (203.448¢)
Consistency limit 7
Distinct consistency limit 7

348 equal divisions of the octave (abbreviated 348edo or 348ed2), also called 348-tone equal temperament (348tet) or 348 equal temperament (348et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 348 equal parts of about 3.45 ¢ each. Each step represents a frequency ratio of 21/348, or the 348th root of 2.

## Theory

348et is consistent to the 7-odd-limit, but the error of the harmonic 3 is quite large, commending itself as a 2.9.5.7.11.13 subgroup temperament.

Using the patent val, it tempers out 2401/2400, 15625/15552, 390625/388962 and 156250000/155649627 and in the 7-limit. It supports quadritikleismic and subneutral.

### Odd harmonics

Approximation of odd harmonics in 348edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +1.49 -0.11 +0.14 -0.46 +0.41 +0.85 +1.39 -1.51 -0.96 +1.63 -0.69
Relative (%) +43.3 -3.1 +4.0 -13.4 +11.8 +24.7 +40.2 -43.7 -27.9 +47.4 -20.0
Steps
(reduced)
552
(204)
808
(112)
977
(281)
1103
(59)
1204
(160)
1288
(244)
1360
(316)
1422
(30)
1478
(86)
1529
(137)
1574
(182)

### Subsets and supersets

Since 348 factors into 22 × 3 × 29, 348edo has subset edos 2, 3, 4, 6, 12, 29, 58, 87, 116, and 174. 696edo, which doubles it, gives a good correction to the harmonic 3.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.9 [-1103 348 [348 1103]] +0.0728 0.0728 2.11
2.9.5 32805/32768, [7 52 -74 [348 1103 808]] +0.0639 0.0608 1.76
2.9.5.7 32805/32768, 250047/250000, [7 9 -2 -11 [348 1103 808 977]] +0.0355 0.0721 2.09
2.9.5.7.11 9801/9800, 32805/32768, 46656/46585, 151263/151250 [348 1103 808 977 1204]] +0.0049 0.0889 2.58
2.9.5.7.11.13 729/728, 1575/1573, 2200/2197, 32805/32768, 31250/31213 [348 1103 808 977 1204 1288]] -0.0343 0.1194 3.46