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 2^1/348, 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	+4	-13	+12	+25	+40	-44	-28	+47	-20
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