351edo

← 350edo

351edo

352edo →

Prime factorization	33 × 13
Step size	3.4188¢
Fifth	205\351 (700.855¢)
Semitones (A1:m2)	31:28 (106¢ : 95.73¢)
Consistency limit	7
Distinct consistency limit	7

351 equal divisions of the octave (abbreviated 351edo or 351ed2), also called 351-tone equal temperament (351tet) or 351 equal temperament (351et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 351 equal parts of about 3.42 ¢ each. Each step represents a frequency ratio of 2^1/351, or the 351st root of 2.

Theory

351et is consistent to the 7-odd-limit with a reasonable approximation to the 11-limit. The equal temperament tempers out 19683/19600, 65625/65536, and 235298/234375 in the 7-limit; 441/440, 24057/24010, 35937/35840, 41503/41472, 43923/43904, and 46656/46585 in the 11-limit. It supports snape.

Odd harmonics

Approximation of odd harmonics in 351edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	-1.10	+0.01	-1.30	+1.22	-0.89	+0.50	-1.09	+1.03	-0.08	+1.01	+0.79
	relative (%)	-32	+0	-38	+36	-26	+15	-32	+30	-2	+30	+23
Steps (reduced)		556 (205)	815 (113)	985 (283)	1113 (60)	1214 (161)	1299 (246)	1371 (318)	1435 (31)	1491 (87)	1542 (138)	1588 (184)

Subsets and supersets

351 factors into 3³ × 13 with subset edos 3, 9, 13, 27, 39, and 117.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[-556 351⟩	[⟨351 556]]	0.3471	0.3472	10.16
2.3.5	[-36 11 8⟩, [-11 26 -13⟩	[⟨351 556 815]]	0.2298	0.3284	9.61
2.3.5.7	19683/19600, 65625/65536, 235298/234375	[⟨351 556 815 985]]	0.2885	0.3021	8.84
2.3.5.7.11	441/440, 19683/19600, 35937/35840, 65625/65536	[⟨351 556 815 985 1214]]	0.2823	0.2705	7.91

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperaments
1	116\351	396.58	98304/78125	Squarschmidt

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct