351edo

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← 350edo351edo352edo →
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 21/351, 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 (%) -32.2 +0.3 -38.2 +35.6 -26.0 +14.6 -31.9 +30.1 -2.3 +29.7 +23.0
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