357edo

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← 356edo357edo358edo →
Prime factorization 3 × 7 × 17
Step size 3.36134¢
Fifth 209\357 (702.521¢)
Semitones (A1:m2) 35:26 (117.6¢ : 87.39¢)
Consistency limit 7
Distinct consistency limit 7

357 equal divisions of the octave (abbreviated 357edo), or 357-tone equal temperament (357tet), 357 equal temperament (357et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 357 equal parts of about 3.36 ¢ each. Each step of 357edo represents a frequency ratio of 21/357, or the 357th root of 2.

Theory

While not highly accurate for its size, 357et is the point where a few important temperaments meet. The equal temperament tempers out 1600000/1594323 (amity comma), and [61 4 -29 (squarschimidt comma) in the 5-limit; 10976/10935 (hemimage comma), 235298/234375 (triwellisma), 250047/250000 (landscape comma), 2100875/2097152 (rainy comma) in the 7-limit; 3025/3024, 5632/5625, 12005/11979 in the 11-limit; 676/675, 1001/1000, 2080/2079, 4096/4095, 4225/4224, 6656/6655 and 10648/10647 in the 13-limit.

It supports 5-limit amity and 7-limit weak extensions calamity and chromat. It provides the optimal patent val for 11- and 13-limit hemichromat, the 159 & 198 temperament. It also supports avicenna, but 270edo is better suited for this purpose.

Prime harmonics

Approximation of prime harmonics in 357edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error absolute (¢) +0.00 +0.57 +0.24 -0.76 -0.06 -0.19 -0.75 +1.65 +0.30 -1.01 +1.18
relative (%) +0 +17 +7 -23 -2 -6 -22 +49 +9 -30 +35
Steps
(reduced)
357
(0)
566
(209)
829
(115)
1002
(288)
1235
(164)
1321
(250)
1459
(31)
1517
(89)
1615
(187)
1734
(306)
1769
(341)

Subsets and supersets

Since 357 factors into 3 × 7 × 17, 357edo has subset edos 3, 7, 17, 21, 51, and 119.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [566 -357 [357 566]] -0.1786 0.1785 5.31
2.3.5 1600000/1594323, [61 4 -29 [357 566 829]] -0.1536 0.1500 4.46
2.3.5.7 10976/10935, 235298/234375, 2100875/2097152 [357 566 829 1002]] -0.0477 0.2248 6.69
2.3.5.7.11 3025/3024, 5632/5625, 10976/10935, 102487/102400 [357 566 829 1002 1235]] -0.0348 0.2027 6.03
2.3.5.7.11.13 676/675, 1001/1000, 3025/3024, 4096/4095, 10976/10935 [357 566 829 1002 1235 1321]] -0.0204 0.1879 5.59

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 101\357 339.50 243/200 Amity (5-limit)
1 118\357 396.64 44/35 Squarschmidt
1 163\357 547.90 48/35 Calamity
3 9\357 30.25 55/54 Hemichromat
3 18\357 60.50 28/27 Chromat (7-limit)
3 41\357 137.82 13/12 Avicenna
3 48\357 161.34 192/175 Pnict

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