# 357edo

 ← 356edo 357edo 358edo →
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 357ed2), also called 357-tone equal temperament (357tet) or 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 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 (%) +0.0 +16.8 +7.2 -22.6 -1.7 -5.7 -22.4 +49.0 +8.8 -29.9 +35.2
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