# 354edo

 ← 353edo 354edo 355edo →
Prime factorization 2 × 3 × 59
Step size 3.38983¢
Fifth 207\354 (701.695¢) (→69\118)
Semitones (A1:m2) 33:27 (111.9¢ : 91.53¢)
Consistency limit 9
Distinct consistency limit 9

354 equal divisions of the octave (abbreviated 354edo or 354ed2), also called 354-tone equal temperament (354tet) or 354 equal temperament (354et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 354 equal parts of about 3.39 ¢ each. Each step represents a frequency ratio of 21/354, or the 354th root of 2.

## Theory

354edo is enfactored in the 5-limit, with the same tuning as 118edo, defined by tempering out the schisma and the parakleisma, but the approximation to higher harmonics are much improved.

In the 7-limit, the equal temperament tempers out 118098/117649 (stearnsma), 250047/250000 (landscape comma), and 703125/702464 (meter); in the 11-limit, 540/539, and 4000/3993; in the 13-limit, 729/728, 1575/1573, 1716/1715, 2080/2079, 4096/4095, and 4225/4224. In the 13-limit, particularly 2.3.5.13 subgroup, one should consider peithoian, as it preserves 5-limit tuning of 118edo while also improving the first harmonic 118edo tunes inconsistently.

354edo provides the optimal patent val for stearnscape, the 72 & 282 temperament, and 13- and 17-limit terminator, the 171 & 183 temperament.

### Prime harmonics

Approximation of prime harmonics in 354edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.26 +0.13 +0.67 +1.22 +0.15 +0.13 +0.79 -1.16 +0.93 +0.73
Relative (%) +0.0 -7.7 +3.7 +19.6 +36.1 +4.4 +3.8 +23.4 -34.1 +27.5 +21.5
Steps
(reduced)
354
(0)
561
(207)
822
(114)
994
(286)
1225
(163)
1310
(248)
1447
(31)
1504
(88)
1601
(185)
1720
(304)
1754
(338)

### Subsets and supersets

Since 354 factors into 2 × 3 × 59, 354edo has subset edos 2, 3, 6, 59, 118, and 177.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3.5.7 32805/32768, 118098/117649, 250047/250000 [354 561 822 994]] -0.0319 0.1432 4.23
2.3.5.7.11 540/539, 4000/3993, 32805/32768, 137781/137500 [354 561 822 994 1225]] -0.0963 0.1817 5.36
2.3.5.7.11.13 540/539, 729/728, 1575/1573, 4096/4095, 31250/31213 [354 561 822 994 1225 1310]] -0.0871 0.1671 4.93
2.3.5.7.11.13.17 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 4096/4095 [354 561 822 994 1225 1310 1447]] -0.0791 0.1559 4.60
2.3.5.7.11.13.17.19 540/539, 729/728, 936/935, 969/968, 1156/1155, 1445/1444, 1521/1520 [354 561 822 994 1225 1310 1447 1504]] -0.0926 0.1509 4.43

### Rank-2 temperaments

Note: 5-limit temperaments supported by 118et are not included.

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
2 128\354
(49\354)
433.90
(166.10)
9/7
(11/10)
Pogo
3 147\354
(29\354)
498.31
(98.31)
4/3
(18/17)
Term / terminator
6 64\354
(5\354)
216.95
(16.95)
17/15
(245/243)
Stearnscape
6 147\354
(29\354)
498.31
(98.31)
4/3
(18/17)
Semiterm
118 167\354
(2\354)
566.101
(6.78)
165/119
(?)
Oganesson

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