354edo
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, it 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
| 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.
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 2 | 49\354 | 166.10 | 11/10 | Pogo |
| 3 | 29\354 | 98.31 | 18/17 | Term / terminator |
| 6 | 5\354 | 16.95 | 245/243 | Stearnscape |
| 6 | 21\354 | 71.19 | 25/24 | Trivish |
| 6 | 29\354 | 98.31 | 18/17 | Semiterm |
| 118 | 2\354 | 6.78 | ? | Oganesson |