158edo
| ← 157edo | 158edo | 159edo → |
158 equal divisions of the octave (abbreviated 158edo or 158ed2), also called 158-tone equal temperament (158tet) or 158 equal temperament (158et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 158 equal parts of about 7.59 ¢ each. Each step represents a frequency ratio of 21/158, or the 158th root of 2.
158edo is inconsistent to the 5-odd-limit and higher limits. It can be treated as a 2.9.5.21.33.39 subgroup temperament, which is every other step of 316edo, and tempers out the same commas: 1716/1715, 2080/2079, 3025/3024, 3136/3125, 4096/4095, 4225/4224, 9801/9800, etc.
Otherwise, it has five mappings possible for the 13-limit: ⟨158 250 367 444 547 585] (patent val), ⟨158 251 367 444 547 585] (158b), ⟨158 250 366 443 546 584] (158cdef), ⟨158 250 367 443 546 585] (158de), and ⟨158 250 367 443 546 584] (158def).
Using the patent val, it tempers out the Würschmidt comma, 393216/390625 and 3486784401/3355443200 in the 5-limit; 225/224, 8748/8575, and 40960000/40353607 in the 7-limit; 441/440, 1375/1372, 4000/3993, and 19683/19208 in the 11-limit, supporting the 11-limit marvolo temperament and giving a good tuning; 144/143, 640/637, 2025/2002, 2200/2197 and 3159/3125 in the 13-limit.
Using the 158de val, it tempers out 126/125, 33075/32768, and 118098/117649 in the 7-limit; 243/242, 385/384, 1617/1600, and 117649/117128 in the 11-limit; 196/195, 351/350, 1287/1280, 1575/1573, and 4455/4394 in the 13-limit. Using the 158def val, it tempers out 676/675, 847/845, 1573/1568, 1701/1690, and 3159/3136 in the 13-limit.
Using the 158cdef val, it tempers out the magic comma, 3125/3072 and the python comma, 43046721/41943040 in the 5-limit; 2401/2400, 19683/19600, and 78125/76832 in the 7-limit; 243/242, 441/440, 4000/3993, and 33275/32768 in the 11-limit; 325/324, 975/968, 1287/1280, 1573/1568, and 1875/1859 in the 13-limit, supporting the 13-limit harry temperament.
Using the 158b val, it tempers out the diaschisma, 2048/2025 and [-1 -33 23⟩ in the 5-limit; 245/243, 6144/6125 and 2500000/2470629 in the 7-limit; 1331/1323, 1375/1372, 2560/2541, and 4375/4356 in the 11-limit; 364/363, 572/567, 625/624, 640/637, and 1625/1617 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -3.22 | +1.03 | +3.33 | +1.15 | +3.11 | +2.51 | -2.19 | +1.37 | -1.31 | +0.11 | +2.11 |
| Relative (%) | -42.4 | +13.5 | +43.8 | +15.2 | +41.0 | +33.1 | -28.9 | +18.1 | -17.3 | +1.4 | +27.7 | |
| Steps (reduced) |
250 (92) |
367 (51) |
444 (128) |
501 (27) |
547 (73) |
585 (111) |
617 (143) |
646 (14) |
671 (39) |
694 (62) |
715 (83) | |
Subsets and supersets
Since 158 factors into 2 × 79, 158edo contains 2edo and 79edo as its subsets. 316edo, which doubles it, provides good correction to the approximation to harmonics 3, 7, and 11.