148edo
148 equal divisions of the octave (abbreviated 148edo or 148ed2), also called 148-tone equal temperament (148tet) or 148 equal temperament (148et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 148 equal parts of about 8.11 ¢ each. Each step represents a frequency ratio of 21/148, or the 148th root of 2.
| ← 147edo | 148edo | 149edo → |
148edo's closest fifth is on the very sharp side, 3.45 cents sharp of just. With better approximations of 9, 11, 15, 17, and 21, it commends itself as a 2.9.15.21.11.17 subgroup system.
The 5-limit patent val still makes sense, and it tempers out 2048/2025, making it a diaschismic system. In the 7-limit, the patent val tempers out 686/675 and 1029/1024, but the alternative mapping ⟨148 235 344 416] (148d) with a sharp rather than a flat 7 tempers out 3136/3125 instead, and provides a better tuning than the patent val tuning of 80edo for 7-, 13-, 17- and 19-limit bidia, the 68 & 80 temperament. In the 11-limit, the patent val tempers out 385/384 and 441/440, and the alternative mapping with the sharp 7 tempers out 176/175, 896/891 and 1375/1372 instead. In the 13-limit, the patent val tempers out 325/324 and 364/363, and the alternative val 325/324 again, as well as 640/637 and 847/845. It provides the optimal patent val for echidnic, the 46 & 102 temperament, in the 11-limit, and the 148f val is an excellent tuning for echidnic in the 13- and 17-limit.
Prime harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.45 | +2.88 | -3.96 | -1.21 | +0.03 | +2.72 | -1.78 | +0.45 | +2.49 | -0.51 | -3.95 |
| Relative (%) | +42.6 | +35.5 | -48.9 | -14.9 | +0.4 | +33.5 | -22.0 | +5.5 | +30.7 | -6.3 | -48.7 | |
| Steps (reduced) |
235 (87) |
344 (48) |
415 (119) |
469 (25) |
512 (68) |
548 (104) |
578 (134) |
605 (13) |
629 (37) |
650 (58) |
669 (77) | |
Subsets and supersets
Since 148 = 4 × 37, 148edo has subset edos 2, 4, 37, and 74.