148edo: Difference between revisions
Adopt template: EDO intro (including getting rid of its approximation of the kleisma since it lacks significance anyway); redundant categories; misc. cleanup |
Re-analyse it as a 2.9.15.21 subgroup system |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro}} 148edo | {{EDO intro}} | ||
148edo's closest fifth is on the very sharp side, 3.45 cents sharp of just. With better approximations of [[9/1|9]], [[11/1|11]], [[15/1|15]], [[17/1|17]], and [[21/1|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 {{val| 148 235 344 416 }} 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- and 13- limit [[bidia]], the 12 & 68 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 11-limit [[echidnic]], the 10 & 46 temperament. | |||
=== Prime harmonics === | === Prime harmonics === | ||
Revision as of 14:43, 18 May 2024
| ← 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] 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- and 13- limit bidia, the 12 & 68 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 11-limit echidnic, the 10 & 46 temperament.
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.