148edo: Difference between revisions
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{{Infobox ET}} | |||
{{ED 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''' }} (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. | |||
148 = 4 | === Harmonics === | ||
{{Harmonics in equal|148|columns=9}} | |||
{{Harmonics in equal|148|columns=9|start=10|title=Approximation of odd harmonics in 148edo (continued)}} | |||
=== Subsets and supersets === | |||
Since 148 = 4 × 37, 148edo has subset edos {{EDOs| 2, 4, 37, and 74 }}. | |||
[[Category:Echidnic]] | |||
[[Category:Bidia]] | |||
Latest revision as of 14:56, 25 October 2025
| ← 147edo | 148edo | 149edo → |
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.
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.
Harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.45 | +2.88 | -3.96 | -1.21 | +0.03 | +2.72 | -1.78 | +0.45 | +2.49 |
| Relative (%) | +42.6 | +35.5 | -48.9 | -14.9 | +0.4 | +33.5 | -22.0 | +5.5 | +30.7 | |
| Steps (reduced) |
235 (87) |
344 (48) |
415 (119) |
469 (25) |
512 (68) |
548 (104) |
578 (134) |
605 (13) |
629 (37) | |
| Harmonic | 21 | 23 | 25 | 27 | 29 | 31 | 33 | 35 | 37 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.51 | -3.95 | -2.36 | +2.24 | +0.15 | -1.79 | +3.48 | -1.09 | +0.01 |
| Relative (%) | -6.3 | -48.7 | -29.1 | +27.7 | +1.9 | -22.1 | +43.0 | -13.4 | +0.1 | |
| Steps (reduced) |
650 (58) |
669 (77) |
687 (95) |
704 (112) |
719 (127) |
733 (141) |
747 (7) |
759 (19) |
771 (31) | |
Subsets and supersets
Since 148 = 4 × 37, 148edo has subset edos 2, 4, 37, and 74.