148418edo
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This page presents a topic of primarily mathematical interest.
While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown. |
148418 equal divisions of the octave (abbreviated 148418edo or 148418ed2), also called 148418-tone equal temperament (148418tet) or 148418 equal temperament (148418et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 148418 equal parts of about 0.00809 ¢ each. Each step represents a frequency ratio of 21/148418, or the 148418th root of 2.
| ← 148417edo | 148418edo | 148419edo → |
148418edo is a very strong higher-limit tuning, distinctly consistent through the 39-odd-limit, and is a zeta peak edo. It marks the first equal temperament with lower relative errors after 16808 in the 23-, 29-, and 31-limit, the first after 83096 in the 37-limit, and the first after 95524 in the 41-limit. Some of the simpler commas it tempers out include 408761/408760, 453376/453375, 509796/509795, 601426/601425, 633556/633555, 709632/709631, 949026/949025, 1154440/1154439, 1163800/1163799, 1255501/1255500, 2023425/2023424, 2307361/2307360, 2697696/2697695, 3897166/3897165, 4096576/4096575, and 5909761/5909760.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00000 | +0.00029 | +0.00061 | -0.00002 | +0.00063 | +0.00112 | -0.00048 | +0.00076 | -0.00015 |
| Relative (%) | +0.0 | +3.6 | +7.6 | -0.3 | +7.8 | +13.8 | -6.0 | +9.4 | -1.8 | |
| Steps (reduced) |
148418 (0) |
235237 (86819) |
344616 (47780) |
416662 (119826) |
513442 (68188) |
549212 (103958) |
606653 (12981) |
630469 (36797) |
671378 (77706) | |
| Harmonic | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00143 | +0.00074 | +0.00283 | -0.00124 | -0.00293 | +0.00026 | -0.00403 | -0.00094 | -0.00141 |
| Relative (%) | +17.7 | +9.2 | +35.0 | -15.3 | -36.2 | +3.2 | -49.8 | -11.6 | -17.4 | |
| Steps (reduced) |
721012 (127340) |
735292 (141620) |
773177 (31087) |
795157 (53067) |
805355 (63265) |
824401 (82311) |
850126 (108036) |
873090 (131000) |
880228 (138138) | |
Subsets and supersets
Since 148418 factors into primes as 2 × 74209, 148418edo has subset edos 2 and 74209.