8269edo: Difference between revisions
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Revision as of 04:17, 9 July 2023
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This page presents a novelty topic.
It may contain ideas which are less likely to find practical applications in music, or numbers or structures that are arbitrary or exceedingly small, large, or complex. Novelty topics are often developed by a single person or a small group. As such, this page may also contain idiosyncratic terms, notation, or conceptual frameworks. |
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| ← 8268edo | 8269edo | 8270edo → |
8269edo is both a zeta peak and zeta integral edo, which has to do with the fact that it is a very strong 19- and 23-limit division. It has a lower 19-limit and a lower 23-limit relative error than any smaller division, a lower 19-limit TE loglfat badness than any smaller division, and a lower 23-limit logflat badness than any excepting 311, 581, 1578 and 2460. While 8539 has received most of the attention in this size range, 8269 is actually a bit better in the 23-limit and nearly as good in the 19-limit. They are rather like twins, including the fact both are primes. A step of 8269edo has also been similarly proposed as an interval size measure, the major tina.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | -0.0080 | -0.0034 | -0.0026 | -0.0058 | +0.0093 | -0.0334 | -0.0163 | -0.0484 | +0.0515 | -0.0362 |
| Relative (%) | +0.0 | -5.5 | -2.3 | -1.8 | -4.0 | +6.4 | -23.0 | -11.3 | -33.4 | +35.5 | -24.9 | |
| Steps (reduced) |
8269 (0) |
13106 (4837) |
19200 (2662) |
23214 (6676) |
28606 (3799) |
30599 (5792) |
33799 (723) |
35126 (2050) |
37405 (4329) |
40171 (7095) |
40966 (7890) | |
Subsets and supersets
8269edo is the 1037th prime edo.