8539edo: Difference between revisions
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{{EDO intro|8539}} | {{EDO intro|8539}} | ||
While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a [[interval size measure|unit of interval measure]], the '''tina'''. This is because it is a very strong higher-limit system, distinctly [[consistent]] through the 27-odd-limit. It is a [[ | While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a [[interval size measure|unit of interval measure]], the '''tina'''. This is because it is a very strong higher-limit system, distinctly [[consistent]] through the 27-odd-limit. It is a [[the Riemann zeta function and tuning #Zeta EDO lists|strict zeta]] tuning, and is also the first [[trivial temperament|non-trivial]] edo to be consistent in the 27-[[odd prime sum limit|odd-prime-sum-limit]]. In the 13-limit, the only smaller systems with a lower logflat badness are {{EDOs| 72, 270, 494, 5585 and 6079 }}; in the 17-limit, that becomes {{EDOs| 72, 494, 1506, 3395 and 7033 }}. In the 19-limit, where it really shines, nothing beats it in terms of logflat badness until [[20203edo|20203]]. | ||
Some of the simpler commas it [[tempering out|tempers out]] include [[123201/123200]] in the 13-limit; [[28561/28560]], [[31213/31212]], [[37180/37179]] in the 17-limit; 27456/27455, 43681/43680, 89376/89375 in the 19-limit; 19551/19550, 21736/21735, 25025/25024, 43264/43263 among others in the 23-limit. | |||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|8539}} | {{Harmonics in equal|8539|columns=12}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Revision as of 13:11, 22 August 2024
| ← 8538edo | 8539edo | 8540edo → |
While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a unit of interval measure, the tina. This is because it is a very strong higher-limit system, distinctly consistent through the 27-odd-limit. It is a strict zeta tuning, and is also the first non-trivial edo to be consistent in the 27-odd-prime-sum-limit. In the 13-limit, the only smaller systems with a lower logflat badness are 72, 270, 494, 5585 and 6079; in the 17-limit, that becomes 72, 494, 1506, 3395 and 7033. In the 19-limit, where it really shines, nothing beats it in terms of logflat badness until 20203.
Some of the simpler commas it tempers out include 123201/123200 in the 13-limit; 28561/28560, 31213/31212, 37180/37179 in the 17-limit; 27456/27455, 43681/43680, 89376/89375 in the 19-limit; 19551/19550, 21736/21735, 25025/25024, 43264/43263 among others in the 23-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | +0.0007 | +0.0079 | -0.0005 | -0.0122 | -0.0077 | +0.0218 | -0.0075 | +0.0428 | -0.0421 | +0.0165 | +0.0671 |
| Relative (%) | +0.0 | +0.5 | +5.6 | -0.4 | -8.7 | -5.5 | +15.5 | -5.3 | +30.4 | -30.0 | +11.8 | +47.8 | |
| Steps (reduced) |
8539 (0) |
13534 (4995) |
19827 (2749) |
23972 (6894) |
29540 (3923) |
31598 (5981) |
34903 (747) |
36273 (2117) |
38627 (4471) |
41482 (7326) |
42304 (8148) |
44484 (1789) | |
Subsets and supersets
8539edo is the 1065th prime edo. On that basis, the tina as a unit of measure could be criticized; however, some people prefer primes for this sort of job, as they do not imply a preference for one smaller edo over another.