8539edo: Difference between revisions
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{{EDO intro|8539}} While it may strike many people as too large to be practical, it 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]] (see [http://www.tonalsoft.com/enc/t/tina.aspx http://www.tonalsoft.com/enc/t/tina.aspx]). This is because it is a very strong higher-limit system, distinctly [[consistent]] through the 27-odd-limit, and is both a [[The Riemann zeta function and Tuning #Zeta EDO lists|zeta peak, zeta integral, and zeta gap]] tuning. 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 17-limit commas it tempers out are 28561/28560, 31213/31212 and 37180/37179; in the 19-limit it tempers out are 27456/27455 and 43681/43680. 8539 is a prime number, and the tina as a unit of measure could be criticized on that basis; however some people prefer primes for this sort of job, as they do not imply a preference for one smaller edo over another. | |||
=== Prime harmonics === | === Prime harmonics === | ||
Revision as of 11:35, 10 February 2023
| ← 8538edo | 8539edo | 8540edo → |
Template:EDO intro While it may strike many people as too large to be practical, it 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 (see http://www.tonalsoft.com/enc/t/tina.aspx). This is because it is a very strong higher-limit system, distinctly consistent through the 27-odd-limit, and is both a zeta peak, zeta integral, and zeta gap tuning. 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 17-limit commas it tempers out are 28561/28560, 31213/31212 and 37180/37179; in the 19-limit it tempers out are 27456/27455 and 43681/43680. 8539 is a prime number, and the tina as a unit of measure could be criticized on that basis; however some people prefer primes for this sort of job, as they do not imply a preference for one smaller edo over another.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | +0.0007 | +0.0079 | -0.0005 | -0.0122 | -0.0077 | +0.0218 | -0.0075 | +0.0428 | -0.0421 | +0.0165 |
| Relative (%) | +0.0 | +0.5 | +5.6 | -0.4 | -8.7 | -5.5 | +15.5 | -5.3 | +30.4 | -30.0 | +11.8 | |
| Steps (reduced) |
8539 (0) |
13534 (4995) |
19827 (2749) |
23972 (6894) |
29540 (3923) |
31598 (5981) |
34903 (747) |
36273 (2117) |
38627 (4471) |
41482 (7326) |
42304 (8148) | |