8539edo: Difference between revisions

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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''' (see [http://www.tonalsoft.com/enc/t/tina.aspx Tonalsoft Encyclopedia | ''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 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 27456/27455 and 43681/43680.

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''' (see [http://www.tonalsoft.com/enc/t/tina.aspx Tonalsoft Encyclopedia | ''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 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 27456/27455 and 43681/43680.

=== Prime harmonics ===

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 {{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''' (see [http://www.tonalsoft.com/enc/t/tina.aspx Tonalsoft Encyclopedia | ''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 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 27456/27455 and 43681/43680.
+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''' (see [http://www.tonalsoft.com/enc/t/tina.aspx Tonalsoft Encyclopedia | ''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 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 27456/27455 and 43681/43680.
 === Prime harmonics ===