8539edo: Difference between revisions

Line 3:

== Theory ==

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]].

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, [[consistency|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 [[odd prime sum limit|27-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; 12168/12167, 16929/16928, 19551/19550, 21736/21735, 25025/25024, 43264/43263 among others in the 23-limit.

Line 10:

=== Prime harmonics ===

{{Harmonics in equal|8539|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 8539edo (continued)}}

=== Subsets and supersets ===

@@ Line 3: / Line 3: @@
 == Theory ==
-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]].
+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, [[consistency|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 [[odd prime sum limit|27-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; 12168/12167, 16929/16928, 19551/19550, 21736/21735, 25025/25024, 43264/43263 among others in the 23-limit.
@@ Line 10: / Line 10: @@
 === Prime harmonics ===
-{{Harmonics in equal|8539|columns=12}}
+{{Harmonics in equal|8539|columns=9}}
+{{Harmonics in equal|8539|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 8539edo (continued)}}
 === Subsets and supersets ===