96ed5: Difference between revisions

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== Theory ==

~~96ed5 is an equal-step tuning system created by dividing the interval of 5/1 into 96 equal parts.~~

This non-octave, non-tritave scale features a well-balanced [[harmonic series segment]] from 5 to 9, and performs exceptionally well across all [[prime harmonics]] from 5 to 23, with the exception of 19.

This non-octave, non-tritave scale features a well-balanced harmonic series segment from 5 to 9, and performs exceptionally well across all prime harmonics from 5 to 23, with the exception of 19.

~~This system can be approximated as 41.34495 EDO, meaning each step of 96ed5 corresponds roughly to three steps of 124edo~~.

~~96ed5 sets a height record on the Riemann zeta function with primes 2 and 3 removed, approximating~~ 41.~~3478 EDO. This record remains unbeaten until approximately 98.62575~~ EDO.

This system can be approximated as 41.34495 EDO, meaning each step of 96ed5 corresponds roughly to three steps of [[124edo]], or [[124ed8]].

~~Additionally~~, ~~96ed5 is related to 186zpi~~.

96ed5 sets a height record on the [[The Riemann zeta function and tuning|Riemann zeta function]] with [[The Riemann zeta function and tuning#Removing primes|primes 2 and 3 removed]], approximating 41.3478 EDO. This record remains unbeaten until approximately 98.62575 EDO (~[[229ed5]]).

Additionally, 96ed5 is related to [[186zpi]].

== Harmonic series ==

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== Optimization ==

The local maxima for the finite Euler product over the primes 5.7.11.13.17.23 is 29.0283 cents.

{{Harmonics in cet|29.0283|columns=15|title=Approximation of harmonics in optimized 96ed5}}

{{Harmonics in cet|29.0283|columns=16|start=16|title=Approximation of harmonics in optimized 96ed5}}

== Intervals ==

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+{{Infobox ET}}
+{{ED intro}}
 == Theory ==
-ed5 is an equal-step tuning system created by dividing the interval of 5/1 into 96 equal parts.
+This non-octave, non-tritave scale features a well-balanced [[harmonic series segment]] from 5 to 9, and performs exceptionally well across all [[prime harmonics]] from 5 to 23, with the exception of 19.
-This non-octave, non-tritave scale features a well-balanced harmonic series segment from 5 to 9, and performs exceptionally well across all prime harmonics from 5 to 23, with the exception of 19.
-This system can be approximated as 41.34495 EDO, meaning each step of 96ed5 corresponds roughly to three steps of 124edo.
-ed5 sets a height record on the Riemann zeta function with primes 2 and 3 removed, approximating 41.3478 EDO. This record remains unbeaten until approximately 98.62575 EDO.
+This system can be approximated as 41.34495 EDO, meaning each step of 96ed5 corresponds roughly to three steps of [[124edo]], or [[124ed8]].
-Additionally, 96ed5 is related to 186zpi.
+ed5 sets a height record on the [[The Riemann zeta function and tuning|Riemann zeta function]] with [[The Riemann zeta function and tuning#Removing primes|primes 2 and 3 removed]], approximating 41.3478 EDO. This record remains unbeaten until approximately 98.62575 EDO (~[[229ed5]]).
+Additionally, 96ed5 is related to [[186zpi]].
 == Harmonic series ==
@@ Line 16: / Line 16: @@
 {{Harmonics in equal|96|5|1|prec=1|columns=15}}
 {{Harmonics in equal|96|5|1|prec=1|columns=16|start=16}}
+== Optimization ==
+The local maxima for the finite Euler product over the primes 5.7.11.13.17.23 is 29.0283 cents.
+{{Harmonics in cet|29.0283|columns=15|title=Approximation of harmonics in optimized 96ed5}}
+{{Harmonics in cet|29.0283|columns=16|start=16|title=Approximation of harmonics in optimized 96ed5}}
+== Intervals ==
+{{Interval table}}
+{{todo|expand}}