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.

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.

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.

Additionally, 96ed5 is related to 186zpi.

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

== Harmonic series ==

== 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.
+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.
+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.
-Additionally, 96ed5 is related to 186zpi.
+Additionally, 96ed5 is related to [[186zpi]].
 == Harmonic series ==
 {{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}}