96ed5: Difference between revisions

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== Theory ==

~~96ed5 is an [[Equal-step tuning|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 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]], or [[124ed8]].

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.

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 ==

~~In the 32-integer-limit and 5.7.11.13.17.23 subgroup, the lowest relative error is 41.346437627379-edo, or 41<1189.94532112775>, or 29.023056612872 cents.~~

~~{{Harmonics in cet|29.023056612872|columns=15|title=Approximation of harmonics in 186zpi}}~~

~~{{Harmonics in cet|29.023056612872|columns=16|start=16|title=Approximation of harmonics in 186zpi}}~~

~~[[Category:Ed5]]~~

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 ==

@@ Line 1: / Line 1: @@
+{{Infobox ET}}
+{{ED intro}}
 == Theory ==
-ed5 is an [[Equal-step tuning|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 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]], or [[124ed8]].
-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.
+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 18: @@
 == Optimization ==
-In the 32-integer-limit and 5.7.11.13.17.23 subgroup, the lowest relative error is 41.346437627379-edo, or 41<1189.94532112775>, or 29.023056612872 cents.
-{{Harmonics in cet|29.023056612872|columns=15|title=Approximation of harmonics in 186zpi}}
-{{Harmonics in cet|29.023056612872|columns=16|start=16|title=Approximation of harmonics in 186zpi}}
-[[Category:Ed5]]
+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}}