96ed5: Difference between revisions

(4 intermediate revisions by 4 users not shown)

Line 6:

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 ==

Line 17:

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 optimized 96ed5}}

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.~~023056612872~~|columns=16|start=16|title=Approximation of harmonics in optimized 96ed5}}

{{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 ==

~~[[Category:Ed5]]~~

@@ Line 6: / Line 6: @@
 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 17: / 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 optimized 96ed5}}
+The local maxima for the finite Euler product over the primes 5.7.11.13.17.23 is 29.0283 cents.
-{{Harmonics in cet|29.023056612872|columns=16|start=16|title=Approximation of harmonics in optimized 96ed5}}
+{{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}}
-[[Category:Ed5]]
+{{todo|expand}}