96ed5 - Revision history

BudjarnLambeth: /* Theory */

2026-02-28T09:15:13Z

Theory

← Older revision		Revision as of 09:15, 28 February 2026
Line 8:		Line 8:
	This system can be approximated as 41.34495 EDO, meaning each step of 96ed5 corresponds roughly to three steps of [[124edo]], or [[124ed8]].		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]].		Additionally, 96ed5 is related to [[186zpi]].

2026-01-27T04:15:26Z

Theory: link 124ed8

← Older revision		Revision as of 04:15, 27 January 2026
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 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.

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

	== Harmonic series ==		== Harmonic series ==

2025-08-01T18:39:07Z

2025-07-05T22:09:13Z

← Older revision		Revision as of 22:09, 5 July 2025
Line 17:		Line 17:

	== Optimization ==		== 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 ==		== Intervals ==

2024-12-22T04:13:32Z

2024-10-06T01:34:19Z

Infobox, intro

← Older revision		Revision as of 01:34, 6 October 2024
Line 1:		Line 1:
			{{Infobox ET}}
			{{ED intro}}

	== Theory ==		== 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 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.

2024-10-06T01:33:34Z

Intervals

← Older revision		Revision as of 01:33, 6 October 2024
Line 19:		Line 19:
	{{Harmonics in cet\|29.023056612872\|columns=15\|title=Approximation of harmonics in optimized 96ed5}}		{{Harmonics in cet\|29.023056612872\|columns=15\|title=Approximation of harmonics in optimized 96ed5}}
	{{Harmonics in cet\|29.023056612872\|columns=16\|start=16\|title=Approximation of harmonics in optimized 96ed5}}		{{Harmonics in cet\|29.023056612872\|columns=16\|start=16\|title=Approximation of harmonics in optimized 96ed5}}

			== Intervals ==
			{{Interval table}}

	[[Category:Ed5]]		[[Category:Ed5]]

2024-09-11T21:13:04Z

Optimization

← Older revision		Revision as of 21:13, 11 September 2024
Line 17:		Line 17:
	== Optimization ==		== 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.		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=15\|title=Approximation of harmonics in optimized 96ed5}}
	{{Harmonics in cet\|29.023056612872\|columns=16\|start=16\|title=Approximation of harmonics in ~~186zpi~~}}		{{Harmonics in cet\|29.023056612872\|columns=16\|start=16\|title=Approximation of harmonics in optimized 96ed5}}

	[[Category:Ed5]]		[[Category:Ed5]]

2024-09-11T21:11:52Z

← Older revision		Revision as of 21:11, 11 September 2024
Line 17:		Line 17:
	== Optimization ==		== 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.		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]]		[[Category:Ed5]]

2024-09-11T20:50:59Z

Optimization

← Older revision		Revision as of 20:50, 11 September 2024
Line 16:		Line 16:

	== Optimization ==		== Optimization ==
	In the 32-integer-limit and 5.7.11.13.17.23 subgroup, the lowest error is 41.346437627379-edo, or 41<1189.94532112775>, or 29.023056612872 cents.		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.
	[[Category:Ed5]]		[[Category:Ed5]]