204ed96 - Revision history

ArrowHead294 at 15:03, 16 July 2025

2025-07-16T15:03:30Z

← Older revision		Revision as of 15:03, 16 July 2025
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	== Theory ==		== Theory ==
	The 96th harmonic is too wide to be a useful equivalence, so 204ed96 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 96/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.79{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 31 is located at 30.978382, which has a step size of 38.737{{c}} and an octave of 1200.837{{c}} (which is stretched by 0.837{{c}}), making 204ed96 extremely close to optimal for 31edo.		The 96th harmonic is far too wide to be a useful equivalence, so 204ed96 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 96/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.79{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 31 is located at 30.978382, which has a step size of 38.737{{c}} and an octave of 1200.837{{c}} (which is stretched by 0.837{{c}}), making 204ed96 extremely close to optimal for 31edo.

	=== Harmonics ===		=== Harmonics ===

ArrowHead294 at 15:02, 16 July 2025

2025-07-16T15:02:13Z

← Older revision		Revision as of 15:02, 16 July 2025
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	{{Harmonics in equal\|204\|96\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 204ed96 (continued)}}		{{Harmonics in equal\|204\|96\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 204ed96 (continued)}}

	~~== See also ==~~
	* [[18edf]] – relative edf		* [[18edf]] – relative edf
	* [[31edo]] – relative edo		* [[31edo]] – relative edo
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	* [[72ed5]] – relative ed5		* [[72ed5]] – relative ed5
	* [[80ed6]] – relative ed6		* [[80ed6]] – relative ed6
			* [[87ed7]] – relative ed7
			* [[107ed11]] – relative ed11
	* [[111ed12]] – relative ed12		* [[111ed12]] – relative ed12
			* [[138ed22]] – relative ed22
			* [[39cET]]

	[[Category:31edo]]		[[Category:31edo]]
	[[Category:Zeta-optimized tunings]]		[[Category:Zeta-optimized tunings]]

FloraC: Text replacement - "{{mathematical interest}}" to "{{Mathematical interest}}"

2025-06-26T11:57:48Z

Text replacement - "{{mathematical interest}}" to "{{Mathematical interest}}"

← Older revision		Revision as of 11:57, 26 June 2025
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	{{~~mathematical~~ interest}}		{{Mathematical interest}}
	{{Infobox ET}}		{{Infobox ET}}
	{{ED intro}}		{{ED intro}}

ArrowHead294: /* Theory */

2025-06-18T15:48:33Z

Theory

← Older revision		Revision as of 15:48, 18 June 2025
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	== Theory ==		== Theory ==
	The 96th harmonic ~~would~~ be ~~extremely wide for an~~ equivalence, so 204ed96 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 96/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.79{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 31 is located at 30.978382, which has a step size of 38.737{{c}} and an octave of 1200.837{{c}} (which is stretched by 0.837{{c}}), making 204ed96 extremely close to optimal for 31edo.		The 96th harmonic is too wide to be a useful equivalence, so 204ed96 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 96/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.79{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 31 is located at 30.978382, which has a step size of 38.737{{c}} and an octave of 1200.837{{c}} (which is stretched by 0.837{{c}}), making 204ed96 extremely close to optimal for 31edo.

	=== Harmonics ===		=== Harmonics ===

ArrowHead294 at 13:28, 18 June 2025

2025-06-18T13:28:06Z

← Older revision		Revision as of 13:28, 18 June 2025
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	[[Category:31edo]]		[[Category:31edo]]
			[[Category:Zeta-optimized tunings]]

ArrowHead294 at 13:17, 18 June 2025

2025-06-18T13:17:34Z

← Older revision		Revision as of 13:17, 18 June 2025
Line 4:		Line 4:

	== Theory ==		== Theory ==
	The 96th harmonic would be extremely wide for an equivalence, so 204ed96 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 96/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.79{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 31 is located at 30.978382, which has a step size of 38.737{{c}} and an octave of 1200.837{{c}} (which is ~~compressed~~ by 2.31{{c}}), making 204ed96 extremely close to optimal for 31edo.		The 96th harmonic would be extremely wide for an equivalence, so 204ed96 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 96/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.79{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 31 is located at 30.978382, which has a step size of 38.737{{c}} and an octave of 1200.837{{c}} (which is stretched by 0.837{{c}}), making 204ed96 extremely close to optimal for 31edo.

	=== Harmonics ===		=== Harmonics ===

ArrowHead294 at 13:29, 10 June 2025

2025-06-10T13:29:27Z

← Older revision		Revision as of 13:29, 10 June 2025
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			{{mathematical interest}}
	{{Infobox ET}}		{{Infobox ET}}
	{{ED intro}}		{{ED intro}}
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	== See also ==		== See also ==
			* [[18edf]] – relative edf
	* [[31edo]] – relative edo		* [[31edo]] – relative edo
	* [[49edt]] – relative edt		* [[49edt]] – relative edt
	* [[72ed5]] – relative ed5		* [[72ed5]] – relative ed5
	* [[80ed6]] – relative ed6		* [[80ed6]] – relative ed6
			* [[111ed12]] – relative ed12

			[[Category:31edo]]

ArrowHead294: Fix info copied over from 79ed12 and other pages

2025-06-09T03:49:13Z

Fix info copied over from 79ed12 and other pages

← Older revision		Revision as of 03:49, 9 June 2025
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	== Theory ==		== Theory ==
	The 96th harmonic would be extremely wide for an equivalence, so 204ed96 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 96/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.79{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 12 is located at 30.978382, which has a step size of 38.737{{c}} and an octave of 1200.837{{c}} (which is compressed by 2.31{{c}}), making 204ed96 extremely close to optimal for 31edo.		The 96th harmonic would be extremely wide for an equivalence, so 204ed96 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 96/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.79{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 31 is located at 30.978382, which has a step size of 38.737{{c}} and an octave of 1200.837{{c}} (which is compressed by 2.31{{c}}), making 204ed96 extremely close to optimal for 31edo.

	=== Harmonics ===		=== Harmonics ===
	{{Harmonics in equal\|79\|12\|1\|intervals=integer\|columns=11}}		{{Harmonics in equal\|204\|96\|1\|intervals=integer\|columns=11}}
	{{Harmonics in equal\|79\|12\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in ~~79ed12~~ (continued)}}		{{Harmonics in equal\|204\|96\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 204ed96 (continued)}}

	~~=== Subsets and supersets ===~~
	~~79ed12 is the 22nd [[prime equal division\|prime ed12]], so it does not contain any nontrivial subset ed12's.~~

	== See also ==		== See also ==

ArrowHead294: Make page

2025-06-09T03:47:22Z

Make page

New page

{{Infobox ET}}
{{ED intro}}

== Theory ==
The 96th harmonic would be extremely wide for an equivalence, so 204ed96 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 96/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning|stretched]] by about 0.79{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch|zeta peak]] around 12 is located at 30.978382, which has a step size of 38.737{{c}} and an octave of 1200.837{{c}} (which is compressed by 2.31{{c}}), making 204ed96 extremely close to optimal for 31edo.

=== Harmonics ===
{{Harmonics in equal|79|12|1|intervals=integer|columns=11}}
{{Harmonics in equal|79|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 79ed12 (continued)}}

=== Subsets and supersets ===
79ed12 is the 22nd [[prime equal division|prime ed12]], so it does not contain any nontrivial subset ed12's.

== See also ==
* [[31edo]] – relative edo
* [[49edt]] – relative edt
* [[72ed5]] – relative ed5
* [[80ed6]] – relative ed6