361ed448 - Revision history

ArrowHead294: /* Theory */

2025-08-10T19:01:45Z

Theory

← Older revision		Revision as of 19:01, 10 August 2025
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	== Theory ==		== Theory ==
	The 448th harmonic is far too wide to be a useful equivalence, so 361ed448 is better thought of as a stretched version of [[41edo]]. Indeed, tuning the 448/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.338{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 41 is located at 40.988078, which has a step size of 29.277{{c}} and an octave of 1200.349{{c}} (which is stretched by 0.~~837~~{{c}}), making 361ed448 extremely close to optimal for 41edo.		The 448th harmonic is far too wide to be a useful equivalence, so 361ed448 is better thought of as a stretched version of [[41edo]]. Indeed, tuning the 448/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.338{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 41 is located at 40.988078, which has a step size of 29.277{{c}} and an octave of 1200.349{{c}} (which is stretched by 0.349{{c}}), making 361ed448 extremely close to optimal for 41edo.

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

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

2025-06-26T11:57:52Z

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}}

FloraC: Text replacement - "{{infobox et}}↵{{ed intro}}" to "{{Infobox ET}} {{ED intro}}"

2025-06-26T11:56:56Z

Text replacement - "{{infobox et}}↵{{ed intro}}" to "{{Infobox ET}} {{ED intro}}"

← Older revision		Revision as of 11:56, 26 June 2025
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	{{mathematical interest}}		{{mathematical interest}}
	{{~~infobox et~~}}		{{Infobox ET}}
	{{ed intro}}		{{ED intro}}

	== Theory ==		== Theory ==

Lériendil at 00:00, 21 June 2025

2025-06-21T00:00:33Z

← Older revision		Revision as of 00:00, 21 June 2025
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	=== Harmonics ===		=== Harmonics ===
	{{Harmonics in equal\|361\|448\|1\|intervals=integer\|columns=11}}		{{Harmonics in equal\|361\|448\|1\|intervals=integer\|columns=11}}
	{{Harmonics in equal\|361\|448\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 361ed448}} (continued)}}		{{Harmonics in equal\|361\|448\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 361ed448 (continued)}}

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

ArrowHead294 at 14:24, 20 June 2025

2025-06-20T14:24:29Z

← Older revision		Revision as of 14:24, 20 June 2025
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	== Theory ==		== Theory ==
	The 448th harmonic is far too wide to be a useful equivalence, so ~~361ed{{lc:448}}~~ is better thought of as a stretched version of [[41edo]]. Indeed, tuning the 448/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.338{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 41 is located at 40.988078, which has a step size of 29.277{{c}} and an octave of 1200.349{{c}} (which is stretched by 0.837{{c}}), making 361ed448 extremely close to optimal for 41edo.		The 448th harmonic is far too wide to be a useful equivalence, so 361ed448 is better thought of as a stretched version of [[41edo]]. Indeed, tuning the 448/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.338{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 41 is located at 40.988078, which has a step size of 29.277{{c}} and an octave of 1200.349{{c}} (which is stretched by 0.837{{c}}), making 361ed448 extremely close to optimal for 41edo.

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

ArrowHead294 at 14:24, 20 June 2025

2025-06-20T14:24:17Z

← Older revision		Revision as of 14:24, 20 June 2025
Line 4:		Line 4:

	== Theory ==		== Theory ==
	The 448th harmonic is far too wide to be a useful equivalence, so ~~361ed448~~ is better thought of as a stretched version of [[41edo]]. Indeed, tuning the 448/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.338{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 41 is located at 40.988078, which has a step size of 29.277{{c}} and an octave of 1200.349{{c}} (which is stretched by 0.837{{c}}), making 361ed448 extremely close to optimal for 41edo.		The 448th harmonic is far too wide to be a useful equivalence, so 361ed{{lc:448}} is better thought of as a stretched version of [[41edo]]. Indeed, tuning the 448/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.338{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 41 is located at 40.988078, which has a step size of 29.277{{c}} and an octave of 1200.349{{c}} (which is stretched by 0.837{{c}}), making 361ed448 extremely close to optimal for 41edo.

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

ArrowHead294 at 14:22, 20 June 2025

2025-06-20T14:22:05Z

← Older revision		Revision as of 14:22, 20 June 2025
Line 4:		Line 4:

	== Theory ==		== Theory ==
	The 448th harmonic is far too wide to be a useful equivalence, so is better thought of as a stretched version of [[41edo]]. Indeed, tuning the 448/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.338{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 41 is located at 40.988078, which has a step size of 29.277{{c}} and an octave of 1200.349{{c}} (which is stretched by 0.837{{c}}), making 361ed448 extremely close to optimal for 41edo.		The 448th harmonic is far too wide to be a useful equivalence, so 361ed448 is better thought of as a stretched version of [[41edo]]. Indeed, tuning the 448/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.338{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 41 is located at 40.988078, which has a step size of 29.277{{c}} and an octave of 1200.349{{c}} (which is stretched by 0.837{{c}}), making 361ed448 extremely close to optimal for 41edo.

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

ArrowHead294: Try

2025-06-20T14:21:54Z

Try

← Older revision		Revision as of 14:21, 20 June 2025
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	== Theory ==		== Theory ==
	The 448th harmonic is far too wide to be a useful equivalence, so ~~361ed448~~ is better thought of as a stretched version of [[41edo]]. Indeed, tuning the 448/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.338{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 41 is located at 40.988078, which has a step size of 29.277{{c}} and an octave of 1200.349{{c}} (which is stretched by 0.837{{c}}), making 361ed448 extremely close to optimal for 41edo.		The 448th harmonic is far too wide to be a useful equivalence, so is better thought of as a stretched version of [[41edo]]. Indeed, tuning the 448/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.338{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 41 is located at 40.988078, which has a step size of 29.277{{c}} and an octave of 1200.349{{c}} (which is stretched by 0.837{{c}}), making 361ed448 extremely close to optimal for 41edo.

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

FloraC: - abuse of regex

2025-06-20T14:15:16Z

- abuse of regex

← Older revision		Revision as of 14:15, 20 June 2025
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	== Theory ==		== Theory ==
	The 448th harmonic is far too wide to be a useful equivalence, so ~~{{#rmatch: {{PAGENAME}}\|/.?(\d+)[Ee][Dd]([^\s]+)/e\|\1ed{{lc:\2}}}}~~ is better thought of as a stretched version of [[41edo]]. Indeed, tuning ~~the448~~/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.338{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 41 is located at 40.988078, which has a step size of 29.277{{c}} and an octave of 1200.349{{c}} (which is stretched by 0.837{{c}}), making ~~{{#rmatch: {{PAGENAME}}\|/.?(\d+)[Ee][Dd]([^\s]+)/e\|\1ed{{lc:\2}}}}~~ extremely close to optimal for 41edo.		The 448th harmonic is far too wide to be a useful equivalence, so 361ed448 is better thought of as a stretched version of [[41edo]]. Indeed, tuning the 448/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.338{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 41 is located at 40.988078, which has a step size of 29.277{{c}} and an octave of 1200.349{{c}} (which is stretched by 0.837{{c}}), making 361ed448 extremely close to optimal for 41edo.

	=== Harmonics ===		=== Harmonics ===
	{{Harmonics in equal\|361\|448\|1\|intervals=integer\|columns=11}}		{{Harmonics in equal\|361\|448\|1\|intervals=integer\|columns=11}}
	{{Harmonics in equal\|361\|448\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in ~~{{#rmatch: {{PAGENAME}}\|/.*?(\d+)[Ee][Dd]([^\s]+)/e\|\1ed{{lc:\2}}~~}} (continued)}}		{{Harmonics in equal\|361\|448\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 361ed448}} (continued)}}

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

ArrowHead294: Make page

2025-06-19T12:49:03Z

Make page

New page

{{mathematical interest}}
{{infobox et}}
{{ed intro}}

== Theory ==
The 448th harmonic is far too wide to be a useful equivalence, so {{#rmatch: {{PAGENAME}}|/.*?(\d+)[Ee][Dd]([^\s]+)/e|\1ed{{lc:\2}}}} is better thought of as a stretched version of [[41edo]]. Indeed, tuning the448/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning|stretched]] by about 0.338{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch|zeta peak]] around 41 is located at 40.988078, which has a step size of 29.277{{c}} and an octave of 1200.349{{c}} (which is stretched by 0.837{{c}}), making {{#rmatch: {{PAGENAME}}|/.*?(\d+)[Ee][Dd]([^\s]+)/e|\1ed{{lc:\2}}}} extremely close to optimal for 41edo.

=== Harmonics ===
{{Harmonics in equal|361|448|1|intervals=integer|columns=11}}
{{Harmonics in equal|361|448|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in {{#rmatch: {{PAGENAME}}|/.*?(\d+)[Ee][Dd]([^\s]+)/e|\1ed{{lc:\2}}}} (continued)}}

== See also ==
* [[24edf]] – relative edf
* [[41edo]] – relative edo
* [[65edt]] – relative edt
* [[95ed5]] – relative ed5
* [[106ed6]] – relative ed6
* [[147ed12]] – relative ed12

[[Category:41edo]]
[[Category:Zeta-optimized tunings]]