361ed448: Difference between revisions

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== 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 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 ==

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 == 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 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 ==