311edo: Difference between revisions

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== Theory ==

311edo is [[consistent]] through the [[41-odd-limit]] and nearly distinctly consistent through the [[27-odd-limit]] with the single exception of [[25/24]]~[[26/25]], [[tempering out]] [[625/624|S25 (625/624)]], and is a [[zeta gap edo]] and a [[zeta peak integer edo]]. It achieves this since all [[harmonic]]s up to and including the 42nd, and all composite harmonics up to and including the 80th, are more in-tune than out-of-tune (but note prime 73 ''is'' tuned accurately, in fact more accurately than all prior primes). Thus all the ratios between those harmonics are mapped consistently, and thus with a maximum error of ~1.~~929¢~~. This means 311edo is an ''extremely'' efficient temperament for approximating the [[harmonic series]] consistently and ''simply'', given how much harmonic content it approximates/represents for its size.

311edo is [[consistent]] through the [[41-odd-limit]] and nearly distinctly consistent through the [[27-odd-limit]] with the single exception of [[25/24]]~[[26/25]], [[tempering out]] [[625/624|S25 (625/624)]], and is a [[zeta gap edo]] and a [[zeta peak integer edo]]. It achieves this since all [[harmonic]]s up to and including the 42nd, and all composite harmonics up to and including the 80th, are more in-tune than out-of-tune (but note prime 73 ''is'' tuned accurately, in fact more accurately than all prior primes). Thus all the ratios between those harmonics are mapped consistently, and thus with a maximum error of ~1.929{{c}}. This means 311edo is an ''extremely'' efficient temperament for approximating the [[harmonic series]] consistently and ''simply'', given how much harmonic content it approximates/represents for its size.

It also maintains [[relative interval error]]s of [[minimal consistent EDOs|no greater than 25%]] on all of the first 42 harmonics of the harmonic series, and is the smallest EDO to maintain less than 25% relative error on the first 32 harmonics. The next edo with less than 25% error on the first 32 harmonics is [[16808edo|16808]], the smallest EDO that approximates the 43rd harmonic while maintaining the same maximum relative errors on the 42nd and lower is [[20567edo|20567]], and the smallest edo that maintains less than 25% relative error on the first 64 harmonics is [[3159811edo|3159811]].

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 == Theory ==
-edo is [[consistent]] through the [[41-odd-limit]] and nearly distinctly consistent through the [[27-odd-limit]] with the single exception of [[25/24]]~[[26/25]], [[tempering out]] [[625/624|S25 (625/624)]], and is a [[zeta gap edo]] and a [[zeta peak integer edo]]. It achieves this since all [[harmonic]]s up to and including the 42nd, and all composite harmonics up to and including the 80th, are more in-tune than out-of-tune (but note prime 73 ''is'' tuned accurately, in fact more accurately than all prior primes). Thus all the ratios between those harmonics are mapped consistently, and thus with a maximum error of ~1.929¢. This means 311edo is an ''extremely'' efficient temperament for approximating the [[harmonic series]] consistently and ''simply'', given how much harmonic content it approximates/represents for its size.
+edo is [[consistent]] through the [[41-odd-limit]] and nearly distinctly consistent through the [[27-odd-limit]] with the single exception of [[25/24]]~[[26/25]], [[tempering out]] [[625/624|S25 (625/624)]], and is a [[zeta gap edo]] and a [[zeta peak integer edo]]. It achieves this since all [[harmonic]]s up to and including the 42nd, and all composite harmonics up to and including the 80th, are more in-tune than out-of-tune (but note prime 73 ''is'' tuned accurately, in fact more accurately than all prior primes). Thus all the ratios between those harmonics are mapped consistently, and thus with a maximum error of ~1.929{{c}}. This means 311edo is an ''extremely'' efficient temperament for approximating the [[harmonic series]] consistently and ''simply'', given how much harmonic content it approximates/represents for its size.
 It also maintains [[relative interval error]]s of [[minimal consistent EDOs|no greater than 25%]] on all of the first 42 harmonics of the harmonic series, and is the smallest EDO to maintain less than 25% relative error on the first 32 harmonics. The next edo with less than 25% error on the first 32 harmonics is [[16808edo|16808]], the smallest EDO that approximates the 43rd harmonic while maintaining the same maximum relative errors on the 42nd and lower is [[20567edo|20567]], and the smallest edo that maintains less than 25% relative error on the first 64 harmonics is [[3159811edo|3159811]].