311edo: Difference between revisions

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== Theory ==

311edo is [[consistent]] through the [[41-odd-limit]] and distinctly consistent through the [[23-odd-limit]], 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 [[625/624|S25]]), 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 valuable from a psychoacoustic perspective as its step is also conincidentally close enough to the [[just-noticeable difference]], which only affirms its efficiency of interval representation.

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 == Theory ==
-edo is [[consistent]] through the [[41-odd-limit]] and distinctly consistent through the [[23-odd-limit]], 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 [[625/624|S25]]), 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 valuable from a psychoacoustic perspective as its step is also conincidentally close enough to the [[just-noticeable difference]], which only affirms its efficiency of interval representation.