311edo: Difference between revisions

Line 5:

== 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]]), 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]]), 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.

311 is also the lowest edo that maintains [[relative interval error]]s of no greater than 25% on all of the first 42 harmonics of the harmonic series, and the next lowest edo that approximates the 43rd harmonic while maintaining the same maximum relative errors on the 42nd and lower is [[20567edo|20567]].

311 is also the lowest edo that maintains [[relative interval error]]s of no greater than 25% on all of the first 42 harmonics of the harmonic series, and the next lowest 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]].

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.

@@ Line 5: / Line 5: @@
 == 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]]), 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]]), 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.
-is also the lowest edo that maintains [[relative interval error]]s of no greater than 25% on all of the first 42 harmonics of the harmonic series, and the next lowest edo that approximates the 43rd harmonic while maintaining the same maximum relative errors on the 42nd and lower is [[20567edo|20567]].
+is also the lowest edo that maintains [[relative interval error]]s of no greater than 25% on all of the first 42 harmonics of the harmonic series, and the next lowest 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]].
 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.