311edo: Difference between revisions

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The '''311 equal divisions of the octave''' ('''311edo'''), or the '''311(-tone) equal temperament''' ('''311tet''', '''311et''') when viewed from a [[regular temperament]] perspective, is a remarkable very high limit equal temperament, [[EDO|~~dividing the octave equally]] into~~ 311 ~~parts of about 3.86 [[cent]]s each.~~

311edo is highly acclaimed for its large consistency limit and efficient and well-tempered just interval representation relative to its size.

== Theory ==

311edo is [[consistent]] through the 41-odd-limit and distinctly consistent through the [[23-odd-limit]], and is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta gap edo]] and a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|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.

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 {{Infobox ET}}
-The '''311 equal divisions of the octave''' ('''311edo'''), or the '''311(-tone) equal temperament''' ('''311tet''', '''311et''') when viewed from a [[regular temperament]] perspective, is a remarkable very high limit equal temperament, [[EDO|dividing the octave equally]] into 311 parts of about 3.86 [[cent]]s each.
+{{EDO intro|311}}
+edo is highly acclaimed for its large consistency limit and efficient and well-tempered just interval representation relative to its size.
 == Theory ==
 edo is [[consistent]] through the 41-odd-limit and distinctly consistent through the [[23-odd-limit]], and is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta gap edo]] and a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|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.