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]]. This is because all [[~~Harmonic|harmonics~~]] up to the 42nd, and all composite harmonics up to the 80th, are tuned more closely than they are mistuned. (Prime 73 is also unusually accurate, more so than all smaller primes.) As a result, all ratios among those harmonics are mapped consistently, with a maximum error of about 1.929 ¢. This means 311edo is a ''serendipitously'' efficient temperament for approximating the [[harmonic series]] and the [[41-limit]] in general, consistently and ''simply'', given how much harmonic content it approximates/represents for its size. The smallest ~~EDO~~ that has a higher [[consistency limit]] is [[20567edo|20567]], being consistent in the 57-odd-limit.

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]]. This is because all [[harmonic]]s up to the 42nd, and all composite harmonics up to the 80th, are tuned more closely than they are mistuned. (Prime 73 is also unusually accurate, more so than all smaller primes.) As a result, all ratios among those harmonics are mapped consistently, with a maximum error of about 1.929{{c}}. This means 311edo is a ''serendipitously'' efficient temperament for approximating the [[harmonic series]] and the [[41-limit]] in general, consistently and ''simply'', given how much harmonic content it approximates/represents for its size. The smallest edo that has a higher [[consistency limit]] is [[20567edo|20567]], being consistent in the 57-odd-limit.

It is also the smallest ~~EDO~~ that is [[purely consistent]] on all the first 32 harmonics (in this case, up to the 42nd harmonic). The next ~~[[EDO]]~~ with less relative error is [[16808edo|16808]]. The smallest edo [[purely consistent]] on the first 64 harmonics is [[3159811edo|3159811]].

It is also the smallest edo that is [[purely consistent]] on all the first 32 harmonics (in this case, up to the 42nd harmonic). The next edo with less relative error is [[16808edo|16808]]. The smallest edo purely consistent on the first 64 harmonics is [[3159811edo|3159811]].

Although it does not do as well as [[270edo]] in the 13-limit, it is still very accurate in the lower limits. It tempers out the [[amity comma]], 1600000/1594323, the [[lafa comma]], {{monzo| 77 -31 -12 }}, the [[vavoom comma]], {{monzo| -68 18 17 }} in the [[5-limit]]; 2401/2400 ([[breedsma]]), 65625/65536 ([[horwell comma]]), and 33554432/33480783 ([[garischisma]]) in the 7-limit; [[3025/3024]], [[4000/3993]], [[6250/6237]], [[12005/11979]], and [[19712/19683]] in the 11-limit; and 625/624, [[1575/1573]], [[2080/2079]], [[2200/2197]], [[4096/4095]], and [[4225/4224]] in the 13-limit. It allows [[petrmic chords|petrmic]] and [[nicolic chords]] in the 15-odd-limit.

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=== Prime harmonics ===

{{Harmonics in equal|311|columns=17|start=~~14|prec=3~~|collapsed=true|title=Approximation of prime harmonics in 311edo (continued)}}

{{Harmonics in equal|311|prec=3|columns=12|start=13|collapsed=true|title=Approximation of prime harmonics in 311edo (continued)}}

=== Subsets and supersets ===

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-{{Infobox ET|Zeta=yes}}
+{{Infobox ET}}
 {{ED intro}}
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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]]. This is because all [[Harmonic|harmonics]] up to the 42nd, and all composite harmonics up to the 80th, are tuned more closely than they are mistuned. (Prime 73 is also unusually accurate, more so than all smaller primes.) As a result, all ratios among those harmonics are mapped consistently, with a maximum error of about 1.929 ¢. This means 311edo is a ''serendipitously'' efficient temperament for approximating the [[harmonic series]] and the [[41-limit]] in general, consistently and ''simply'', given how much harmonic content it approximates/represents for its size. The smallest EDO that has a higher [[consistency limit]] is [[20567edo|20567]], being consistent in the 57-odd-limit.
+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]]. This is because all [[harmonic]]s up to the 42nd, and all composite harmonics up to the 80th, are tuned more closely than they are mistuned. (Prime 73 is also unusually accurate, more so than all smaller primes.) As a result, all ratios among those harmonics are mapped consistently, with a maximum error of about 1.929{{c}}. This means 311edo is a ''serendipitously'' efficient temperament for approximating the [[harmonic series]] and the [[41-limit]] in general, consistently and ''simply'', given how much harmonic content it approximates/represents for its size. The smallest edo that has a higher [[consistency limit]] is [[20567edo|20567]], being consistent in the 57-odd-limit.
-It is also the smallest EDO that is [[purely consistent]] on all the first 32 harmonics (in this case, up to the 42nd harmonic). The next [[EDO]] with less relative error is [[16808edo|16808]]. The smallest edo [[purely consistent]] on the first 64 harmonics is [[3159811edo|3159811]].
+It is also the smallest edo that is [[purely consistent]] on all the first 32 harmonics (in this case, up to the 42nd harmonic). The next edo with less relative error is [[16808edo|16808]]. The smallest edo purely consistent on the first 64 harmonics is [[3159811edo|3159811]].
 Although it does not do as well as [[270edo]] in the 13-limit, it is still very accurate in the lower limits. It tempers out the [[amity comma]], 1600000/1594323, the [[lafa comma]], {{monzo| 77 -31 -12 }}, the [[vavoom comma]], {{monzo| -68 18 17 }} in the [[5-limit]]; 2401/2400 ([[breedsma]]), 65625/65536 ([[horwell comma]]), and 33554432/33480783 ([[garischisma]]) in the 7-limit; [[3025/3024]], [[4000/3993]], [[6250/6237]], [[12005/11979]], and [[19712/19683]] in the 11-limit; and 625/624, [[1575/1573]], [[2080/2079]], [[2200/2197]], [[4096/4095]], and [[4225/4224]] in the 13-limit. It allows [[petrmic chords|petrmic]] and [[nicolic chords]] in the 15-odd-limit.
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 === Prime harmonics ===
-{{Harmonics in equal|311|prec=3|columns=13}}
+{{Harmonics in equal|311|prec=3|columns=12}}
-{{Harmonics in equal|311|columns=17|start=14|prec=3|collapsed=true|title=Approximation of prime harmonics in 311edo (continued)}}
+{{Harmonics in equal|311|prec=3|columns=12|start=13|collapsed=true|title=Approximation of prime harmonics in 311edo (continued)}}
 === Subsets and supersets ===