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]] ({{S|25}}), 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~~ more ~~in tune~~ than ~~out of tune~~. (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 [[17461edo|17461]], being consistent in the [[45-odd-limit]], though one may prefer [[20567edo|20567]], as it is consistent in the [[57-odd-limit]].

311edo is [[consistent]] through the [[41-odd-limit]] and nearly distinctly consistent through the [[27-odd-limit]] except for [[25/24]][[~]][[26/25]], [[tempering out]] [[625/624]] ({{S|25}}), 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, have no more than ±25% error. (Prime 73 is also unusually accurate, more so than all smaller primes.) As a result, all ratios among those harmonics are mapped consistently, with errors lower than 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.

~~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~~]].

The smallest edo that comes closest to having a higher consistency is [[1600edo]] ([[37-odd-limit]]), almost consistent in the [[43-odd-limit]] except for two intervals ([[39/25]], [[50/39]]). The next edo with a truly higher [[consistency limit]] is [[17461edo|17461]] ([[45-odd-limit]]), though one may prefer [[20567edo|20567]] ([[57-odd-limit]]).

~~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~~.

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

Beyond the 13-limit, primes [[17/1|17]] and [[23/1|23]] are 311edo's first notable improvements over 270edo's approximation. It tempers out [[595/594]], [[833/832]], [[1156/1155]], [[1225/1224]], [[1275/1274]], [[2058/2057]], [[2431/2430]] in the 17-limit; [[969/968]], [[1216/1215]], [[1445/1444]], [[1540/1539]], [[1729/1728]] in the 19-limit; and [[760/759]], [[875/874]], [[1105/1104]], [[1197/1196]], [[1288/1287]], [[1496/1495]] in the 23-limit.

Although it does not do as well as [[270edo]] in the 13-limit, it is still very accurate in the lower limits, having around +0.15 cents more of ajusted error than 270edo (0.44c). 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.

Beyond the 13-limit, primes [[17/1|17]] and [[23/1|23]] are 311edo's first notable improvements over 270edo's approximation. It tempers out [[595/594]], [[833/832]], [[1156/1155]], [[1225/1224]], [[1275/1274]], [[2058/2057]], [[2431/2430]] in the 17-limit; [[969/968]], [[1216/1215]], [[1445/1444]], [[1540/1539]], [[1729/1728]] in the 19-limit; and [[760/759]], [[875/874]], [[1105/1104]], [[1197/1196]], [[1288/1287]], [[1496/1495]] in the 23-limit. Their edo sum, [[581edo]], is also a very strong 23-limit temperament.

It is valuable from a psychoacoustic perspective as its step is also coincidentally above the melodic [[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 nearly distinctly consistent through the [[27-odd-limit]] with the single exception of [[25/24]][[~]][[26/25]], [[tempering out]] [[625/624]] ({{S|25}}), 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 more in tune than out of tune. (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 [[17461edo|17461]], being consistent in the [[45-odd-limit]], though one may prefer [[20567edo|20567]], as it is consistent in the [[57-odd-limit]].
+edo is [[consistent]] through the [[41-odd-limit]] and nearly distinctly consistent through the [[27-odd-limit]] except for [[25/24]][[~]][[26/25]], [[tempering out]] [[625/624]] ({{S|25}}), 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, have no more than ±25% error. (Prime 73 is also unusually accurate, more so than all smaller primes.) As a result, all ratios among those harmonics are mapped consistently, with errors lower than 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.
-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]].
+The smallest edo that comes closest to having a higher consistency is [[1600edo]] ([[37-odd-limit]]), almost consistent in the [[43-odd-limit]] except for two intervals ([[39/25]], [[50/39]]). The next edo with a truly higher [[consistency limit]] is [[17461edo|17461]] ([[45-odd-limit]]), though one may prefer [[20567edo|20567]] ([[57-odd-limit]]).
-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.
+It is also the smallest edo that is [[purely consistent]] on all the first 32 harmonics (in this case, up to the 42nd). The next edo with less relative error is [[16808edo|16808]]. The smallest edo purely consistent on the first 64 harmonics is [[3159811edo|3159811]].
-Beyond the 13-limit, primes [[17/1|17]] and [[23/1|23]] are 311edo's first notable improvements over 270edo's approximation. It tempers out [[595/594]], [[833/832]], [[1156/1155]], [[1225/1224]], [[1275/1274]], [[2058/2057]], [[2431/2430]] in the 17-limit; [[969/968]], [[1216/1215]], [[1445/1444]], [[1540/1539]], [[1729/1728]] in the 19-limit; and [[760/759]], [[875/874]], [[1105/1104]], [[1197/1196]], [[1288/1287]], [[1496/1495]] in the 23-limit.
+Although it does not do as well as [[270edo]] in the 13-limit, it is still very accurate in the lower limits, having around +0.15 cents more of ajusted error than 270edo (0.44c). 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.
+Beyond the 13-limit, primes [[17/1|17]] and [[23/1|23]] are 311edo's first notable improvements over 270edo's approximation. It tempers out [[595/594]], [[833/832]], [[1156/1155]], [[1225/1224]], [[1275/1274]], [[2058/2057]], [[2431/2430]] in the 17-limit; [[969/968]], [[1216/1215]], [[1445/1444]], [[1540/1539]], [[1729/1728]] in the 19-limit; and [[760/759]], [[875/874]], [[1105/1104]], [[1197/1196]], [[1288/1287]], [[1496/1495]] in the 23-limit. Their edo sum, [[581edo]], is also a very strong 23-limit temperament.
 It is valuable from a psychoacoustic perspective as its step is also coincidentally above the melodic [[just-noticeable difference]], which only affirms its efficiency of interval representation.