311edo: Difference between revisions

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It is also the lowest edo that maintains [[relative interval error]]s of [[minimal consistent EDOs|no greater than 25%]] on all of the first 42 harmonics of the harmonic series. 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]].

It is still very accurate in the lower limits. Although it does not do as well as [[270edo]] in the 13-limit, it makes for an interesting comparison. The equal temperament 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]], and [[19712/19683]] in the 11-limit; and 625/624, [[1575/1573]], [[2080/2079]], [[2200/2197]], [[4096/4095]], [[4225/4224]] in the 13-limit. It allows [[petrmic chords|petrmic]] and [[nicolic chords]] in the 15-odd-limit.

It is still very accurate in the lower limits. Although it does not do as well as [[270edo]] in the 13-limit, it makes for an interesting comparison. The equal temperament 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.

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 It is also the lowest edo that maintains [[relative interval error]]s of [[minimal consistent EDOs|no greater than 25%]] on all of the first 42 harmonics of the harmonic series. 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]].
-It is still very accurate in the lower limits. Although it does not do as well as [[270edo]] in the 13-limit, it makes for an interesting comparison. The equal temperament 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]], and [[19712/19683]] in the 11-limit; and 625/624, [[1575/1573]], [[2080/2079]], [[2200/2197]], [[4096/4095]], [[4225/4224]] in the 13-limit. It allows [[petrmic chords|petrmic]] and [[nicolic chords]] in the 15-odd-limit.
+It is still very accurate in the lower limits. Although it does not do as well as [[270edo]] in the 13-limit, it makes for an interesting comparison. The equal temperament 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.