311edo: Difference between revisions
ArrowHead294 (talk | contribs) m →Theory |
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{{EDO intro|311}} | {{EDO intro|311}} | ||
311edo is highly acclaimed for its large consistency limit and efficient and well-tempered just interval representation relative to its size. | 311edo is highly acclaimed for its large [[consistency]] limit and efficient and well-tempered just interval representation relative to its size. | ||
== Theory == | == 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]] | 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]]. 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. | ||
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. | |||
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. | |||
It 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. | |||
=== Prime harmonics === | === Prime harmonics === | ||
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| 1600000/1594323, {{monzo| -59 5 22 }} | |||
| {{mapping| 311 493 722 }} | | {{mapping| 311 493 722 }} | ||
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<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | <nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | ||
=== Commas === | |||
Some 41-limit [[comma]]s it tempers out are [[595/594]], [[625/624]], 697/696, 703/702, 714/713, 760/759, [[784/783]], 820/819, [[833/832]], 875/874, 900/899, 925/924, 931/930, 962/961, 969/968, 1000/999, 1015/1014, 1024/1023, [[1025/1024]], 1036/1035, 1045/1044, 1054/1053, 1105/1104, 1148/1147, [[1156/1155]], 1184/1183, 1189/1188, 1190/1189, 1197/1196, 1210/1209, [[1216/1215]], [[1225/1224]], [[1275/1274]], 1288/1287, 1312/1311, 1332/1331, 1353/1352, 1365/1364, 1369/1368, 1444/1443, [[1445/1444]], 1450/1449, 1480/1479, 1496/1495, 1519/1518, 1520/1519, 1540/1539, 1596/1595, 1600/1599, 1625/1624, 1665/1664, 1666/1665, 1681/1680, 1683/1682, 1702/1701, [[1729/1728]], 1768/1767, 1805/1804, 1860/1859, 1886/1885, 1887/1886, 1925/1924, 2002/2001, 2016/2015, 2025/2024, [[2058/2057]], [[2080/2079]], 2091/2090, 2109/2108, 2146/2145, 2176/2175, 2185/2184, 2205/2204, 2233/2232, 2255/2254, 2295/2294, 2296/2295, 2300/2299, [[2401/2400]], [[2431/2430]], [[2432/2431]], 2465/2464, [[2500/2499]], 2542/2541, 2553/2552, 2584/2583, [[2601/2600]], 2625/2624, 2640/2639, 2646/2645, 2665/2664, 2737/2736, 2738/2737, 2755/2754, 2784/2783, 2850/2849, 2926/2925, and 2945/2944. | |||
== Detemperaments == | == Detemperaments == | ||