311edo: Difference between revisions

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== 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.

311edo 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.

Some 41-limit [[comma]]s it [[tempering out|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.

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=== Subsets and supersets ===

311edo is the 64th [[prime edo]].

As an interval size measure, one step of 311edo is called ''gene'', named after [[Gene Ward Smith]].

== Intervals ==

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 == 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.
+edo 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.
 Some 41-limit [[comma]]s it [[tempering out|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.
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 === Subsets and supersets ===
 edo is the 64th [[prime edo]].
+As an interval size measure, one step of 311edo is called ''gene'', named after [[Gene Ward Smith]].
 == Intervals ==