311edo: Difference between revisions
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''' | {{Infobox ET | ||
| Prime factorization = 311 (prime) | |||
| Step size = 3.85852¢ | |||
| Fifth = 182\311 (702.25¢) | |||
| Semitones = 30:23 (115.76¢ : 88.75¢) | |||
| Consistency = 41 | |||
}} | |||
The '''311 equal divisions of the octave''' ('''311edo'''), or the '''311(-tone) equal temperament''' ('''311tet''', '''311et''') when viewed from a [[regular temperament]] perspective, a remarkable very high limit equal temperament, [[EDO|dividing the octave equally]] into 311 parts of about 3.86 [[cent]]s each. | |||
== Theory == | == Theory == | ||
311edo is [[consistent]] through the 41-odd-limit and uniquely 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 except for the prime harmonics greater than 41 (but not including the prime 73 which ''is'' tuned accurately, in fact more accurately than all prior primes), all harmonics up to and including the 80th are more in-tune than out-of-tune with 311edo and 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. | |||
Some 41-limit [[comma]]s it [[ | 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. | ||
311edo is the 64th [[prime edo]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
Revision as of 14:45, 28 December 2021
| ← 310edo | 311edo | 312edo → |
The 311 equal divisions of the octave (311edo), or the 311(-tone) equal temperament (311tet, 311et) when viewed from a regular temperament perspective, a remarkable very high limit equal temperament, dividing the octave equally into 311 parts of about 3.86 cents each.
Theory
311edo is consistent through the 41-odd-limit and uniquely consistent through the 23-odd-limit and is a zeta gap edo and a zeta peak integer edo. It achieves this since except for the prime harmonics greater than 41 (but not including the prime 73 which is tuned accurately, in fact more accurately than all prior primes), all harmonics up to and including the 80th are more in-tune than out-of-tune with 311edo and 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.
Some 41-limit commas 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.
311edo is the 64th prime edo.
Prime harmonics
Script error: No such module "primes_in_edo".
Intervals
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [493 -311⟩ | [⟨311 493]] | -0.0933 | 0.0933 | 2.42 |
| 2.3.5 | 1600000/1594323, [-59 5 22⟩ | [⟨311 493 722]] | +0.0040 | 0.1573 | 4.08 |
| 2.3.5.7 | 2401/2400, 65625/65536, 1600000/1594323 | [⟨311 493 722 873]] | +0.0331 | 0.1453 | 3.76 |
| 2.3.5.7.11 | 2401/2400, 3025/3024, 4000/3993, 19712/19683 | [⟨311 493 722 873 1076]] | +0.0004 | 0.1454 | 3.77 |
| 2.3.5.7.11.13 | 625/624, 1575/1573, 2080/2079, 2200/2197, 2401/2400 | [⟨311 493 722 873 1076 1151]] | -0.0280 | 0.1472 | 3.81 |
| 2.3.5.7.11.13.17 | 595/594, 625/624, 833/832, 1156/1155, 1575/1573, 2200/2197 | [⟨311 493 722 873 1076 1151 1271]] | +0.0031 | 0.1561 | 4.05 |
| 2.3.5.7.11.13.17.19 | 595/594, 625/624, 833/832, 969/968, 1156/1155, 1216/1215, 1575/1573 | [⟨311 493 722 873 1076 1151 1271 1321]] | +0.0146 | 0.1492 | 3.87 |
| 2.3.5.7.11.13.17.19.23 | 595/594, 625/624, 760/759, 833/832, 875/874, 969/968, 1105/1104, 1156/1155 | [⟨311 493 722 873 1076 1151 1271 1321 1407]] | -0.0033 | 0.1496 | 3.88 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 10\311 | 38.59 | 45/44 | Hemitert |
| 1 | 11\311 | 42.44 | 40/39 | Humorous |
| 1 | 17\311 | 65.59 | 27/26 | Luminal |
| 1 | 20\311 | 77.17 | 256/245 | Tertiaseptal |
| 1 | 22\311 | 84.89 | 21/20 | Amicable / amical / amorous |
| 1 | 29\311 | 111.90 | 16/15 | Vavoom |
| 1 | 35\311 | 135.05 | 27/25 | Superlimmal |
| 1 | 43\311 | 165.92 | 11/10 | Satin |
| 1 | 67\311 | 258.52 | [-32 13 5⟩ | Lafa |
| 1 | 88\311 | 339.55 | 243/200 | Amity / paramity |
| 1 | 91\311 | 351.13 | 49/40 | Newt |
| 1 | 108\311 | 416.72 | 14/11 | Unthirds |
| 1 | 129\311 | 497.75 | 4/3 | Gary |
| 1 | 143\311 | 551.77 | 11/8 | Emka / emkay |