311edo: Difference between revisions
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=== Notation === | === Notation === | ||
One possible notation uses / and \ (lifts and drops) to stand for 5 edosteps. | One possible notation uses / and \ (lifts and drops) to stand for 5 edosteps. Double is abbreviated as "dub-": | ||
0\311 = P1 = perfect unison | 0\311 = P1 = perfect unison | ||
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2\311 = ^^1 = dup unison | 2\311 = ^^1 = dup unison | ||
3\311 = vv/1 = | 3\311 = vv/1 = dudlift unison | ||
4\311 = v/1 = downlift unison | 4\311 = v/1 = downlift unison | ||
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7\311 = ^^/1 = duplift unison | 7\311 = ^^/1 = duplift unison | ||
8\311 = vv//1 = | 8\311 = vv//1 = dud-dublift unison | ||
9\311 = v//1 = down- | 9\311 = v//1 = down-dublift unison | ||
10\311 = //1 = | 10\311 = //1 = dublift unison | ||
11\311 = ^//1 = up- | 11\311 = ^//1 = up-dublift unison = vv\\m2 = dud-dubdropminor second | ||
12\311 = v\m2 = down- | 12\311 = ^^//1 = dup-dublift unison = v\\m2 = down-dubdropminor second | ||
13\311 = \m2 = | 13\311 = \\m2 = dubdropminor second | ||
14\311 = ^\m2 = up- | 14\311 = ^\\m2 = up-dubdropminor second | ||
15\311 = ^^\m2 = dup- | 15\311 = ^^\\m2 = dup-dubdropminor second | ||
16\311 = vv\m2 = | 16\311 = vv\m2 = duddropminor second | ||
... | ... | ||
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25\311 = ^^m2 = dupminor second | 25\311 = ^^m2 = dupminor second | ||
26\311 = vv/m2 = | 26\311 = vv/m2 = dudliftminor second | ||
27\311 = v/m2 = downliftminor second | 27\311 = v/m2 = downliftminor second | ||
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30\311 = ^^/m2 = dupliftminor second | 30\311 = ^^/m2 = dupliftminor second | ||
31\311 = vv\~2 = | 31\311 = vv\~2 = duddropmid second | ||
32\311 = v\~2 = downdropmid second | 32\311 = v\~2 = downdropmid second | ||
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35\311 = ^^\~2 = dupdropmid second | 35\311 = ^^\~2 = dupdropmid second | ||
36\311 = vv~2 = | 36\311 = vv~2 = dudmid second | ||
37\311 = v~2 = downmid second | 37\311 = v~2 = downmid second | ||
Revision as of 18:35, 18 November 2022
| ← 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, is 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
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.296 | -0.462 | -0.337 | +0.451 | +0.630 | -0.775 | -0.407 | +0.665 | +0.648 | +0.945 | -0.540 | -0.767 |
| Relative (%) | +0.0 | +7.7 | -12.0 | -8.7 | +11.7 | +16.3 | -20.1 | -10.5 | +17.2 | +16.8 | +24.5 | -14.0 | -19.9 | |
| Steps (reduced) |
311 (0) |
493 (182) |
722 (100) |
873 (251) |
1076 (143) |
1151 (218) |
1271 (27) |
1321 (77) |
1407 (163) |
1511 (267) |
1541 (297) |
1620 (65) |
1666 (111) | |
Intervals
Notation
One possible notation uses / and \ (lifts and drops) to stand for 5 edosteps. Double is abbreviated as "dub-":
0\311 = P1 = perfect unison
1\311 = ^1 = up unison
2\311 = ^^1 = dup unison
3\311 = vv/1 = dudlift unison
4\311 = v/1 = downlift unison
5\311 = /1 = lift unison
6\311 = ^/1 = uplift unison
7\311 = ^^/1 = duplift unison
8\311 = vv//1 = dud-dublift unison
9\311 = v//1 = down-dublift unison
10\311 = //1 = dublift unison
11\311 = ^//1 = up-dublift unison = vv\\m2 = dud-dubdropminor second
12\311 = ^^//1 = dup-dublift unison = v\\m2 = down-dubdropminor second
13\311 = \\m2 = dubdropminor second
14\311 = ^\\m2 = up-dubdropminor second
15\311 = ^^\\m2 = dup-dubdropminor second
16\311 = vv\m2 = duddropminor second
...
24\311 = ^m2 = upminor second
25\311 = ^^m2 = dupminor second
26\311 = vv/m2 = dudliftminor second
27\311 = v/m2 = downliftminor second
28\311 = /m2 = liftminor second
29\311 = ^/m2 = upliftminor second
30\311 = ^^/m2 = dupliftminor second
31\311 = vv\~2 = duddropmid second
32\311 = v\~2 = downdropmid second
33\311 = \~2 = dropmid second
34\311 = ^\~2 = updropmid second
35\311 = ^^\~2 = dupdropmid second
36\311 = vv~2 = dudmid second
37\311 = v~2 = downmid second
38\311 = ~2 = mid second
etc.
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, 23/22 | Tertiaseptal / tertiaseptia |
| 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 | 133\311 | 513.18 | 35/26 | Trinity |
| 1 | 143\311 | 551.77 | 11/8 | Emka / emkay |
| 1 | 155\311 | 598.08 | 847/600 | Vydubychi |