311edo
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 distinctly consistent through the 23-odd-limit, and is a zeta gap edo and a zeta peak integer edo. It achieves this since all harmonics 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.
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.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 | +1.666 |
| 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 | +43.2 | |
| 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) |
1688 (133) | |
Subsets and supersets
311edo is the 64th prime edo.
Intervals
Notation
Sagittal
Sagittal in textual form.
| Steps | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Symbol | |( | )|( | )~| | (|( | ~~| | /| | |) | |\ | (| | (|( | ~|\ | //| | /|) | /|\ | /|\( |
| Steps | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| Symbol | (|) | (|\ | )||( | )~|| | ~||( | )||~ | /|| | ||) | ||\ | ~||\ | (||( | ~||\ | //|| | /||) | /||\ |
Syntonic-rastmic subchroma notation
Syntonic-rastmic subchroma notation in textual form.
| Steps | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Symbol | > | / | /> | ↑\ | ↑< | ↑ | ↑> | ↑/ | ↑/> | ↑↑\ | ↑↑< | ↑↑ | ↑↑> | t< | t |
| Steps | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| Symbol | t> | #↓↓< | #↓↓ | #↓↓> | #↓↓/ | #↓\< | #↓\ | #↓< | #↓ | #↓> | #↓/ | #\< | #\ | #< | # |
Ups and downs 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
17\311 = v\m2 = downdropminor second
18\311 = \m2 = dropminor second
19\311 = ^\m2 = updropminor second
20\311 = ^^\m2 = dupdropminor second
21\311 = vvm2 = dudminor second
22\311 = vm2 = downminor second
23\311 = m2 = minor 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 8ve |
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 | 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 | Emkay |
| 1 | 155\311 | 598.08 | 847/600 | Vydubychi |
Ringer scales
There are two known Ringer scales based on 311edo. Both consistently map the complete mode 234 of the harmonic series using non-patent vals of 311edo, which is believed to be the highest possible complete harmonic series mode mapped by a 311-form.
Ringer 311[+61]
|
Scale as chord: 936:940:941:943:944:948:950:952:954:956:958:960:962: |
Reduced to mode 234: 234:235:941/4:943/4:236:237:475/2:238:477/2:239:479/2:240:481/2: |
Ringer 311[+61, −67]
|
Scale as chord: 936:940:941:943:944:948:950:952:954:956:958:960:962: |
Reduced to mode 234: 234:235:941/4:943/4:236:237:475/2:238:477/2:239:479/2:240:481/2: |