311edo
| ← 310edo | 311edo | 312edo → |
311 equal divisions of the octave (abbreviated 311edo or 311ed2), also called 311-tone equal temperament (311tet) or 311 equal temperament (311et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 311 equal parts of about 3.86 ¢ each. Each step represents a frequency ratio of 21/311, or the 311th root of 2.
311edo is highly acclaimed for its large consistency limit and efficient and well-tempered just interval representation relative to its size.
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 (tempering out S25), 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.
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 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 | +8 | -12 | -9 | +12 | +16 | -20 | -11 | +17 | +17 | +24 | -14 | -20 | +43 | |
| 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.
As an interval size measure, one step of 311edo is called gene, named after Gene Ward Smith.
Intervals
The 41-limit add-73 add-89 add-101 add-109 add-113 123-odd-limit is represented very close to completely consistently, and as aforementioned, the 77-odd-limit subset of that odd-limit is perfectly consistent, to which a variety of odds can be added that keep perfect consistency, but for comprehensiveness and practical use as a temperament approximating the low-to-mid end of the harmonic series, we consider a larger odd-limit than that which seeks to be more complete.
There are 884 interval pairs in that odd limit (the 41-limit add-73 add-89 add-101 add-109 add-113 123-odd-limit), where "pairs" refers to that each interval has an octave complement with equal and opposite error. That odd limit can be described explicitly as the tonality diamond of {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 45, 49, 51, 55, 57, 63, 65, 69, 73, 75, 77, 81, 85, 87, 89, 91, 93, 95, 99, 101, 105, 109, 111, 113, 115, 117, 119, 121, 123}. We can also express that odd-limit as the 123-odd-limit minus only the following twelve prime odds: {43, 47, 53, 59, 61, 67, 71, 79, 83, 97, 103, 107}.
Of those 884 interval pairs, only 42 interval pairs (< 4.8%) are inconsistent, not mapped to the nearest interval of 311edo but to the second-nearest interval. Reduced to the lower half of the octave, these intervals, from smallest to largest, are: 101/100, 100/99, 82/81, 121/119, 119/117, 95/93, 87/85, 124/119, 85/81, 101/95, 100/93, 85/78, 93/85, 119/108, 93/82, 81/70, 138/119, 136/117, 99/85, 117/100, 95/81, 119/101, 101/85, 81/68, 140/117, 119/99, 117/95, 85/69, 100/81, 108/85, 119/93, 85/66, 156/119, 93/70, 162/119, 93/68, 119/87, 85/62, 117/85, 140/101, 164/117, 170/121.
Of them, only 6 interval pairs (119/117, 85/81, 93/85, 101/85, 119/93, 117/85) are more than 10% inconsistent, which is to say, all 36 of the other inconsistent intervals have less than 60% of a step of 311edo of error relative to where they are mapped in 311edo by the patent val, which is to say less than 3/5 = 60% relative interval error, which is equal to 2.3 ¢. The 6 highest-error intervals mentioned instead have less than 2/3 = 67% relative interval error.
The below table was generated by a simple Python 3 script to print it in plaintext using Godtone's code to simplify certain steps.
It should be noted that while almost all intervals shown in the table are intervals of the 123-odd-limit restricted to the aforementioned prime subgroup, the square-particulars up to S41 = (41/40)/(42/41) were added manually for completeness and reference in understanding the mapping of the 41-odd-limit by 311edo. Therefore, the very beginning of the table (from 0\311 to 3\311 inclusive) is the only part that is not algorithmically generated.
| Genes* | Cents | Marks | Approximate Intervals† |
|---|---|---|---|
| 0 | 0.0 | P1 | 1/1, S41, S40, S38, S37, S35 = S49*S50, S34, S32, S30, S28, S25, |
| 1 | 3.85 | S39, S36, S33, S31, S29, S27, S26 = S13/S15, S24, S23, S22, S21, S20, S19, S17 | |
| 2 | 7.71 | S18, S16, S15, S14, 170/169 | |
| 3 | 11.57 | S13 = 169/168, S12 = 144/143, 171/170 | |
| 4 | 15.43 | 124/123, 121/120, 120/119, 117/116, 116/115, 115/114, 114/113, 113/112, 112/111, 111/110, 110/109, 109/108, 105/104, 102/101, 100/99 | |
| 5 | 19.29 | 101/100, 99/98, 96/95, 93/92, 92/91, 91/90, 90/89, 89/88, 88/87, 85/84, 82/81 | |
| 6 | 23.15 | 81/80, 78/77, 77/76, 76/75, 75/74, 74/73, 73/72, 70/69 | |
| 7 | 27.0 | 69/68, 66/65, 65/64, 64/63, 63/62, 123/121, 119/117 | |
| 8 | 30.86 | sd2 | 121/119, 117/115, 58/57, 115/113, 57/56, 113/111, 56/55, 111/109, 55/54 |
| 9 | 34.72 | 52/51, 51/50, 101/99, 50/49, 49/48, 95/93 | |
| 10 | 38.58 | 93/91, 46/45, 91/89, 45/44, 89/87 | |
| 11 | 42.44 | 87/85, 42/41, 124/121, 41/40, 40/39, 119/116 | |
| 12 | 46.3 | 39/38, 116/113, 77/75, 115/112, 38/37, 113/110, 75/73, 112/109, 37/36 | |
| 13 | 50.16 | 36/35, 35/34, 104/101, 34/33 | |
| 14 | 54.01 | 101/98, 33/32, 98/95, 65/63, 32/31, 95/92 | |
| 15 | 57.87 | sA1 | 31/30, 123/119, 92/89, 91/88, 121/117, 30/29, 119/115 |
| 16 | 61.73 | 88/85, 117/113, 29/28, 115/111, 57/55, 85/82, 113/109, 28/27 | |
| 17 | 65.59 | 109/105, 27/26, 80/77, 105/101 | |
| 18 | 69.45 | 26/25, 77/74, 128/123, 51/49, 76/73, 126/121, 25/24 | |
| 19 | 73.31 | 124/119, 99/95, 73/70, 121/116, 24/23, 119/114, 95/91 | |
| 20 | 77.17 | 117/112, 93/89, 116/111, 23/22, 114/109, 91/87, 68/65, 113/108 | |
| 21 | 81.02 | 89/85, 22/21, 109/104, 65/62, 85/81 | |
| 22 | 84.88 | 21/20, 104/99, 41/39 | |
| 23 | 88.74 | m2 | 81/77, 101/96, 121/115, 20/19, 119/113, 98/93 |
| 24 | 92.6 | 39/37, 58/55, 77/73, 96/91, 115/109, 19/18 | |
| 25 | 96.46 | 93/88, 130/123, 37/35, 92/87, 55/52, 128/121, 73/69 | |
| 26 | 100.32 | 18/17, 89/84, 124/117, 123/116, 35/33 | |
| 27 | 104.18 | 87/82, 52/49, 121/114, 69/65, 120/113, 17/16 | |
| 28 | 108.03 | 101/95, 117/110, 116/109, 33/31, 115/108, 82/77, 49/46 | |
| 29 | 111.89 | 81/76, 16/15, 111/104, 95/89 | |
| 30 | 115.75 | A1 | 78/73, 109/102, 31/29, 108/101, 77/72, 123/115 |
| 31 | 119.61 | 91/85, 121/113, 15/14, 119/111, 74/69 | |
| 32 | 123.47 | 44/41, 117/109, 73/68, 102/95, 29/27, 130/121, 100/93 | |
| 33 | 127.33 | 128/119, 99/92, 113/105, 14/13 | |
| 34 | 131.18 | 69/64, 124/115, 55/51, 96/89, 41/38, 109/101, 68/63, 95/88 | |
| 35 | 135.04 | 27/25, 121/112, 40/37, 119/110 | |
| 36 | 138.9 | 92/85, 13/12 | |
| 37 | 142.76 | 89/82, 38/35, 101/93, 63/58, 88/81, 113/104, 25/23 | |
| 38 | 146.62 | N2 | 87/80, 62/57, 99/91, 37/34, 123/113, 49/45, 110/101, 85/78 |
| 39 | 150.48 | 109/100, 121/111, 12/11, 119/109, 95/87 | |
| 40 | 154.34 | 130/119, 82/75, 35/32, 128/117 | |
| 41 | 158.19 | 93/85, 81/74, 104/95, 23/21, 126/115, 80/73, 57/52, 34/31 | |
| 42 | 162.05 | 124/113, 45/41, 101/92, 56/51, 123/112, 89/81, 100/91, 111/101 | |
| 43 | 165.91 | 11/10, 120/109, 109/99, 98/89, 76/69, 119/108 | |
| 44 | 169.77 | 54/49, 75/68, 32/29, 85/77 | |
| 45 | 173.63 | 116/105, 21/19, 136/123, 115/104, 73/66 | |
| 46 | 177.49 | d3 | 31/28, 72/65, 113/102, 41/37, 51/46, 112/101 |
| 47 | 181.35 | 132/119, 81/73, 91/82, 101/91, 111/100, 121/109, 10/9 | |
| 48 | 185.2 | 109/98, 99/89, 89/80, 69/62, 128/115, 49/44 | |
| 49 | 189.06 | 39/35, 126/113, 29/26, 77/69 | |
| 50 | 192.92 | 124/111, 19/17, 123/110, 104/93, 85/76, 113/101 | |
| 51 | 196.78 | 28/25, 121/108, 65/58, 102/91, 37/33 | |
| 52 | 200.64 | 46/41, 101/90, 55/49, 64/57, 73/65, 82/73, 91/81, 100/89, 136/121 | |
| 53 | 204.5 | M2 | 9/8, 98/87 |
| 54 | 208.36 | 62/55, 115/102, 44/39, 123/109, 114/101, 35/31 | |
| 55 | 212.21 | 96/85, 87/77, 113/100, 26/23, 95/84, 112/99 | |
| 56 | 216.07 | 77/68, 111/98, 128/113, 17/15 | |
| 57 | 219.93 | 93/82, 101/89, 42/37, 109/96, 92/81, 25/22 | |
| 58 | 223.79 | 108/95, 58/51, 91/80, 124/109, 33/29, 140/123, 74/65, 115/101, 41/36 | |
| 59 | 227.65 | 57/50, 65/57, 138/121, 73/64, 89/78, 105/92, 113/99 | |
| 60 | 231.51 | 8/7, 119/104 | |
| 61 | 235.36 | sd3 | 87/76, 63/55, 55/48, 102/89 |
| 62 | 239.22 | 39/34, 109/95, 101/88, 132/115, 31/27, 116/101, 85/74, 100/87 | |
| 63 | 243.08 | 23/20, 130/113, 84/73, 38/33 | |
| 64 | 246.94 | 121/105, 98/85, 113/98, 128/111, 15/13 | |
| 65 | 250.8 | 52/45, 89/77, 126/109, 37/32, 140/121 | |
| 66 | 254.66 | 81/70, 22/19, 117/101, 95/82, 73/63, 51/44, 80/69 | |
| 67 | 258.52 | 138/119, 29/25, 65/56, 101/87, 36/31, 115/99, 136/117 | |
| 68 | 262.37 | sA2 | 93/80, 57/49, 121/104, 64/55, 85/73 |
| 69 | 266.23 | 99/85, 7/6 | |
| 70 | 270.09 | 132/113, 111/95, 104/89, 90/77, 76/65 | |
| 71 | 273.95 | 117/100, 48/41, 89/76, 130/111, 41/35, 116/99, 75/64, 109/93, 34/29, 95/81 | |
| 72 | 277.81 | 88/75, 115/98, 27/23, 128/109, 74/63 | |
| 73 | 281.67 | 87/74, 20/17, 113/96, 73/62, 119/101 | |
| 74 | 285.53 | 33/28, 112/95, 46/39, 105/89, 85/72 | |
| 75 | 289.38 | 124/105, 13/11, 136/115, 123/104, 110/93 | |
| 76 | 293.24 | m3 | 58/49, 45/38, 77/65, 109/92, 32/27 |
| 77 | 297.1 | 121/102, 89/75, 108/91, 146/123, 19/16, 120/101, 82/69 | |
| 78 | 300.96 | 101/85, 44/37, 113/95, 69/58, 119/100, 144/121, 25/21 | |
| 79 | 304.82 | 81/68, 87/73, 31/26, 130/109, 68/57, 105/88, 37/31 | |
| 80 | 308.68 | 117/98, 92/77, 49/41, 104/87, 55/46, 140/117 | |
| 81 | 312.54 | 91/76, 109/91, 115/96, 121/101 | |
| 82 | 316.39 | 6/5, 119/99 | |
| 83 | 320.25 | A2 | 101/84, 89/74, 77/64, 148/123, 136/113, 65/54, 112/93 |
| 84 | 324.11 | 88/73, 41/34, 76/63, 111/92, 146/121, 35/29 | |
| 85 | 327.97 | 99/82, 93/77, 29/24, 110/91, 75/62, 98/81 | |
| 86 | 331.83 | 121/100, 144/119, 23/19, 132/109, 109/90, 63/52, 40/33 | |
| 87 | 335.69 | 91/75, 108/89, 17/14, 113/93 | |
| 88 | 339.54 | 62/51, 45/37, 73/60, 28/23, 123/101, 95/78 | |
| 89 | 343.4 | 39/32, 128/105, 89/73, 50/41, 111/91 | |
| 90 | 347.26 | 116/95, 138/113, 11/9, 148/121 | |
| 91 | 351.12 | N3 | 104/85, 93/76, 60/49, 109/89, 49/40, 136/111, 38/31 |
| 92 | 354.98 | 92/75, 146/119, 27/22, 124/101, 70/57, 113/92 | |
| 93 | 358.84 | 91/74, 123/100, 16/13, 85/69 | |
| 94 | 362.7 | 117/95, 101/82, 69/56, 90/73, 37/30, 95/77, 100/81 | |
| 95 | 366.55 | 121/98, 21/17, 152/123, 110/89, 89/72, 68/55, 115/93 | |
| 96 | 370.41 | 99/80, 26/21, 109/88, 140/113, 57/46, 119/96, 150/121 | |
| 97 | 374.27 | 31/25, 36/29, 113/91, 77/62, 41/33 | |
| 98 | 378.13 | 87/70, 46/37, 148/119, 51/41, 56/45 | |
| 99 | 381.99 | d4 | 81/65, 91/73, 96/77, 101/81, 111/89, 116/93, 126/101, 136/109, 146/117 |
| 100 | 385.85 | 5/4 | |
| 101 | 389.71 | 154/123, 144/115, 124/99, 119/95, 114/91, 109/87 | |
| 102 | 393.56 | 69/55, 64/51, 123/98, 113/90, 152/121, 49/39 | |
| 103 | 397.42 | 93/74, 44/35, 39/31, 112/89, 73/58, 34/27 | |
| 104 | 401.28 | 63/50, 92/73, 121/96, 150/119, 29/23, 140/111, 111/88, 82/65 | |
| 105 | 405.14 | 101/80, 24/19, 115/91, 91/72, 110/87, 148/117 | |
| 106 | 409.0 | M3 | 62/49, 81/64, 138/109, 19/15, 128/101 |
| 107 | 412.86 | 52/41, 33/26, 146/115, 113/89, 80/63 | |
| 108 | 416.72 | 108/85, 89/70, 117/92, 14/11 | |
| 109 | 420.57 | 121/95, 93/73, 144/113, 65/51, 116/91, 51/40, 88/69, 37/29 | |
| 110 | 424.43 | 152/119, 23/18, 119/93 | |
| 111 | 428.29 | 87/68, 32/25, 105/82, 73/57, 114/89, 41/32, 50/39 | |
| 112 | 432.15 | 109/85, 77/60, 95/74, 104/81, 113/88, 140/109 | |
| 113 | 436.01 | 9/7, 148/115, 130/101, 112/87, 85/66 | |
| 114 | 439.87 | sd4 | 58/45, 156/121, 49/38, 89/69, 40/31 |
| 115 | 443.72 | 31/24, 146/113, 115/89, 84/65, 128/99, 75/58, 119/92 | |
| 116 | 447.58 | 22/17, 123/95, 101/78, 136/105, 57/44, 35/27 | |
| 117 | 451.44 | 48/37, 109/84, 74/57, 100/77, 113/87, 152/117 | |
| 118 | 455.3 | 13/10, 160/123, 121/93, 95/73, 82/63 | |
| 119 | 459.16 | 99/76, 116/89, 73/56, 30/23 | |
| 120 | 463.02 | 124/95, 111/85, 64/49, 81/62, 98/75, 115/88, 132/101, 17/13 | |
| 121 | 466.88 | sA3 | 89/68, 72/55, 55/42, 148/113, 38/29 |
| 122 | 470.73 | 156/119, 101/77, 21/16, 130/99 | |
| 123 | 474.59 | 46/35, 117/89, 96/73, 121/92, 146/111, 25/19, 154/117 | |
| 124 | 478.45 | 54/41, 112/85, 29/22, 120/91, 91/69, 95/72 | |
| 125 | 482.31 | 33/25, 144/109, 37/28, 152/115, 115/87, 119/90, 160/121, 41/31 | |
| 126 | 486.17 | 45/34, 49/37, 102/77 | |
| 127 | 490.03 | 126/95, 65/49, 69/52, 73/55, 150/113, 77/58, 85/64 | |
| 128 | 493.89 | 93/70, 101/76, 109/82, 113/85, 117/88, 121/91 | |
| 129 | 497.74 | P4 | 4/3 |
| 130 | 501.6 | 123/92, 119/89 | |
| 131 | 505.46 | 99/74, 91/68, 87/65, 162/121, 154/115, 75/56, 146/109 | |
| 132 | 509.32 | 114/85, 55/41, 51/38, 98/73 | |
| 133 | 513.18 | 121/90, 160/119, 39/29, 152/113, 113/84, 74/55, 109/81, 35/26, 136/101 | |
| 134 | 517.04 | 101/75, 66/49, 128/95, 31/23, 120/89, 89/66, 85/63 | |
| 135 | 520.9 | 27/20, 104/77, 77/57, 50/37, 123/91, 73/54, 119/88 | |
| 136 | 524.75 | A3 | 23/17, 111/82, 88/65, 65/48, 42/31, 164/121 |
| 137 | 528.61 | 99/73, 156/115, 19/14, 148/109, 110/81 | |
| 138 | 532.47 | 87/64, 121/89, 34/25, 49/36 | |
| 139 | 536.33 | 162/119, 109/80, 124/91, 154/113, 15/11 | |
| 140 | 540.19 | 116/85, 101/74, 56/41, 138/101, 41/30, 160/117, 119/87 | |
| 141 | 544.05 | 93/68, 26/19, 115/84, 89/65, 152/111, 63/46, 100/73, 37/27, 85/62 | |
| 142 | 547.9 | 48/35, 70/51, 136/99 | |
| 143 | 551.76 | 11/8, 150/109, 128/93, 95/69 | |
| 144 | 555.62 | sA4 | 117/85, 62/45, 113/82, 164/119, 51/37, 91/66, 40/29 |
| 145 | 559.48 | 69/50, 156/113, 29/21, 105/76, 76/55, 123/89, 170/123, 112/81 | |
| 146 | 563.34 | 101/73, 18/13, 140/101 | |
| 147 | 567.2 | 104/75, 154/111, 111/80, 68/49, 168/121, 25/18 | |
| 148 | 571.06 | 132/95, 57/41, 146/105, 89/64, 121/87, 32/23 | |
| 149 | 574.91 | 39/28, 124/89, 46/33, 152/109, 113/81 | |
| 150 | 578.77 | 81/58, 88/63, 95/68, 102/73, 109/78, 123/88, 130/93 | |
| 151 | 582.63 | 7/5, 164/117 | |
| 152 | 586.49 | d5 | 115/82, 108/77, 101/72, 87/62, 80/57, 73/52, 170/121 |
| 153 | 590.35 | 52/37, 45/32, 128/91, 38/27 | |
| 154 | 594.21 | 69/49, 162/115, 31/22, 148/105, 55/39 | |
| 155 | 598.07 | 24/17, 113/80, 89/63, 154/109, 65/46, 41/29, 140/99 | |
| 156 | 601.92 | 99/70, 58/41, 92/65, 109/77, 126/89, 160/113, 17/12 | |
| 157 | 605.78 | 78/55, 105/74, 44/31, 115/81, 98/69 | |
| 158 | 609.64 | 27/19, 91/64, 64/45, 37/26 | |
| 159 | 613.5 | A4 | 121/85, 104/73, 57/40, 124/87, 144/101, 77/54, 164/115 |
| 160 | 617.36 | 117/82, 10/7 | |
| 161 | 621.22 | 93/65, 176/123, 156/109, 73/51, 136/95, 63/44, 116/81 | |
| 162 | 625.08 | 162/113, 109/76, 33/23, 89/62, 56/39 | |
| 163 | 628.93 | 23/16, 174/121, 128/89, 105/73, 82/57, 95/66 | |
| 164 | 632.79 | 36/25, 121/84, 49/34, 160/111, 111/77, 75/52 | |
| 165 | 636.65 | 101/70, 13/9, 146/101 | |
| 166 | 640.51 | 81/56, 123/85, 178/123, 55/38, 152/105, 42/29, 113/78, 100/69 | |
| 167 | 644.37 | sd5 | 29/20, 132/91, 74/51, 119/82, 164/113, 45/31, 170/117 |
| 168 | 648.23 | 138/95, 93/64, 109/75, 16/11 | |
| 169 | 652.09 | 99/68, 51/35, 35/24 | |
| 170 | 655.94 | 124/85, 54/37, 73/50, 92/63, 111/76, 130/89, 168/115, 19/13, 136/93 | |
| 171 | 659.8 | 174/119, 117/80, 60/41, 101/69, 41/28, 148/101, 85/58 | |
| 172 | 663.66 | 22/15, 113/77, 91/62, 160/109, 119/81 | |
| 173 | 667.52 | 72/49, 25/17, 178/121, 128/87 | |
| 174 | 671.38 | 81/55, 109/74, 28/19, 115/78, 146/99 | |
| 175 | 675.24 | d6 | 121/82, 31/21, 96/65, 65/44, 164/111, 34/23 |
| 176 | 679.09 | 176/119, 108/73, 182/123, 37/25, 114/77, 77/52, 40/27 | |
| 177 | 682.95 | 126/85, 132/89, 89/60, 46/31, 95/64, 49/33, 150/101 | |
| 178 | 686.81 | 101/68, 52/35, 162/109, 55/37, 168/113, 113/76, 58/39, 119/80, 180/121 | |
| 179 | 690.67 | 73/49, 76/51, 82/55, 85/57 | |
| 180 | 694.53 | 109/73, 112/75, 115/77, 121/81, 130/87, 136/91, 148/99 | |
| 181 | 698.39 | 178/119, 184/123 | |
| 182 | 702.25 | P5 | 3/2 |
| 183 | 706.1 | 182/121, 176/117, 170/113, 164/109, 152/101, 140/93 | |
| 184 | 709.96 | 128/85, 116/77, 113/75, 110/73, 104/69, 98/65, 95/63 | |
| 185 | 713.82 | 77/51, 74/49, 68/45 | |
| 186 | 717.68 | 62/41, 121/80, 180/119, 174/115, 115/76, 56/37, 109/72, 50/33 | |
| 187 | 721.54 | 144/95, 138/91, 91/60, 44/29, 85/56, 41/27 | |
| 188 | 725.4 | 117/77, 38/25, 111/73, 184/121, 73/48, 178/117, 35/23 | |
| 189 | 729.26 | 99/65, 32/21, 154/101, 119/78 | |
| 190 | 733.11 | sd6 | 29/19, 113/74, 84/55, 55/36, 136/89 |
| 191 | 736.97 | 26/17, 101/66, 176/115, 75/49, 124/81, 49/32, 170/111, 95/62 | |
| 192 | 740.83 | 23/15, 112/73, 89/58, 152/99 | |
| 193 | 744.69 | 63/41, 146/95, 186/121, 123/80, 20/13 | |
| 194 | 748.55 | 117/76, 174/113, 77/50, 57/37, 168/109, 37/24 | |
| 195 | 752.41 | 54/35, 88/57, 105/68, 156/101, 190/123, 17/11 | |
| 196 | 756.27 | 184/119, 116/75, 99/64, 65/42, 178/115, 113/73, 48/31 | |
| 197 | 760.12 | sA5 | 31/20, 138/89, 76/49, 121/78, 45/29 |
| 198 | 763.98 | 132/85, 87/56, 101/65, 115/74, 14/9 | |
| 199 | 767.84 | 109/70, 176/113, 81/52, 148/95, 120/77, 170/109 | |
| 200 | 771.7 | 39/25, 64/41, 89/57, 114/73, 164/105, 25/16, 136/87 | |
| 201 | 775.56 | 186/119, 36/23, 119/76 | |
| 202 | 779.42 | 58/37, 69/44, 80/51, 91/58, 102/65, 113/72, 146/93, 190/121 | |
| 203 | 783.27 | 11/7, 184/117, 140/89, 85/54 | |
| 204 | 787.13 | 63/40, 178/113, 115/73, 52/33, 41/26 | |
| 205 | 790.99 | m6 | 101/64, 30/19, 109/69, 128/81, 49/31 |
| 206 | 794.85 | 117/74, 87/55, 144/91, 182/115, 19/12, 160/101 | |
| 207 | 798.71 | 65/41, 176/111, 111/70, 46/29, 119/75, 192/121, 73/46, 100/63 | |
| 208 | 802.57 | 27/17, 116/73, 89/56, 62/39, 35/22, 148/93 | |
| 209 | 806.43 | 78/49, 121/76, 180/113, 196/123, 51/32, 110/69 | |
| 210 | 810.28 | 174/109, 91/57, 190/119, 99/62, 115/72, 123/77 | |
| 211 | 814.14 | 8/5 | |
| 212 | 818.0 | A5 | 117/73, 109/68, 101/63, 93/58, 178/111, 162/101, 77/48, 146/91, 130/81 |
| 213 | 821.86 | 45/28, 82/51, 119/74, 37/23, 140/87 | |
| 214 | 825.72 | 66/41, 124/77, 182/113, 29/18, 50/31 | |
| 215 | 829.58 | 121/75, 192/119, 92/57, 113/70, 176/109, 21/13, 160/99 | |
| 216 | 833.44 | 186/115, 55/34, 144/89, 89/55, 123/76, 34/21, 196/121 | |
| 217 | 837.29 | 81/50, 154/95, 60/37, 73/45, 112/69, 164/101, 190/117 | |
| 218 | 841.15 | 138/85, 13/8, 200/123, 148/91 | |
| 219 | 845.01 | 184/113, 57/35, 101/62, 44/27, 119/73, 75/46 | |
| 220 | 848.87 | N6 | 31/19, 111/68, 80/49, 178/109, 49/30, 152/93, 85/52 |
| 221 | 852.73 | 121/74, 18/11, 113/69, 95/58 | |
| 222 | 856.59 | 182/111, 41/25, 146/89, 105/64, 64/39 | |
| 223 | 860.45 | 156/95, 202/123, 23/14, 120/73, 74/45, 51/31 | |
| 224 | 864.3 | 186/113, 28/17, 89/54, 150/91 | |
| 225 | 868.16 | 33/20, 104/63, 180/109, 109/66, 38/23, 119/72, 200/121 | |
| 226 | 872.02 | 81/49, 124/75, 91/55, 48/29, 154/93, 164/99 | |
| 227 | 875.88 | 58/35, 121/73, 184/111, 63/38, 68/41, 73/44 | |
| 228 | 879.74 | d7 | 93/56, 108/65, 113/68, 123/74, 128/77, 148/89, 168/101 |
| 229 | 883.6 | 198/119, 5/3 | |
| 230 | 887.45 | 202/121, 192/115, 182/109, 152/91 | |
| 231 | 891.31 | 117/70, 92/55, 87/52, 82/49, 77/46, 196/117 | |
| 232 | 895.17 | 62/37, 176/105, 57/34, 109/65, 52/31, 146/87, 136/81 | |
| 233 | 899.03 | 42/25, 121/72, 200/119, 116/69, 190/113, 37/22, 170/101 | |
| 234 | 902.89 | 69/41, 101/60, 32/19, 123/73, 91/54, 150/89, 204/121 | |
| 235 | 906.75 | M6 | 27/16, 184/109, 130/77, 76/45, 49/29 |
| 236 | 910.61 | 93/55, 208/123, 115/68, 22/13, 105/62 | |
| 237 | 914.46 | 144/85, 178/105, 39/23, 95/56, 56/33 | |
| 238 | 918.32 | 202/119, 124/73, 192/113, 17/10, 148/87 | |
| 239 | 922.18 | 63/37, 109/64, 46/27, 196/115, 75/44 | |
| 240 | 926.04 | 162/95, 29/17, 186/109, 128/75, 99/58, 70/41, 111/65, 152/89, 41/24, 200/117 | |
| 241 | 929.9 | 65/38, 77/45, 89/52, 190/111, 113/66 | |
| 242 | 933.76 | 12/7, 170/99 | |
| 243 | 937.62 | sd7 | 146/85, 55/32, 208/121, 98/57, 160/93 |
| 244 | 941.47 | 117/68, 198/115, 31/18, 174/101, 112/65, 50/29, 119/69 | |
| 245 | 945.33 | 69/40, 88/51, 126/73, 164/95, 202/117, 19/11, 140/81 | |
| 246 | 949.19 | 121/70, 64/37, 109/63, 154/89, 45/26 | |
| 247 | 953.05 | 26/15, 111/64, 196/113, 85/49, 210/121 | |
| 248 | 956.91 | 33/19, 73/42, 113/65, 40/23 | |
| 249 | 960.77 | 87/50, 148/85, 101/58, 54/31, 115/66, 176/101, 190/109, 68/39 | |
| 250 | 964.63 | sA6 | 89/51, 96/55, 110/63, 152/87 |
| 251 | 968.48 | 208/119, 7/4 | |
| 252 | 972.34 | 198/113, 184/105, 156/89, 128/73, 121/69, 114/65, 100/57 | |
| 253 | 976.2 | 72/41, 202/115, 65/37, 123/70, 58/33, 109/62, 160/91, 51/29, 95/54 | |
| 254 | 980.06 | 44/25, 81/46, 192/109, 37/21, 178/101, 164/93 | |
| 255 | 983.92 | 30/17, 113/64, 196/111, 136/77 | |
| 256 | 987.78 | 99/56, 168/95, 23/13, 200/113, 154/87, 85/48 | |
| 257 | 991.63 | 62/35, 101/57, 218/123, 39/22, 204/115, 55/31 | |
| 258 | 995.49 | m7 | 87/49, 16/9 |
| 259 | 999.35 | 121/68, 89/50, 162/91, 73/41, 130/73, 57/32, 98/55, 180/101, 41/23 | |
| 260 | 1003.21 | 66/37, 91/51, 116/65, 216/121, 25/14 | |
| 261 | 1007.07 | 202/113, 152/85, 93/52, 220/123, 34/19, 111/62 | |
| 262 | 1010.93 | 138/77, 52/29, 113/63, 70/39 | |
| 263 | 1014.79 | 88/49, 115/64, 124/69, 160/89, 178/99, 196/109 | |
| 264 | 1018.64 | 9/5, 218/121, 200/111, 182/101, 164/91, 146/81, 119/66 | |
| 265 | 1022.5 | A6 | 101/56, 92/51, 74/41, 204/113, 65/36, 56/31 |
| 266 | 1026.36 | 132/73, 208/115, 123/68, 38/21, 105/58 | |
| 267 | 1030.22 | 154/85, 29/16, 136/75, 49/27 | |
| 268 | 1034.08 | 216/119, 69/38, 89/49, 198/109, 109/60, 20/11 | |
| 269 | 1037.94 | 202/111, 91/50, 162/89, 224/123, 51/28, 184/101, 82/45, 113/62 | |
| 270 | 1041.8 | 31/17, 104/57, 73/40, 115/63, 42/23, 95/52, 148/81, 170/93 | |
| 271 | 1045.65 | 117/64, 64/35, 75/41, 119/65 | |
| 272 | 1049.51 | 174/95, 218/119, 11/6, 222/121, 200/109 | |
| 273 | 1053.37 | N7 | 156/85, 101/55, 90/49, 226/123, 68/37, 182/99, 57/31, 160/87 |
| 274 | 1057.23 | 46/25, 208/113, 81/44, 116/63, 186/101, 35/19, 164/89 | |
| 275 | 1061.09 | 24/13, 85/46 | |
| 276 | 1064.95 | 220/119, 37/20, 224/121, 50/27 | |
| 277 | 1068.81 | 176/95, 63/34, 202/109, 76/41, 89/48, 102/55, 115/62, 128/69 | |
| 278 | 1072.66 | 13/7, 210/113, 184/99, 119/64 | |
| 279 | 1076.52 | 93/50, 121/65, 54/29, 95/51, 136/73, 218/117, 41/22 | |
| 280 | 1080.38 | 69/37, 222/119, 28/15, 226/121, 170/91 | |
| 281 | 1084.24 | d8 | 230/123, 144/77, 101/54, 58/31, 204/109, 73/39 |
| 282 | 1088.1 | 178/95, 208/111, 15/8, 152/81 | |
| 283 | 1091.96 | 92/49, 77/41, 216/115, 62/33, 109/58, 220/117, 190/101 | |
| 284 | 1095.81 | 32/17, 113/60, 130/69, 228/121, 49/26, 164/87 | |
| 285 | 1099.67 | 66/35, 232/123, 117/62, 168/89, 17/9 | |
| 286 | 1103.53 | 138/73, 121/64, 104/55, 87/46, 70/37, 123/65, 176/93 | |
| 287 | 1107.39 | 36/19, 218/115, 91/48, 146/77, 55/29, 74/39 | |
| 288 | 1111.25 | M7 | 93/49, 226/119, 19/10, 230/121, 192/101, 154/81 |
| 289 | 1115.11 | 78/41, 99/52, 40/21 | |
| 290 | 1118.97 | 162/85, 124/65, 208/109, 21/11, 170/89 | |
| 291 | 1122.82 | 216/113, 65/34, 174/91, 109/57, 44/23, 111/58, 178/93, 224/117 | |
| 292 | 1126.68 | 182/95, 228/119, 23/12, 232/121, 140/73, 190/99, 119/62 | |
| 293 | 1130.54 | 48/25, 121/63, 73/38, 98/51, 123/64, 148/77, 25/13 | |
| 294 | 1134.4 | 202/105, 77/40, 52/27, 210/109 | |
| 295 | 1138.26 | 27/14, 218/113, 164/85, 110/57, 222/115, 56/29, 226/117, 85/44 | |
| 296 | 1142.12 | sd8 | 230/119, 29/15, 234/121, 176/91, 89/46, 238/123, 60/31 |
| 297 | 1145.98 | 184/95, 31/16, 126/65, 95/49, 64/33, 196/101 | |
| 298 | 1149.83 | 33/17, 101/52, 68/35, 35/18 | |
| 299 | 1153.69 | 72/37, 109/56, 146/75, 220/113, 37/19, 224/115, 150/77, 113/58, 76/39 | |
| 300 | 1157.55 | 232/119, 39/20, 80/41, 121/62, 41/21, 170/87 | |
| 301 | 1161.41 | 174/89, 88/45, 178/91, 45/23, 182/93 | |
| 302 | 1165.27 | 186/95, 96/49, 49/25, 198/101, 100/51, 51/26 | |
| 303 | 1169.13 | sA7 | 108/55, 218/111, 55/28, 222/113, 112/57, 226/115, 57/29, 230/117, 238/121 |
| 304 | 1172.99 | 234/119, 242/123, 124/63, 63/32, 128/65, 65/33, 136/69 | |
| 305 | 1176.84 | 69/35, 144/73, 73/37, 148/75, 75/38, 152/77, 77/39, 160/81 | |
| 306 | 1180.7 | 81/41, 168/85, 87/44, 176/89, 89/45, 180/91, 91/46, 184/93, 95/48, 196/99, 200/101 | |
| 307 | 1184.56 | 99/50, 101/51, 208/105, 216/109, 109/55, 220/111, 111/56, 224/113, 113/57, 228/115, 115/58, 232/117, 119/60, 240/121, 123/62 | |
| 308 | 1188.42 | ||
| 309 | 1192.28 | ||
| 310 | 1196.14 | ||
| 311 | 1200.0 | P8 | 2/1 |
* gene is the interval size measure for 311edo, named after Gene Ward Smith
† odd harmonics and subharmonics are in bold and linked, inconsistent intervals in italics, all 23-limit intervals linked)
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 |
- 311et has lower relative errors than any previous equal temperaments in the 23-limit and beyond. In the 23-limit it beats 282 and is bettered by 373g in terms of absolute error, and by 581 in terms of relative error.
- 311et is also notable in the 17- and 19-limit, with lower absolute errors than any previous equal temperaments, beating 270 in both subgroups and is bettered by 354 in the 17-limit, and by 400 in the 19-limit.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | 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 | 142\311 | 547.92 | 48/35 | Calamity |
| 1 | 143\311 | 551.77 | 11/8 | Emkay |
| 1 | 155\311 | 598.08 | 572/405 | Vydubychi |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Detemperaments
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: |
Music
- Etude in C, Op. 1, No. 1 (2022)
- Baoyu（𨰻𨰻） (2023) – for electric organs tuned in 311edo