311edo
| ← 310edo | 311edo | 312edo → |
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 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.
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 (%) | +0.0 | +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.
As an interval size measure, one step of 311edo is called gene, named after Gene Ward Smith.
Intervals
| genes | intervals of 41-limit add-73 add-89 add-101 add-109 add-113 (/2^n and 2^n/ in bold & linked, inconsistent in italics, else, if 23-limit, linked) |
| 0 | 1/1, S41, S40, S38, S37, S35 = S49*S50, S34, S32, S30, S28, S25, |
| 1 | S39, S36, S33, S31, S29, S27, S26 = S13/S15, S24, S23, S22, S21, S20, S19, S17 |
| 2 | S18, S16, S15, S14, 170/169 |
| 3 | S13 = 169/168, S12 = 144/143, 171/170 |
| 4 | 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 | 101/100, 99/98, 96/95, 93/92, 92/91, 91/90, 90/89, 89/88, 88/87, 85/84, 82/81 |
| 6 | 81/80, 78/77, 77/76, 76/75, 75/74, 74/73, 73/72, 70/69 |
| 7 | 69/68, 66/65, 65/64, 64/63, 63/62, 123/121, 119/117 |
| 8 | 121/119, 117/115, 58/57, 115/113, 57/56, 113/111, 56/55, 111/109, 55/54 |
| 9 | 52/51, 51/50, 101/99, 50/49, 49/48, 95/93 |
| 10 | 93/91, 46/45, 91/89, 45/44, 89/87 |
| 11 | 87/85, 42/41, 124/121, 41/40, 40/39, 119/116 |
| 12 | 39/38, 116/113, 77/75, 115/112, 38/37, 113/110, 75/73, 112/109, 37/36 |
| 13 | 36/35, 35/34, 104/101, 34/33 |
| 14 | 101/98, 33/32, 98/95, 65/63, 32/31, 95/92 |
| 15 | 31/30, 123/119, 92/89, 91/88, 121/117, 30/29, 119/115 |
| 16 | 88/85, 117/113, 29/28, 115/111, 57/55, 85/82, 113/109, 28/27 |
| 17 | 109/105, 27/26, 80/77, 105/101 |
| 18 | 26/25, 77/74, 128/123, 51/49, 76/73, 126/121, 25/24 |
| 19 | 124/119, 99/95, 73/70, 121/116, 24/23, 119/114, 95/91 |
| 20 | 117/112, 93/89, 116/111, 23/22, 114/109, 91/87, 68/65, 113/108 |
| 21 | 89/85, 22/21, 109/104, 65/62, 85/81 |
| 22 | 21/20, 104/99, 41/39 |
| 23 | 81/77, 101/96, 121/115, 20/19, 119/113, 98/93 |
| 24 | 39/37, 58/55, 77/73, 96/91, 115/109, 19/18 |
| 25 | 93/88, 130/123, 37/35, 92/87, 55/52, 128/121, 73/69 |
| 26 | 18/17, 89/84, 124/117, 123/116, 35/33 |
| 27 | 87/82, 52/49, 121/114, 69/65, 120/113, 17/16 |
| 28 | 101/95, 117/110, 116/109, 33/31, 115/108, 82/77, 49/46 |
| 29 | 81/76, 16/15, 111/104, 95/89 |
| 30 | 78/73, 109/102, 31/29, 108/101, 77/72, 123/115 |
| 31 | 91/85, 121/113, 15/14, 119/111, 74/69 |
| 32 | 44/41, 117/109, 73/68, 102/95, 29/27, 130/121, 100/93 |
| 33 | 128/119, 99/92, 113/105, 14/13 |
| 34 | 69/64, 124/115, 55/51, 96/89, 41/38, 109/101, 68/63, 95/88 |
| 35 | 27/25, 121/112, 40/37, 119/110 |
| 36 | 92/85, 13/12 |
| 37 | 89/82, 38/35, 101/93, 63/58, 88/81, 113/104, 25/23 |
| 38 | 87/80, 62/57, 99/91, 37/34, 123/113, 49/45, 110/101, 85/78 |
| 39 | 109/100, 121/111, 12/11, 119/109, 95/87 |
| 40 | 130/119, 82/75, 35/32, 128/117 |
| 41 | 93/85, 81/74, 104/95, 23/21, 126/115, 80/73, 57/52, 34/31 |
| 42 | 124/113, 45/41, 101/92, 56/51, 123/112, 89/81, 100/91, 111/101 |
| 43 | 11/10, 120/109, 109/99, 98/89, 76/69, 119/108 |
| 44 | 54/49, 75/68, 32/29, 85/77 |
| 45 | 116/105, 21/19, 136/123, 115/104, 73/66 |
| 46 | 31/28, 72/65, 113/102, 41/37, 51/46, 112/101 |
| 47 | 132/119, 81/73, 91/82, 101/91, 111/100, 121/109, 10/9 |
| 48 | 109/98, 99/89, 89/80, 69/62, 128/115, 49/44 |
| 49 | 39/35, 126/113, 29/26, 77/69 |
| 50 | 124/111, 19/17, 123/110, 104/93, 85/76, 113/101 |
| 51 | 28/25, 121/108, 65/58, 102/91, 37/33 |
| 52 | 46/41, 101/90, 55/49, 64/57, 73/65, 82/73, 91/81, 100/89, 136/121 |
| 53 | 9/8, 98/87 |
| 54 | 62/55, 115/102, 44/39, 123/109, 114/101, 35/31 |
| 55 | 96/85, 87/77, 113/100, 26/23, 95/84, 112/99 |
| 56 | 77/68, 111/98, 128/113, 17/15 |
| 57 | 93/82, 101/89, 42/37, 109/96, 92/81, 25/22 |
| 58 | 108/95, 58/51, 91/80, 124/109, 33/29, 140/123, 74/65, 115/101, 41/36 |
| 59 | 57/50, 65/57, 138/121, 73/64, 89/78, 105/92, 113/99 |
| 60 | 8/7, 119/104 |
| 61 | 87/76, 63/55, 55/48, 102/89 |
| 62 | 39/34, 109/95, 101/88, 132/115, 31/27, 116/101, 85/74, 100/87 |
| 63 | 23/20, 130/113, 84/73, 38/33 |
| 64 | 121/105, 98/85, 113/98, 128/111, 15/13 |
| 65 | 52/45, 89/77, 126/109, 37/32, 140/121 |
| 66 | 81/70, 22/19, 117/101, 95/82, 73/63, 51/44, 80/69 |
| 67 | 138/119, 29/25, 65/56, 101/87, 36/31, 115/99, 136/117 |
| 68 | 93/80, 57/49, 121/104, 64/55, 85/73 |
| 69 | 99/85, 7/6 |
| 70 | 132/113, 111/95, 104/89, 90/77, 76/65 |
| 71 | 117/100, 48/41, 89/76, 130/111, 41/35, 116/99, 75/64, 109/93, 34/29, 95/81 |
| 72 | 88/75, 115/98, 27/23, 128/109, 74/63 |
| 73 | 87/74, 20/17, 113/96, 73/62, 119/101 |
| 74 | 33/28, 112/95, 46/39, 105/89, 85/72 |
| 75 | 124/105, 13/11, 136/115, 123/104, 110/93 |
| 76 | 58/49, 45/38, 77/65, 109/92, 32/27 |
| 77 | 121/102, 89/75, 108/91, 146/123, 19/16, 120/101, 82/69 |
| 78 | 101/85, 44/37, 113/95, 69/58, 119/100, 144/121, 25/21 |
| 79 | 81/68, 87/73, 31/26, 130/109, 68/57, 105/88, 37/31 |
| 80 | 117/98, 92/77, 49/41, 104/87, 55/46, 140/117 |
| 81 | 91/76, 109/91, 115/96, 121/101 |
| 82 | 6/5, 119/99 |
| 83 | 101/84, 89/74, 77/64, 148/123, 136/113, 65/54, 112/93 |
| 84 | 88/73, 41/34, 76/63, 111/92, 146/121, 35/29 |
| 85 | 99/82, 93/77, 29/24, 110/91, 75/62, 98/81 |
| 86 | 121/100, 144/119, 23/19, 132/109, 109/90, 63/52, 40/33 |
| 87 | 91/75, 108/89, 17/14, 113/93 |
| 88 | 62/51, 45/37, 73/60, 28/23, 123/101, 95/78 |
| 89 | 39/32, 128/105, 89/73, 50/41, 111/91 |
| 90 | 116/95, 138/113, 11/9, 148/121 |
| 91 | 104/85, 93/76, 60/49, 109/89, 49/40, 136/111, 38/31 |
| 92 | 92/75, 146/119, 27/22, 124/101, 70/57, 113/92 |
| 93 | 91/74, 123/100, 16/13, 85/69 |
| 94 | 117/95, 101/82, 69/56, 90/73, 37/30, 95/77, 100/81 |
| 95 | 121/98, 21/17, 152/123, 110/89, 89/72, 68/55, 115/93 |
| 96 | 99/80, 26/21, 109/88, 140/113, 57/46, 119/96, 150/121 |
| 97 | 31/25, 36/29, 113/91, 77/62, 41/33 |
| 98 | 87/70, 46/37, 148/119, 51/41, 56/45 |
| 99 | 81/65, 91/73, 96/77, 101/81, 111/89, 116/93, 126/101, 136/109, 146/117 |
| 100 | 5/4 |
| 101 | 154/123, 144/115, 124/99, 119/95, 114/91, 109/87 |
| 102 | 69/55, 64/51, 123/98, 113/90, 152/121, 49/39 |
| 103 | 93/74, 44/35, 39/31, 112/89, 73/58, 34/27 |
| 104 | 63/50, 92/73, 121/96, 150/119, 29/23, 140/111, 111/88, 82/65 |
| 105 | 101/80, 24/19, 115/91, 91/72, 110/87, 148/117 |
| 106 | 62/49, 81/64, 138/109, 19/15, 128/101 |
| 107 | 52/41, 33/26, 146/115, 113/89, 80/63 |
| 108 | 108/85, 89/70, 117/92, 14/11 |
| 109 | 121/95, 93/73, 144/113, 65/51, 116/91, 51/40, 88/69, 37/29 |
| 110 | 152/119, 23/18, 119/93 |
| 111 | 87/68, 32/25, 105/82, 73/57, 114/89, 41/32, 50/39 |
| 112 | 109/85, 77/60, 95/74, 104/81, 113/88, 140/109 |
| 113 | 9/7, 148/115, 130/101, 112/87, 85/66 |
| 114 | 58/45, 156/121, 49/38, 89/69, 40/31 |
| 115 | 31/24, 146/113, 115/89, 84/65, 128/99, 75/58, 119/92 |
| 116 | 22/17, 123/95, 101/78, 136/105, 57/44, 35/27 |
| 117 | 48/37, 109/84, 74/57, 100/77, 113/87, 152/117 |
| 118 | 13/10, 160/123, 121/93, 95/73, 82/63 |
| 119 | 99/76, 116/89, 73/56, 30/23 |
| 120 | 124/95, 111/85, 64/49, 81/62, 98/75, 115/88, 132/101, 17/13 |
| 121 | 89/68, 72/55, 55/42, 148/113, 38/29 |
| 122 | 156/119, 101/77, 21/16, 130/99 |
| 123 | 46/35, 117/89, 96/73, 121/92, 146/111, 25/19, 154/117 |
| 124 | 54/41, 112/85, 29/22, 120/91, 91/69, 95/72 |
| 125 | 33/25, 144/109, 37/28, 152/115, 115/87, 119/90, 160/121, 41/31 |
| 126 | 45/34, 49/37, 102/77 |
| 127 | 126/95, 65/49, 69/52, 73/55, 150/113, 77/58, 85/64 |
| 128 | 93/70, 101/76, 109/82, 113/85, 117/88, 121/91 |
| 129 | 4/3 |
| 130 | 123/92, 119/89 |
| 131 | 99/74, 91/68, 87/65, 162/121, 154/115, 75/56, 146/109 |
| 132 | 114/85, 55/41, 51/38, 98/73 |
| 133 | 121/90, 160/119, 39/29, 152/113, 113/84, 74/55, 109/81, 35/26, 136/101 |
| 134 | 101/75, 66/49, 128/95, 31/23, 120/89, 89/66, 85/63 |
| 135 | 27/20, 104/77, 77/57, 50/37, 123/91, 73/54, 119/88 |
| 136 | 23/17, 111/82, 88/65, 65/48, 42/31, 164/121 |
| 137 | 99/73, 156/115, 19/14, 148/109, 110/81 |
| 138 | 87/64, 121/89, 34/25, 49/36 |
| 139 | 162/119, 109/80, 124/91, 154/113, 15/11 |
| 140 | 116/85, 101/74, 56/41, 138/101, 41/30, 160/117, 119/87 |
| 141 | 93/68, 26/19, 115/84, 89/65, 152/111, 63/46, 100/73, 37/27, 85/62 |
| 142 | 48/35, 70/51, 136/99 |
| 143 | 11/8, 150/109, 128/93, 95/69 |
| 144 | 117/85, 62/45, 113/82, 164/119, 51/37, 91/66, 40/29 |
| 145 | 69/50, 156/113, 29/21, 105/76, 76/55, 123/89, 170/123, 112/81 |
| 146 | 101/73, 18/13, 140/101 |
| 147 | 104/75, 154/111, 111/80, 68/49, 168/121, 25/18 |
| 148 | 132/95, 57/41, 146/105, 89/64, 121/87, 32/23 |
| 149 | 39/28, 124/89, 46/33, 152/109, 113/81 |
| 150 | 81/58, 88/63, 95/68, 102/73, 109/78, 123/88, 130/93 |
| 151 | 7/5, 164/117 |
| 152 | 115/82, 108/77, 101/72, 87/62, 80/57, 73/52, 170/121 |
| 153 | 52/37, 45/32, 128/91, 38/27 |
| 154 | 69/49, 162/115, 31/22, 148/105, 55/39 |
| 155 | 24/17, 113/80, 89/63, 154/109, 65/46, 41/29, 140/99 |
| 156 | 99/70, 58/41, 92/65, 109/77, 126/89, 160/113, 17/12 |
| 157 | 78/55, 105/74, 44/31, 115/81, 98/69 |
| 158 | 27/19, 91/64, 64/45, 37/26 |
| 159 | 121/85, 104/73, 57/40, 124/87, 144/101, 77/54, 164/115 |
| 160 | 117/82, 10/7 |
| 161 | 93/65, 176/123, 156/109, 73/51, 136/95, 63/44, 116/81 |
| 162 | 162/113, 109/76, 33/23, 89/62, 56/39 |
| 163 | 23/16, 174/121, 128/89, 105/73, 82/57, 95/66 |
| 164 | 36/25, 121/84, 49/34, 160/111, 111/77, 75/52 |
| 165 | 101/70, 13/9, 146/101 |
| 166 | 81/56, 123/85, 178/123, 55/38, 152/105, 42/29, 113/78, 100/69 |
| 167 | 29/20, 132/91, 74/51, 119/82, 164/113, 45/31, 170/117 |
| 168 | 138/95, 93/64, 109/75, 16/11 |
| 169 | 99/68, 51/35, 35/24 |
| 170 | 124/85, 54/37, 73/50, 92/63, 111/76, 130/89, 168/115, 19/13, 136/93 |
| 171 | 174/119, 117/80, 60/41, 101/69, 41/28, 148/101, 85/58 |
| 172 | 22/15, 113/77, 91/62, 160/109, 119/81 |
| 173 | 72/49, 25/17, 178/121, 128/87 |
| 174 | 81/55, 109/74, 28/19, 115/78, 146/99 |
| 175 | 121/82, 31/21, 96/65, 65/44, 164/111, 34/23 |
| 176 | 176/119, 108/73, 182/123, 37/25, 114/77, 77/52, 40/27 |
| 177 | 126/85, 132/89, 89/60, 46/31, 95/64, 49/33, 150/101 |
| 178 | 101/68, 52/35, 162/109, 55/37, 168/113, 113/76, 58/39, 119/80, 180/121 |
| 179 | 73/49, 76/51, 82/55, 85/57 |
| 180 | 109/73, 112/75, 115/77, 121/81, 130/87, 136/91, 148/99 |
| 181 | 178/119, 184/123 |
| 182 | 3/2 |
| 183 | 182/121, 176/117, 170/113, 164/109, 152/101, 140/93 |
| 184 | 128/85, 116/77, 113/75, 110/73, 104/69, 98/65, 95/63 |
| 185 | 77/51, 74/49, 68/45 |
| 186 | 62/41, 121/80, 180/119, 174/115, 115/76, 56/37, 109/72, 50/33 |
| 187 | 144/95, 138/91, 91/60, 44/29, 85/56, 41/27 |
| 188 | 117/77, 38/25, 111/73, 184/121, 73/48, 178/117, 35/23 |
| 189 | 99/65, 32/21, 154/101, 119/78 |
| 190 | 29/19, 113/74, 84/55, 55/36, 136/89 |
| 191 | 26/17, 101/66, 176/115, 75/49, 124/81, 49/32, 170/111, 95/62 |
| 192 | 23/15, 112/73, 89/58, 152/99 |
| 193 | 63/41, 146/95, 186/121, 123/80, 20/13 |
| 194 | 117/76, 174/113, 77/50, 57/37, 168/109, 37/24 |
| 195 | 54/35, 88/57, 105/68, 156/101, 190/123, 17/11 |
| 196 | 184/119, 116/75, 99/64, 65/42, 178/115, 113/73, 48/31 |
| 197 | 31/20, 138/89, 76/49, 121/78, 45/29 |
| 198 | 132/85, 87/56, 101/65, 115/74, 14/9 |
| 199 | 109/70, 176/113, 81/52, 148/95, 120/77, 170/109 |
| 200 | 39/25, 64/41, 89/57, 114/73, 164/105, 25/16, 136/87 |
| 201 | 186/119, 36/23, 119/76 |
| 202 | 58/37, 69/44, 80/51, 91/58, 102/65, 113/72, 146/93, 190/121 |
| 203 | 11/7, 184/117, 140/89, 85/54 |
| 204 | 63/40, 178/113, 115/73, 52/33, 41/26 |
| 205 | 101/64, 30/19, 109/69, 128/81, 49/31 |
| 206 | 117/74, 87/55, 144/91, 182/115, 19/12, 160/101 |
| 207 | 65/41, 176/111, 111/70, 46/29, 119/75, 192/121, 73/46, 100/63 |
| 208 | 27/17, 116/73, 89/56, 62/39, 35/22, 148/93 |
| 209 | 78/49, 121/76, 180/113, 196/123, 51/32, 110/69 |
| 210 | 174/109, 91/57, 190/119, 99/62, 115/72, 123/77 |
| 211 | 8/5 |
| 212 | 117/73, 109/68, 101/63, 93/58, 178/111, 162/101, 77/48, 146/91, 130/81 |
| 213 | 45/28, 82/51, 119/74, 37/23, 140/87 |
| 214 | 66/41, 124/77, 182/113, 29/18, 50/31 |
| 215 | 121/75, 192/119, 92/57, 113/70, 176/109, 21/13, 160/99 |
| 216 | 186/115, 55/34, 144/89, 89/55, 123/76, 34/21, 196/121 |
| 217 | 81/50, 154/95, 60/37, 73/45, 112/69, 164/101, 190/117 |
| 218 | 138/85, 13/8, 200/123, 148/91 |
| 219 | 184/113, 57/35, 101/62, 44/27, 119/73, 75/46 |
| 220 | 31/19, 111/68, 80/49, 178/109, 49/30, 152/93, 85/52 |
| 221 | 121/74, 18/11, 113/69, 95/58 |
| 222 | 182/111, 41/25, 146/89, 105/64, 64/39 |
| 223 | 156/95, 202/123, 23/14, 120/73, 74/45, 51/31 |
| 224 | 186/113, 28/17, 89/54, 150/91 |
| 225 | 33/20, 104/63, 180/109, 109/66, 38/23, 119/72, 200/121 |
| 226 | 81/49, 124/75, 91/55, 48/29, 154/93, 164/99 |
| 227 | 58/35, 121/73, 184/111, 63/38, 68/41, 73/44 |
| 228 | 93/56, 108/65, 113/68, 123/74, 128/77, 148/89, 168/101 |
| 229 | 198/119, 5/3 |
| 230 | 202/121, 192/115, 182/109, 152/91 |
| 231 | 117/70, 92/55, 87/52, 82/49, 77/46, 196/117 |
| 232 | 62/37, 176/105, 57/34, 109/65, 52/31, 146/87, 136/81 |
| 233 | 42/25, 121/72, 200/119, 116/69, 190/113, 37/22, 170/101 |
| 234 | 69/41, 101/60, 32/19, 123/73, 91/54, 150/89, 204/121 |
| 235 | 27/16, 184/109, 130/77, 76/45, 49/29 |
| 236 | 93/55, 208/123, 115/68, 22/13, 105/62 |
| 237 | 144/85, 178/105, 39/23, 95/56, 56/33 |
| 238 | 202/119, 124/73, 192/113, 17/10, 148/87 |
| 239 | 63/37, 109/64, 46/27, 196/115, 75/44 |
| 240 | 162/95, 29/17, 186/109, 128/75, 99/58, 70/41, 111/65, 152/89, 41/24, 200/117 |
| 241 | 65/38, 77/45, 89/52, 190/111, 113/66 |
| 242 | 12/7, 170/99 |
| 243 | 146/85, 55/32, 208/121, 98/57, 160/93 |
| 244 | 117/68, 198/115, 31/18, 174/101, 112/65, 50/29, 119/69 |
| 245 | 69/40, 88/51, 126/73, 164/95, 202/117, 19/11, 140/81 |
| 246 | 121/70, 64/37, 109/63, 154/89, 45/26 |
| 247 | 26/15, 111/64, 196/113, 85/49, 210/121 |
| 248 | 33/19, 73/42, 113/65, 40/23 |
| 249 | 87/50, 148/85, 101/58, 54/31, 115/66, 176/101, 190/109, 68/39 |
| 250 | 89/51, 96/55, 110/63, 152/87 |
| 251 | 208/119, 7/4 |
| 252 | 198/113, 184/105, 156/89, 128/73, 121/69, 114/65, 100/57 |
| 253 | 72/41, 202/115, 65/37, 123/70, 58/33, 109/62, 160/91, 51/29, 95/54 |
| 254 | 44/25, 81/46, 192/109, 37/21, 178/101, 164/93 |
| 255 | 30/17, 113/64, 196/111, 136/77 |
| 256 | 99/56, 168/95, 23/13, 200/113, 154/87, 85/48 |
| 257 | 62/35, 101/57, 218/123, 39/22, 204/115, 55/31 |
| 258 | 87/49, 16/9 |
| 259 | 121/68, 89/50, 162/91, 73/41, 130/73, 57/32, 98/55, 180/101, 41/23 |
| 260 | 66/37, 91/51, 116/65, 216/121, 25/14 |
| 261 | 202/113, 152/85, 93/52, 220/123, 34/19, 111/62 |
| 262 | 138/77, 52/29, 113/63, 70/39 |
| 263 | 88/49, 115/64, 124/69, 160/89, 178/99, 196/109 |
| 264 | 9/5, 218/121, 200/111, 182/101, 164/91, 146/81, 119/66 |
| 265 | 101/56, 92/51, 74/41, 204/113, 65/36, 56/31 |
| 266 | 132/73, 208/115, 123/68, 38/21, 105/58 |
| 267 | 154/85, 29/16, 136/75, 49/27 |
| 268 | 216/119, 69/38, 89/49, 198/109, 109/60, 20/11 |
| 269 | 202/111, 91/50, 162/89, 224/123, 51/28, 184/101, 82/45, 113/62 |
| 270 | 31/17, 104/57, 73/40, 115/63, 42/23, 95/52, 148/81, 170/93 |
| 271 | 117/64, 64/35, 75/41, 119/65 |
| 272 | 174/95, 218/119, 11/6, 222/121, 200/109 |
| 273 | 156/85, 101/55, 90/49, 226/123, 68/37, 182/99, 57/31, 160/87 |
| 274 | 46/25, 208/113, 81/44, 116/63, 186/101, 35/19, 164/89 |
| 275 | 24/13, 85/46 |
| 276 | 220/119, 37/20, 224/121, 50/27 |
| 277 | 176/95, 63/34, 202/109, 76/41, 89/48, 102/55, 115/62, 128/69 |
| 278 | 13/7, 210/113, 184/99, 119/64 |
| 279 | 93/50, 121/65, 54/29, 95/51, 136/73, 218/117, 41/22 |
| 280 | 69/37, 222/119, 28/15, 226/121, 170/91 |
| 281 | 230/123, 144/77, 101/54, 58/31, 204/109, 73/39 |
| 282 | 178/95, 208/111, 15/8, 152/81 |
| 283 | 92/49, 77/41, 216/115, 62/33, 109/58, 220/117, 190/101 |
| 284 | 32/17, 113/60, 130/69, 228/121, 49/26, 164/87 |
| 285 | 66/35, 232/123, 117/62, 168/89, 17/9 |
| 286 | 138/73, 121/64, 104/55, 87/46, 70/37, 123/65, 176/93 |
| 287 | 36/19, 218/115, 91/48, 146/77, 55/29, 74/39 |
| 288 | 93/49, 226/119, 19/10, 230/121, 192/101, 154/81 |
| 289 | 78/41, 99/52, 40/21 |
| 290 | 162/85, 124/65, 208/109, 21/11, 170/89 |
| 291 | 216/113, 65/34, 174/91, 109/57, 44/23, 111/58, 178/93, 224/117 |
| 292 | 182/95, 228/119, 23/12, 232/121, 140/73, 190/99, 119/62 |
| 293 | 48/25, 121/63, 73/38, 98/51, 123/64, 148/77, 25/13 |
| 294 | 202/105, 77/40, 52/27, 210/109 |
| 295 | 27/14, 218/113, 164/85, 110/57, 222/115, 56/29, 226/117, 85/44 |
| 296 | 230/119, 29/15, 234/121, 176/91, 89/46, 238/123, 60/31 |
| 297 | 184/95, 31/16, 126/65, 95/49, 64/33, 196/101 |
| 298 | 33/17, 101/52, 68/35, 35/18 |
| 299 | 72/37, 109/56, 146/75, 220/113, 37/19, 224/115, 150/77, 113/58, 76/39 |
| 300 | 232/119, 39/20, 80/41, 121/62, 41/21, 170/87 |
| 301 | 174/89, 88/45, 178/91, 45/23, 182/93 |
| 302 | 186/95, 96/49, 49/25, 198/101, 100/51, 51/26 |
| 303 | 108/55, 218/111, 55/28, 222/113, 112/57, 226/115, 57/29, 230/117, 238/121 |
| 304 | 234/119, 242/123, 124/63, 63/32, 128/65, 65/33, 136/69 |
| 305 | 69/35, 144/73, 73/37, 148/75, 75/38, 152/77, 77/39, 160/81 |
| 306 | 81/41, 168/85, 87/44, 176/89, 89/45, 180/91, 91/46, 184/93, 95/48, 196/99, 200/101 |
| 307 | 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 | |
| 309 | |
| 310 | |
| 311 | 2/1 |
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* | 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