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
| Steps | Cents | Approximate ratios | Ups and downs notation |
|---|---|---|---|
| 0 | 0 | 1/1 | D |
| 1 | 3.86 | ^D, ^8E♭♭ | |
| 2 | 7.72 | ^^D, ^9E♭♭ | |
| 3 | 11.58 | ^3D, ^10E♭♭ | |
| 4 | 15.43 | ^4D, ^11E♭♭ | |
| 5 | 19.29 | 88/87, 91/90, 92/91, 93/92 | ^5D, ^12E♭♭ |
| 6 | 23.15 | 75/74, 76/75, 77/76 | ^6D, ^13E♭♭ |
| 7 | 27.01 | 64/63, 65/64, 66/65 | ^7D, ^14E♭♭ |
| 8 | 30.87 | 56/55, 57/56 | ^8D, v15E♭ |
| 9 | 34.73 | 50/49, 51/50 | ^9D, v14E♭ |
| 10 | 38.59 | 45/44, 46/45 | ^10D, v13E♭ |
| 11 | 42.44 | 41/40 | ^11D, v12E♭ |
| 12 | 46.3 | 38/37 | ^12D, v11E♭ |
| 13 | 50.16 | 35/34 | ^13D, v10E♭ |
| 14 | 54.02 | 65/63 | ^14D, v9E♭ |
| 15 | 57.88 | 91/88 | ^15D, v8E♭ |
| 16 | 61.74 | 57/55, 85/82 | v14D♯, v7E♭ |
| 17 | 65.59 | 27/26, 80/77 | v13D♯, v6E♭ |
| 18 | 69.45 | 51/49, 77/74 | v12D♯, v5E♭ |
| 19 | 73.31 | 24/23 | v11D♯, v4E♭ |
| 20 | 77.17 | 23/22, 91/87 | v10D♯, v3E♭ |
| 21 | 81.03 | 22/21 | v9D♯, vvE♭ |
| 22 | 84.89 | 21/20 | v8D♯, vE♭ |
| 23 | 88.75 | 20/19 | v7D♯, E♭ |
| 24 | 92.6 | 58/55, 96/91 | v6D♯, ^E♭ |
| 25 | 96.46 | 37/35, 55/52, 92/87 | v5D♯, ^^E♭ |
| 26 | 100.32 | v4D♯, ^3E♭ | |
| 27 | 104.18 | 86/81 | v3D♯, ^4E♭ |
| 28 | 108.04 | 33/31 | vvD♯, ^5E♭ |
| 29 | 111.9 | 16/15 | vD♯, ^6E♭ |
| 30 | 115.76 | 31/29, 77/72 | D♯, ^7E♭ |
| 31 | 119.61 | 15/14 | ^D♯, ^8E♭ |
| 32 | 123.47 | 29/27 | ^^D♯, ^9E♭ |
| 33 | 127.33 | ^3D♯, ^10E♭ | |
| 34 | 131.19 | 41/38, 55/51 | ^4D♯, ^11E♭ |
| 35 | 135.05 | 40/37, 93/86 | ^5D♯, ^12E♭ |
| 36 | 138.91 | 13/12 | ^6D♯, ^13E♭ |
| 37 | 142.77 | 38/35, 63/58 | ^7D♯, ^14E♭ |
| 38 | 146.62 | 37/34 | ^8D♯, v15E |
| 39 | 150.48 | 12/11 | ^9D♯, v14E |
| 40 | 154.34 | 82/75 | ^10D♯, v13E |
| 41 | 158.2 | ^11D♯, v12E | |
| 42 | 162.06 | 56/51 | ^12D♯, v11E |
| 43 | 165.92 | ^13D♯, v10E | |
| 44 | 169.77 | 32/29, 75/68 | ^14D♯, v9E |
| 45 | 173.63 | 21/19, 94/85 | ^15D♯, v8E |
| 46 | 177.49 | 41/37, 72/65 | v14D𝄪, v7E |
| 47 | 181.35 | v13D𝄪, v6E | |
| 48 | 185.21 | 69/62 | v12D𝄪, v5E |
| 49 | 189.07 | 29/26 | v11D𝄪, v4E |
| 50 | 192.93 | 19/17 | v10D𝄪, v3E |
| 51 | 196.78 | 28/25, 65/58 | v9D𝄪, vvE |
| 52 | 200.64 | 64/57 | v8D𝄪, vE |
| 53 | 204.5 | 9/8 | E |
| 54 | 208.36 | 44/39 | ^E, ^8F♭ |
| 55 | 212.22 | 26/23 | ^^E, ^9F♭ |
| 56 | 216.08 | 17/15 | ^3E, ^10F♭ |
| 57 | 219.94 | 42/37, 92/81 | ^4E, ^11F♭ |
| 58 | 223.79 | 33/29 | ^5E, ^12F♭ |
| 59 | 227.65 | 65/57 | ^6E, ^13F♭ |
| 60 | 231.51 | 8/7 | ^7E, ^14F♭ |
| 61 | 235.37 | 55/48, 63/55 | ^8E, v15F |
| 62 | 239.23 | 31/27 | ^9E, v14F |
| 63 | 243.09 | ^10E, v13F | |
| 64 | 246.95 | ^11E, v12F | |
| 65 | 250.8 | 37/32, 52/45 | ^12E, v11F |
| 66 | 254.66 | 95/82 | ^13E, v10F |
| 67 | 258.52 | 36/31, 65/56 | ^14E, v9F |
| 68 | 262.38 | 57/49, 64/55 | ^15E, v8F |
| 69 | 266.24 | 7/6 | v14E♯, v7F |
| 70 | 270.1 | 76/65, 90/77 | v13E♯, v6F |
| 71 | 273.95 | 41/35, 75/64 | v12E♯, v5F |
| 72 | 277.81 | 27/23 | v11E♯, v4F |
| 73 | 281.67 | 20/17 | v10E♯, v3F |
| 74 | 285.53 | 46/39 | v9E♯, vvF |
| 75 | 289.39 | 13/11 | v8E♯, vF |
| 76 | 293.25 | 45/38, 77/65 | F |
| 77 | 297.11 | 19/16 | ^F, ^8G♭♭ |
| 78 | 300.96 | 69/58 | ^^F, ^9G♭♭ |
| 79 | 304.82 | 31/26 | ^3F, ^10G♭♭ |
| 80 | 308.68 | 49/41, 92/77 | ^4F, ^11G♭♭ |
| 81 | 312.54 | ^5F, ^12G♭♭ | |
| 82 | 316.4 | ^6F, ^13G♭♭ | |
| 83 | 320.26 | 77/64 | ^7F, ^14G♭♭ |
| 84 | 324.12 | 41/34, 76/63 | ^8F, v15G♭ |
| 85 | 327.97 | 29/24 | ^9F, v14G♭ |
| 86 | 331.83 | 63/52 | ^10F, v13G♭ |
| 87 | 335.69 | 17/14 | ^11F, v12G♭ |
| 88 | 339.55 | ^12F, v11G♭ | |
| 89 | 343.41 | 50/41 | ^13F, v10G♭ |
| 90 | 347.27 | 11/9 | ^14F, v9G♭ |
| 91 | 351.13 | 49/40, 60/49 | ^15F, v8G♭ |
| 92 | 354.98 | 27/22 | v14F♯, v7G♭ |
| 93 | 358.84 | 16/13 | v13F♯, v6G♭ |
| 94 | 362.7 | 37/30 | v12F♯, v5G♭ |
| 95 | 366.56 | v11F♯, v4G♭ | |
| 96 | 370.42 | v10F♯, v3G♭ | |
| 97 | 374.28 | 36/29 | v9F♯, vvG♭ |
| 98 | 378.14 | 51/41, 56/45 | v8F♯, vG♭ |
| 99 | 381.99 | 96/77 | v7F♯, G♭ |
| 100 | 385.85 | 5/4 | v6F♯, ^G♭ |
| 101 | 389.71 | v5F♯, ^^G♭ | |
| 102 | 393.57 | 64/51 | v4F♯, ^3G♭ |
| 103 | 397.43 | 39/31 | v3F♯, ^4G♭ |
| 104 | 401.29 | 29/23 | vvF♯, ^5G♭ |
| 105 | 405.14 | 91/72 | vF♯, ^6G♭ |
| 106 | 409 | 19/15 | F♯, ^7G♭ |
| 107 | 412.86 | 33/26 | ^F♯, ^8G♭ |
| 108 | 416.72 | ^^F♯, ^9G♭ | |
| 109 | 420.58 | 51/40, 65/51, 88/69 | ^3F♯, ^10G♭ |
| 110 | 424.44 | 23/18 | ^4F♯, ^11G♭ |
| 111 | 428.3 | ^5F♯, ^12G♭ | |
| 112 | 432.15 | 77/60, 95/74 | ^6F♯, ^13G♭ |
| 113 | 436.01 | ^7F♯, ^14G♭ | |
| 114 | 439.87 | 49/38, 58/45 | ^8F♯, v15G |
| 115 | 443.73 | 31/24, 84/65 | ^9F♯, v14G |
| 116 | 447.59 | 57/44 | ^10F♯, v13G |
| 117 | 451.45 | 74/57 | ^11F♯, v12G |
| 118 | 455.31 | ^12F♯, v11G | |
| 119 | 459.16 | ^13F♯, v10G | |
| 120 | 463.02 | 81/62 | ^14F♯, v9G |
| 121 | 466.88 | 55/42, 72/55 | ^15F♯, v8G |
| 122 | 470.74 | 21/16 | v14F𝄪, v7G |
| 123 | 474.6 | 25/19 | v13F𝄪, v6G |
| 124 | 478.46 | 29/22 | v12F𝄪, v5G |
| 125 | 482.32 | 37/28 | v11F𝄪, v4G |
| 126 | 486.17 | 49/37 | v10F𝄪, v3G |
| 127 | 490.03 | 69/52, 77/58 | v9F𝄪, vvG |
| 128 | 493.89 | v8F𝄪, vG | |
| 129 | 497.75 | 4/3 | G |
| 130 | 501.61 | ^G, ^8A♭♭ | |
| 131 | 505.47 | 75/56 | ^^G, ^9A♭♭ |
| 132 | 509.32 | 51/38 | ^3G, ^10A♭♭ |
| 133 | 513.18 | 39/29, 74/55 | ^4G, ^11A♭♭ |
| 134 | 517.04 | 31/23 | ^5G, ^12A♭♭ |
| 135 | 520.9 | 50/37, 77/57 | ^6G, ^13A♭♭ |
| 136 | 524.76 | 65/48, 88/65 | ^7G, ^14A♭♭ |
| 137 | 528.62 | 19/14 | ^8G, v15A♭ |
| 138 | 532.48 | 34/25 | ^9G, v14A♭ |
| 139 | 536.33 | 15/11 | ^10G, v13A♭ |
| 140 | 540.19 | 41/30, 56/41 | ^11G, v12A♭ |
| 141 | 544.05 | 63/46 | ^12G, v11A♭ |
| 142 | 547.91 | 70/51 | ^13G, v10A♭ |
| 143 | 551.77 | 11/8 | ^14G, v9A♭ |
| 144 | 555.63 | 51/37, 91/66 | ^15G, v8A♭ |
| 145 | 559.49 | 76/55 | v14G♯, v7A♭ |
| 146 | 563.34 | 18/13 | v13G♯, v6A♭ |
| 147 | 567.2 | 68/49 | v12G♯, v5A♭ |
| 148 | 571.06 | 57/41 | v11G♯, v4A♭ |
| 149 | 574.92 | 46/33 | v10G♯, v3A♭ |
| 150 | 578.78 | 81/58, 88/63, 95/68 | v9G♯, vvA♭ |
| 151 | 582.64 | 7/5 | v8G♯, vA♭ |
| 152 | 586.5 | 80/57, 87/62 | v7G♯, A♭ |
| 153 | 590.35 | 45/32 | v6G♯, ^A♭ |
| 154 | 594.21 | 31/22 | v5G♯, ^^A♭ |
| 155 | 598.07 | 65/46 | v4G♯, ^3A♭ |
| 156 | 601.93 | 92/65 | v3G♯, ^4A♭ |
| 157 | 605.79 | 44/31 | vvG♯, ^5A♭ |
| 158 | 609.65 | 64/45, 91/64 | vG♯, ^6A♭ |
| 159 | 613.5 | 57/40 | G♯, ^7A♭ |
| 160 | 617.36 | 10/7 | ^G♯, ^8A♭ |
| 161 | 621.22 | 63/44 | ^^G♯, ^9A♭ |
| 162 | 625.08 | 33/23 | ^3G♯, ^10A♭ |
| 163 | 628.94 | 82/57 | ^4G♯, ^11A♭ |
| 164 | 632.8 | 49/34 | ^5G♯, ^12A♭ |
| 165 | 636.66 | 13/9 | ^6G♯, ^13A♭ |
| 166 | 640.51 | 55/38 | ^7G♯, ^14A♭ |
| 167 | 644.37 | 74/51 | ^8G♯, v15A |
| 168 | 648.23 | 16/11 | ^9G♯, v14A |
| 169 | 652.09 | 51/35 | ^10G♯, v13A |
| 170 | 655.95 | 92/63 | ^11G♯, v12A |
| 171 | 659.81 | 41/28, 60/41 | ^12G♯, v11A |
| 172 | 663.67 | 22/15, 91/62 | ^13G♯, v10A |
| 173 | 667.52 | 25/17 | ^14G♯, v9A |
| 174 | 671.38 | 28/19 | ^15G♯, v8A |
| 175 | 675.24 | 65/44, 96/65 | v14G𝄪, v7A |
| 176 | 679.1 | 37/25, 77/52 | v13G𝄪, v6A |
| 177 | 682.96 | 46/31 | v12G𝄪, v5A |
| 178 | 686.82 | 55/37, 58/39 | v11G𝄪, v4A |
| 179 | 690.68 | 76/51 | v10G𝄪, v3A |
| 180 | 694.53 | v9G𝄪, vvA | |
| 181 | 698.39 | v8G𝄪, vA | |
| 182 | 702.25 | 3/2 | A |
| 183 | 706.11 | ^A, ^8B♭♭ | |
| 184 | 709.97 | ^^A, ^9B♭♭ | |
| 185 | 713.83 | 74/49, 77/51 | ^3A, ^10B♭♭ |
| 186 | 717.68 | 56/37 | ^4A, ^11B♭♭ |
| 187 | 721.54 | 44/29, 91/60 | ^5A, ^12B♭♭ |
| 188 | 725.4 | 38/25 | ^6A, ^13B♭♭ |
| 189 | 729.26 | 32/21 | ^7A, ^14B♭♭ |
| 190 | 733.12 | 55/36, 84/55 | ^8A, v15B♭ |
| 191 | 736.98 | 75/49 | ^9A, v14B♭ |
| 192 | 740.84 | ^10A, v13B♭ | |
| 193 | 744.69 | ^11A, v12B♭ | |
| 194 | 748.55 | 57/37 | ^12A, v11B♭ |
| 195 | 752.41 | 88/57 | ^13A, v10B♭ |
| 196 | 756.27 | 48/31, 65/42 | ^14A, v9B♭ |
| 197 | 760.13 | 45/29, 76/49 | ^15A, v8B♭ |
| 198 | 763.99 | v14A♯, v7B♭ | |
| 199 | 767.85 | 81/52 | v13A♯, v6B♭ |
| 200 | 771.7 | v12A♯, v5B♭ | |
| 201 | 775.56 | 36/23 | v11A♯, v4B♭ |
| 202 | 779.42 | 69/44, 80/51, 91/58 | v10A♯, v3B♭ |
| 203 | 783.28 | v9A♯, vvB♭ | |
| 204 | 787.14 | 52/33 | v8A♯, vB♭ |
| 205 | 791 | 30/19 | v7A♯, B♭ |
| 206 | 794.86 | v6A♯, ^B♭ | |
| 207 | 798.71 | 46/29 | v5A♯, ^^B♭ |
| 208 | 802.57 | 62/39 | v4A♯, ^3B♭ |
| 209 | 806.43 | 51/32 | v3A♯, ^4B♭ |
| 210 | 810.29 | 91/57 | vvA♯, ^5B♭ |
| 211 | 814.15 | 8/5 | vA♯, ^6B♭ |
| 212 | 818.01 | 77/48, 93/58 | A♯, ^7B♭ |
| 213 | 821.86 | 45/28, 82/51 | ^A♯, ^8B♭ |
| 214 | 825.72 | 29/18 | ^^A♯, ^9B♭ |
| 215 | 829.58 | ^3A♯, ^10B♭ | |
| 216 | 833.44 | ^4A♯, ^11B♭ | |
| 217 | 837.3 | 60/37 | ^5A♯, ^12B♭ |
| 218 | 841.16 | 13/8 | ^6A♯, ^13B♭ |
| 219 | 845.02 | 44/27 | ^7A♯, ^14B♭ |
| 220 | 848.87 | 49/30, 80/49 | ^8A♯, v15B |
| 221 | 852.73 | 18/11 | ^9A♯, v14B |
| 222 | 856.59 | 41/25 | ^10A♯, v13B |
| 223 | 860.45 | ^11A♯, v12B | |
| 224 | 864.31 | 28/17 | ^12A♯, v11B |
| 225 | 868.17 | ^13A♯, v10B | |
| 226 | 872.03 | 48/29, 91/55 | ^14A♯, v9B |
| 227 | 875.88 | 63/38, 68/41 | ^15A♯, v8B |
| 228 | 879.74 | v14A𝄪, v7B | |
| 229 | 883.6 | v13A𝄪, v6B | |
| 230 | 887.46 | v12A𝄪, v5B | |
| 231 | 891.32 | 77/46, 82/49, 87/52 | v11A𝄪, v4B |
| 232 | 895.18 | 52/31 | v10A𝄪, v3B |
| 233 | 899.04 | v9A𝄪, vvB | |
| 234 | 902.89 | 32/19, 91/54 | v8A𝄪, vB |
| 235 | 906.75 | 76/45 | B |
| 236 | 910.61 | 22/13 | ^B, ^8C♭ |
| 237 | 914.47 | 39/23, 95/56 | ^^B, ^9C♭ |
| 238 | 918.33 | 17/10 | ^3B, ^10C♭ |
| 239 | 922.19 | 46/27 | ^4B, ^11C♭ |
| 240 | 926.05 | 70/41 | ^5B, ^12C♭ |
| 241 | 929.9 | 65/38, 77/45 | ^6B, ^13C♭ |
| 242 | 933.76 | 12/7 | ^7B, ^14C♭ |
| 243 | 937.62 | 55/32 | ^8B, v15C |
| 244 | 941.48 | 31/18 | ^9B, v14C |
| 245 | 945.34 | ^10B, v13C | |
| 246 | 949.2 | 45/26, 64/37 | ^11B, v12C |
| 247 | 953.05 | 85/49 | ^12B, v11C |
| 248 | 956.91 | ^13B, v10C | |
| 249 | 960.77 | 54/31 | ^14B, v9C |
| 250 | 964.63 | 96/55 | ^15B, v8C |
| 251 | 968.49 | 7/4 | v14B♯, v7C |
| 252 | 972.35 | v13B♯, v6C | |
| 253 | 976.21 | 58/33 | v12B♯, v5C |
| 254 | 980.06 | 37/21, 81/46 | v11B♯, v4C |
| 255 | 983.92 | 30/17 | v10B♯, v3C |
| 256 | 987.78 | 23/13 | v9B♯, vvC |
| 257 | 991.64 | 39/22 | v8B♯, vC |
| 258 | 995.5 | 16/9 | C |
| 259 | 999.36 | 57/32 | ^C, ^8D♭♭ |
| 260 | 1003.22 | 25/14 | ^^C, ^9D♭♭ |
| 261 | 1007.07 | 34/19, 93/52 | ^3C, ^10D♭♭ |
| 262 | 1010.93 | 52/29 | ^4C, ^11D♭♭ |
| 263 | 1014.79 | ^5C, ^12D♭♭ | |
| 264 | 1018.65 | ^6C, ^13D♭♭ | |
| 265 | 1022.51 | 65/36, 74/41 | ^7C, ^14D♭♭ |
| 266 | 1026.37 | 38/21, 85/47 | ^8C, v15D♭ |
| 267 | 1030.23 | 29/16 | ^9C, v14D♭ |
| 268 | 1034.08 | ^10C, v13D♭ | |
| 269 | 1037.94 | 51/28 | ^11C, v12D♭ |
| 270 | 1041.8 | ^12C, v11D♭ | |
| 271 | 1045.66 | 75/41 | ^13C, v10D♭ |
| 272 | 1049.52 | 11/6 | ^14C, v9D♭ |
| 273 | 1053.38 | 68/37 | ^15C, v8D♭ |
| 274 | 1057.23 | 35/19 | v14C♯, v7D♭ |
| 275 | 1061.09 | 24/13 | v13C♯, v6D♭ |
| 276 | 1064.95 | 37/20 | v12C♯, v5D♭ |
| 277 | 1068.81 | 76/41 | v11C♯, v4D♭ |
| 278 | 1072.67 | v10C♯, v3D♭ | |
| 279 | 1076.53 | 54/29, 95/51 | v9C♯, vvD♭ |
| 280 | 1080.39 | 28/15 | v8C♯, vD♭ |
| 281 | 1084.24 | 58/31 | v7C♯, D♭ |
| 282 | 1088.1 | 15/8 | v6C♯, ^D♭ |
| 283 | 1091.96 | 62/33 | v5C♯, ^^D♭ |
| 284 | 1095.82 | 81/43 | v4C♯, ^3D♭ |
| 285 | 1099.68 | v3C♯, ^4D♭ | |
| 286 | 1103.54 | 70/37, 87/46 | vvC♯, ^5D♭ |
| 287 | 1107.4 | 55/29, 91/48 | vC♯, ^6D♭ |
| 288 | 1111.25 | 19/10 | C♯, ^7D♭ |
| 289 | 1115.11 | 40/21 | ^C♯, ^8D♭ |
| 290 | 1118.97 | 21/11 | ^^C♯, ^9D♭ |
| 291 | 1122.83 | 44/23 | ^3C♯, ^10D♭ |
| 292 | 1126.69 | 23/12 | ^4C♯, ^11D♭ |
| 293 | 1130.55 | ^5C♯, ^12D♭ | |
| 294 | 1134.41 | 52/27, 77/40 | ^6C♯, ^13D♭ |
| 295 | 1138.26 | ^7C♯, ^14D♭ | |
| 296 | 1142.12 | ^8C♯, v15D | |
| 297 | 1145.98 | 95/49 | ^9C♯, v14D |
| 298 | 1149.84 | 68/35 | ^10C♯, v13D |
| 299 | 1153.7 | 37/19 | ^11C♯, v12D |
| 300 | 1157.56 | 80/41 | ^12C♯, v11D |
| 301 | 1161.41 | 45/23, 88/45 | ^13C♯, v10D |
| 302 | 1165.27 | 49/25 | ^14C♯, v9D |
| 303 | 1169.13 | 55/28 | ^15C♯, v8D |
| 304 | 1172.99 | 63/32, 65/33 | v14C𝄪, v7D |
| 305 | 1176.85 | 75/38 | v13C𝄪, v6D |
| 306 | 1180.71 | 87/44, 91/46 | v12C𝄪, v5D |
| 307 | 1184.57 | v11C𝄪, v4D | |
| 308 | 1188.42 | v10C𝄪, v3D | |
| 309 | 1192.28 | v9C𝄪, vvD | |
| 310 | 1196.14 | v8C𝄪, vD | |
| 311 | 1200 | 2/1 | D |
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 | 142\311 | 547.92 | 48/35 | Calamity |
| 1 | 143\311 | 551.77 | 11/8 | Emkay |
| 1 | 155\311 | 598.08 | 847/600 | Vydubychi |
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: |