77edo
77 tone Equal Temperament
77-tET or 77-EDO divides the octave into 77 steps of size 15.5844 cents each. With fifths less than a cent flat, major thirds a bit over three cents sharp and 7/4s less flat than that, it represents an excellent tuning choice for both valentine, the 31&46 temperament, and starling, the 126/125 planar temperament, giving the optimal patent val for 11-limit valentine and its 13-limit extensions dwynwen and valentino, as well as 11-limit starling and oxpecker temperaments. It also gives the optimal patent val for various members of the unicorn family, with a generator of 4\77 instead of the 5\77 which gives valentine. These are 7-limit alicorn and 11- and 13-limit camahueto.
77et tempers out 32805/32768 in the 5-limit, 126/125, 1029/1024 and 6144/6125 in the 7-limit, 121/120, 176/175, 385/384 and 441/440 in the 11-limit, and 196/195, 351/350, 352/351, 676/675 and 729/728 in the 13-limit.
Intervals
| Degree | Cents Value | 7mus | Approximate Ratios
in the 13-limit |
|---|---|---|---|
| 0 | 1/1 | ||
| 1 | 15.584 | 19.948 (13.F2B16) | 81/80, 99/98 |
| 2 | 31.169 | 39.896 (27.E5616) | 64/63, 49/48 |
| 3 | 46.753 | 59.844 (3B.D8216) | 33/32, 36/35 |
| 4 | 62.338 | 79.792 (4F.CAD16) | 28/27, 26/25 |
| 5 | 77.922 | 99.74 (63.BD816) | 21/20, 25/24 |
| 6 | 93.5065 | 119.688 (77.B0316) | 135/128 |
| 7 | 109.091 | 139.636 (8B.A2F16) | 16/15 |
| 8 | 124.675 | 159.584 (9F.95A16) | 15/14 |
| 9 | 140.26 | 179.5325 (B3.88516) | 13/12 |
| 10 | 155.844 | 199.4805 (C7.7B16) | 12/11, 11/10 |
| 11 | 171.429 | 219.429 (DB.6DB16) | 72/65 |
| 12 | 187.013 | 239.377 (EF.60716) | 10/9 |
| 13 | 202.597 | 259.325 (103.53216) | 9/8 |
| 14 | 218.182 | 279.273 (117.45D16) | 256/225 |
| 15 | 233.766 | 299.221 (12B.38816) | 8/7 |
| 16 | 249.351 | 319.169 (13F.2B416) | 15/13 |
| 17 | 264.935 | 339.117 (153.1DF16) | 7/6 |
| 18 | 280.5195 | 359.065 (167.10A16) | 33/28 |
| 19 | 296.104 | 379.013 (17B.03516) | 32/27, 13/11 |
| 20 | 311.688 | 398.961 (18E.F616) | 6/5 |
| 21 | 327.273 | 418.909 (1A2.E8B16) | 98/81 |
| 22 | 342.857 | 438.857 (1B6.DB716) | 11/9, 39/32 |
| 23 | 358.442 | 458.805 (1CA.CE216) | 16/13 |
| 24 | 374.026 | 478.753 (1DE.C0D16) | 56/45, 26/21 |
| 25 | 389.61 | 498.701 (1F2.B38816) | 5/4 |
| 26 | 405.195 | 518.649 (206.A6416) | 33/26, 81/64 |
| 27 | 420.779 | 538.597 (21A.98F16) | 14/11, 32/25 |
| 28 | 436.364 | 558.5455 (22E.8BA16) | 9/7 |
| 29 | 451.948 | 578.4935 (242.7E516) | 13/10 |
| 30 | 467.5325 | 598.442 (256.71116) | 21/16 |
| 31 | 483.117 | 618.39 (26A.63C16) | 120/91 |
| 32 | 498.701 | 638.338 (27E.56716) | 4/3 |
| 33 | 514.286 | 658.286 (292.49216) | 27/20 |
| 34 | 529.87 | 678.234 (2A6.3BD816) | 49/36 |
| 35 | 545.4545 | 698.182 (2BA.2E916) | 11/8, 15/11 |
| 36 | 561.039 | 718.13 (2CE.21416) | 18/13 |
| 37 | 576.623 | 738.078 (2E2.13F16) | 7/5 |
| 38 | 592.208 | 758.026 (2F6.06A16) | 45/32 |
| 39 | 607.792 | 777.974 (309.F9616) | 64/45 |
| 40 | 623.377 | 797.922 (31D.EC116) | 10/7 |
| 41 | 638.961 | 817.87 (331.DEC16) | 13/9 |
| 42 | 654.5455 | 837.818 (345.D1716) | 16/11, 22/15 |
| 43 | 670.13 | 857.766 (359.C42816) | 72/49 |
| 44 | 685.714 | 877.714 (36D.B6E16) | 40/27 |
| 45 | 701.299 | 897.662 (381.A9916) | 3/2 |
| 46 | 716.883 | 917.61 (395.9C416) | 91/60 |
| 47 | 732.4675 | 937.558 (3A9.8EF16) | 32/21 |
| 48 | 748.052 | 957.5065 (3BD.81B16) | 20/13 |
| 49 | 763.636 | 977.4545 (3D1.74616) | 14/9 |
| 50 | 779.221 | 997.403 (3E5.67116) | 11/7, 25/16 |
| 51 | 794.805 | 1017.351 (3F9.59C16) | 52/33, 128/81 |
| 52 | 810.39 | 1037.299 (40D.4C7816) | 8/5 |
| 53 | 825.974 | 1057.247 (421.3F316) | 45/28, 21/13 |
| 54 | 841.558 | 1077.195 (435.31E16) | 13/8 |
| 55 | 857.143 | 1097.143 (449.24916) | 18/11, 64/39 |
| 56 | 872.727 | 1117.091 (45D.17416) | 81/49 |
| 57 | 888.312 | 1137.039 (471.0A16) | 5/3 |
| 58 | 903.896 | 1156.987 (484.FCB16) | 27/16, 22/13 |
| 59 | 919.4805 | 1176.935 (498.EF616) | 56/33 |
| 60 | 935.065 | 1196.883 (4AC.E2116) | 12/7 |
| 61 | 950.649 | 1216.831 (4C0.D4C16) | 26/15 |
| 62 | 966.234 | 1236.77 (4D4.C7816) | 7/4 |
| 63 | 981.818 | 1256.727 (4E8.BA316) | 225/128 |
| 64 | 997.403 | 1276.675 (4FC.ACE16) | 16/9 |
| 65 | 1012.987 | 1296.623 (510.9F916) | 9/5 |
| 66 | 1028.571 | 1316.571 (524.92416) | 65/36 |
| 67 | 1044.156 | 1336.5195 (538.8516) | 11/6, 20/11 |
| 68 | 1059.74 | 1356.4675 (54C.77B16) | 24/13 |
| 69 | 1075.325 | 1376.416 (560.6A616) | 28/15 |
| 70 | 1090.909 | 1396.364 (574.5D116) | 15/8 |
| 71 | 1106.4935 | 1416.312 (588.4FD16) | 256/135 |
| 72 | 1122.078 | 1436.26 (59C.42616) | 40/21, 48/25 |
| 73 | 1137.662 | 1456.208 (5B0.35316) | 27/14, 25/13 |
| 74 | 1153.247 | 1476.156 (5C4.27E16) | 64/33, 35/18 |
| 75 | 1168.831 | 1496.104 (5D8.1AA16) | 63/32, 96/49 |
| 76 | 1184.416 | 1516.052 (5EC.0D516) | 160/81, 196/99 |
| 77 | 1200 | 1536 (60016) | 2/1 |
Linear temperaments
| Periods
per Octave |
Generator | Temperaments |
|---|---|---|
| 1 | 1\77 | |
| 1 | 2\77 | |
| 1 | 3\77 | |
| 1 | 4\77 | Unicorn/alicorn/camahueto/qilin |
| 1 | 5\77 | Valentine |
| 1 | 6\77 | |
| 1 | 8\77 | |
| 1 | 9\77 | Tsaharuk |
| 1 | 10\77 | |
| 1 | 12\77 | |
| 1 | 13\77 | |
| 1 | 15\77 | Guiron |
| 1 | 16\77 | Hemischis |
| 1 | 17\77 | |
| 1 | 18\77 | |
| 1 | 19\77 | |
| 1 | 20\77 | |
| 1 | 23\77 | Restles (77&87) |
| 1 | 24\77 | |
| 1 | 25\77 | |
| 1 | 26\77 | |
| 1 | 27\77 | |
| 1 | 29\77 | |
| 1 | 30\77 | |
| 1 | 31\77 | Hemiseven |
| 1 | 32\77 | Hendecatonic, grackle |
| 1 | 34\77 | Mabila (34&43) |
| 1 | 36\77 | |
| 1 | 37\77 | |
| 1 | 38\77 | |
| 7 | 1\77 | "Absurdity" (7&84) |
| 7 | 2\77 | |
| 7 | 3\77 | |
| 7 | 4\77 | |
| 7 | 5\77 | |
| 11 | 1\77 | |
| 11 | 2\77 | |
| 11 | 3\77 |
Music
Star 1-GrimaldiA+Bmod by Joel Grant Taylor
A Seed Planted A Seed Planted by Jake Freivald, in an organ version of Claudi Meneghin.