77edo

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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

Approximate Ratios

in the 13-limit

0 0.000 1/1
1 15.584 81/80, 99/98
2 31.169 64/63, 49/48
3 46.753 33/32, 36/35
4 62.338 28/27, 26/25
5 77.922 21/20, 25/24
6 93.506 135/128
7 109.091 16/15
8 124.675 15/14
9 140.260 13/12
10 155.844 12/11, 11/10
11 171.429 72/65
12 187.013 10/9
13 202.597 9/8
14 218.182 256/225
15 233.766 8/7
16 249.351 15/13
17 264.935 7/6
18 280.519 33/28
19 296.104 32/27, 13/11
20 311.688 6/5
21 327.273 98/81
22 342.857 11/9, 39/32
23 358.442 16/13
24 374.026 56/45, 26/21
25 389.610 5/4
26 405.195 33/26, 81/64
27 420.779 14/11, 32/25
28 436.364 9/7
29 451.948 13/10
30 467.532 21/16
31 483.117 120/91
32 498.701 4/3
33 514.286 27/20
34 529.870 49/36
35 545.455 11/8, 15/11
36 561.039 18/13
37 576.623 7/5
38 592.208 45/32
39 607.792 64/45
40 623.377 10/7
41 638.961 13/9
42 654.545 16/11, 22/15
43 670.130 72/49
44 685.714 40/27
45 701.299 3/2
46 716.883 91/60
47 732.468 32/21
48 748.052 20/13
49 763.636 14/9
50 779.221 11/7, 25/16
51 794.805 52/33, 128/81
52 810.390 8/5
53 825.974 45/28, 21/13
54 841.558 13/8
55 857.143 18/11, 64/39
56 872.727 81/49
57 888.312 5/3
58 903.896 27/16, 22/13
59 919.481 56/33
60 935.065 12/7
61 950.649 26/15
62 966.234 7/4
63 981.818 225/128
64 997.403 16/9
65 1012.987 9/5
66 1028.571 65/36
67 1044.156 11/6, 20/11
68 1059.740 24/13
69 1075.325 28/15
70 1090.909 15/8
71 1106.494 256/135
72 1122.078 40/21, 48/25
73 1137.662 27/14, 25/13
74 1153.247 64/33, 35/18
75 1168.831 63/32, 96/49
76 1184.416 160/81, 196/99
77 1200.000 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

77et Star by Chris Vaisvil

A Seed Planted A Seed Planted by Jake Freivald, in an organ version of Claudi Meneghin.