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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.