# 77edo

 ← 76edo 77edo 78edo →
Prime factorization 7 × 11
Step size 15.5844¢
Fifth 45\77 (701.299¢)
Semitones (A1:m2) 7:6 (109.1¢ : 93.51¢)
Consistency limit 9
Distinct consistency limit 9

77 equal divisions of the octave (abbreviated 77edo or 77ed2), also called 77-tone equal temperament (77tet) or 77 equal temperament (77et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 77 equal parts of about 15.6 ¢ each. Each step represents a frequency ratio of 21/77, or the 77th root of 2.

## Theory

With harmonic 3 less than a cent flat, harmonic 5 a bit over three cents sharp and 7's less flat than that, 77edo 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 grackle and 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.

77edo is an excellent edo for Carlos Alpha, since the difference between 5 steps of 77edo and 1 step of Carlos Alpha is only -0.042912 cents.

### Prime harmonics

Approximation of prime harmonics in 77edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error absolute (¢) +0.00 -0.66 +3.30 -2.59 -5.86 +1.03 +4.14 -1.41 -4.90 -1.01 -7.37
relative (%) +0 -4 +21 -17 -38 +7 +27 -9 -31 -6 -47
Steps
(reduced)
77
(0)
122
(45)
179
(25)
216
(62)
266
(35)
285
(54)
315
(7)
327
(19)
348
(40)
374
(66)
381
(73)

## Intervals

Degree Cents 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

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [-122 77 [77 122]] +0.207 0.207 1.33
2.3.5 32805/32768, 1594323/1562500 [77 122 179]] -0.336 0.785 5.04
2.3.5.7 126/125, 1029/1024, 10976/10935 [77 122 179 216]] -0.021 0.872 5.59
2.3.5.7.11 121/120, 126/125, 176/175, 10976/10935 [77 122 179 216 266]] +0.322 1.039 6.66
2.3.5.7.11.13 121/120, 126/125, 176/175, 196/195, 676/675 [77 122 179 216 266 285]] +0.222 0.974 6.25

### Rank-2 temperaments

Periods
per 8ve
Generator
(Reduced)
Cents
(Reduced)
Associated Ratio
(Reduced)
Temperaments
1 4\77 62.34 28/27 Unicorn / alicorn / camahueto / qilin
1 5\77 77.92 21/20 Valentine
1 9\77 140.26 13/12 Tsaharuk
1 15\77 233.77 8/7 Guiron
1 16\77 249.35 15/13 Hemischis (77e)
1 20\77 311.69 6/5 Oolong
1 23\77 358.44 16/13 Restles
1 31\77 483.12 45/34 Hemiseven
1 32\77 498.70 4/3 Grackle
1 34\77 529.87 512/375 Tuskaloosa
Muscogee
7 32\77
(1\77)
498.70
(15.58)
4/3
(81/80)
Absurdity
11 32\77
(3\77)
498.70
(46.75)
4/3
(36/35)
Hendecatonic

## Music

Jake Freivald
Joel Grant Taylor
Chris Vaisvil