# 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.

The 17 and 19 are tuned fairly well, making it consistent to the no-11 21-odd-limit. The equal temperament tempers out 256/255 in the 17-limit; and 171/170, 361/360, 513/512, and 1216/1215 in the 19-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 (%) +0.0 -4.2 +21.2 -16.6 -37.6 +6.6 +26.5 -9.0 -31.4 -6.5 -47.3
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*
0 0.000 1/1
1 15.584 81/80, 91/90, 99/98, 105/104
2 31.169 49/48, 55/54, 64/63, 65/64, 100/99
3 46.753 33/32, 36/35, 40/39, 45/44, 50/49
4 62.338 26/25, 27/26, 28/27
5 77.922 21/20, 22/21, 25/24
6 93.506 18/17, 19/18, 20/19
7 109.091 16/15, 17/16
8 124.675 14/13, 15/14
9 140.260 13/12
10 155.844 11/10, 12/11
11 171.429 21/19
12 187.013 10/9
13 202.597 9/8
14 218.182 17/15
15 233.766 8/7
16 249.351 15/13, 22/19
17 264.935 7/6
18 280.519 20/17
19 296.104 13/11, 19/16, 32/27
20 311.688 6/5
21 327.273 98/81
22 342.857 11/9, 17/14
23 358.442 16/13, 21/17
24 374.026 26/21, 56/45
25 389.610 5/4
26 405.195 19/15, 24/19, 33/26
27 420.779 14/11, 32/25
28 436.364 9/7
29 451.948 13/10
30 467.532 17/13, 21/16
31 483.117 120/91
32 498.701 4/3
33 514.286 27/20
34 529.870 19/14
35 545.455 11/8, 15/11, 26/19
36 561.039 18/13
37 576.623 7/5
38 592.208 24/17, 38/27, 45/32

* as a 19-limit temperament

## 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* Cents* Associated
Ratio*
Temperaments
1 4\77 62.34 28/27 Unicorn / alicorn (77e) / camahueto (77) / qilin (77)
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

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

## Music

Jake Freivald
Joel Grant Taylor
Chris Vaisvil