77edo

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← 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 less flat than that, 77edo represents an excellent tuning choice for both valentine (hence also Carlos Alpha), the 31 & 46 temperament, and starling, the rank-3 temperament tempering out 126/125, giving the optimal patent val for 11-limit valentine and its 13-limit extension valentino, as well as 11-limit starling and oxpecker temperaments. For desirers of purer/more convincing harmonies of 19, it's also a great choice for nestoria (the extension of schismic to prime 19) so that ~16:19:24 can be heard to concord in isolation. 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); it is a very good choice for full-subgroup unicorn. These are 7-limit alicorn and 11- and 13-limit camahueto.

77et tempers out the schisma (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.

It also does surprisingly well (for its size) in a large range of very high odd-limits (41 to 125 range).

Prime harmonics

Approximation of prime harmonics in 77edo
Harmonic 2 3 5 7 11 13 17 19 23
Error Absolute (¢) +0.00 -0.66 +3.30 -2.59 -5.86 +1.03 +4.14 -1.41 -4.90
Relative (%) +0.0 -4.2 +21.2 -16.6 -37.6 +6.6 +26.5 -9.0 -31.4
Steps
(reduced)
77
(0)
122
(45)
179
(25)
216
(62)
266
(35)
285
(54)
315
(7)
327
(19)
348
(40)
Approximation of prime harmonics in 77edo (continued)
Harmonic 29 31 37 41 43 47 53 59 61
Error Absolute (¢) -1.01 -7.37 -1.99 +7.30 +2.77 +4.62 -0.78 +0.57 +5.19
Relative (%) -6.5 -47.3 -12.8 +46.8 +17.8 +29.7 -5.0 +3.6 +33.3
Steps
(reduced)
374
(66)
381
(73)
401
(16)
413
(28)
418
(33)
428
(43)
441
(56)
453
(68)
457
(72)

Subsets and supersets

Since 77 factors into primes as 7 × 11, 77edo contains 7edo and 11edo as subset edos.

Intervals

# Cents Approximate ratios* Ups and downs notation
0 0.0 1/1 D
1 15.6 81/80, 91/90, 99/98, 105/104 ^D, ^^E♭♭
2 31.2 49/48, 55/54, 64/63, 65/64, 100/99 ^^D, ^3E♭♭
3 46.8 33/32, 36/35, 40/39, 45/44, 50/49 ^3D, v3E♭
4 62.3 26/25, 27/26, 28/27 v3D♯, vvE♭
5 77.9 21/20, 22/21, 25/24 vvD♯, vE♭
6 93.5 18/17, 19/18, 20/19 vD♯, E♭
7 109.1 16/15, 17/16 D♯, ^E♭
8 124.7 14/13, 15/14 ^D♯, ^^E♭
9 140.3 13/12 ^^D♯, ^3E♭
10 155.8 11/10, 12/11 ^3D♯, v3E
11 171.4 21/19 v3D𝄪, vvE
12 187.0 10/9 vvD𝄪, vE
13 202.6 9/8 E
14 218.2 17/15 ^E, ^^F♭
15 233.8 8/7 ^^E, ^3F♭
16 249.4 15/13, 22/19 ^3E, v3F
17 264.9 7/6 v3E♯, vvF
18 280.5 20/17 vvE♯, vF
19 296.1 13/11, 19/16, 32/27 F
20 311.7 6/5 ^F, ^^G♭♭
21 327.3 98/81 ^^F, ^3G♭♭
22 342.9 11/9, 17/14 ^3F, v3G♭
23 358.4 16/13, 21/17 v3F♯, vvG♭
24 374.0 26/21, 56/45 vvF♯, vG♭
25 389.6 5/4 vF♯, G♭
26 405.2 19/15, 24/19, 33/26 F♯, ^G♭
27 420.8 14/11, 32/25 ^F♯, ^^G♭
28 436.4 9/7 ^^F♯, ^3G♭
29 451.9 13/10 ^3F♯, v3G
30 467.5 17/13, 21/16 v3F𝄪, vvG
31 483.1 120/91 vvF𝄪, vG
32 498.7 4/3 G
33 514.3 27/20 ^G, ^^A♭♭
34 529.9 19/14 ^^G, ^3A♭♭
35 545.5 11/8, 15/11, 26/19 ^3G, v3A♭
36 561.0 18/13 v3G♯, vvA♭
37 576.6 7/5 vvG♯, vA♭
38 592.2 24/17, 38/27, 45/32 vG♯, A♭

* As a 19-limit temperament

Notation

Ups and downs notation

Using Helmholtz–Ellis accidentals, 77edo can be notated using ups and downs notation:

Step Offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Sharp Symbol
Heji18.svg
Heji19.svg
Heji20.svg
Heji21.svg
Heji22.svg
Heji23.svg
Heji24.svg
Heji25.svg
Heji26.svg
Heji27.svg
Heji28.svg
Heji29.svg
Heji30.svg
Heji31.svg
Heji32.svg
Heji33.svg
Heji34.svg
Heji35.svg
Flat Symbol
Heji17.svg
Heji16.svg
Heji15.svg
Heji14.svg
Heji13.svg
Heji12.svg
Heji11.svg
Heji10.svg
Heji9.svg
Heji8.svg
Heji7.svg
Heji6.svg
Heji5.svg
Heji4.svg
Heji3.svg
Heji2.svg
Heji1.svg

Sagittal notation

Evo flavor

77-EDO Evo Sagittal.svgSagittal notationPeriodic table of EDOs with sagittal notation81/8064/6333/32

Revo flavor

77-EDO Revo Sagittal.svgSagittal notationPeriodic table of EDOs with sagittal notation81/8064/6333/32

Approximation to JI

Zeta peak index

Tuning Strength Closest edo Integer limit
ZPI Steps per octave Step size (cents) Height Integral Gap Edo Octave (cents) Consistent Distinct
414zpi 76.9918536925042 15.5860645308353 8.194847 1.311364 17.029289 77edo 1200.12696887432 10 10

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

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
ratio*
Temperament
1 4\77 62.3 28/27 Unicorn / alicorn (77e) / camahueto (77) / qilin (77)
1 5\77 77.9 21/20 Valentine
1 9\77 140.3 13/12 Tsaharuk
1 15\77 233.8 8/7 Guiron
1 16\77 249.4 15/13 Hemischis (77e)
1 20\77 311.7 6/5 Oolong
1 23\77 358.4 16/13 Restles
1 31\77 483.1 45/34 Hemiseven
1 32\77 498.7 4/3 Grackle
1 34\77 529.9 512/375 Tuskaloosa / muscogee
1 36\77 561.0 18/13 Demivalentine
7 32\77
(1\77)
498.7
(15.6)
4/3
(81/80)
Absurdity
11 32\77
(3\77)
498.7
(46.8)
4/3
(36/35)
Hendecatonic

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

Music

Jake Freivald
Joel Grant Taylor
Chris Vaisvil