# 76edo

 ← 75edo 76edo 77edo →
Prime factorization 22 × 19
Step size 15.7895¢
Fifth 44\76 (694.737¢) (→11\19)
Semitones (A1:m2) 4:8 (63.16¢ : 126.3¢)
Dual sharp fifth 45\76 (710.526¢)
Dual flat fifth 44\76 (694.737¢) (→11\19)
Dual major 2nd 13\76 (205.263¢)
Consistency limit 7
Distinct consistency limit 7

76 equal divisions of the octave (abbreviated 76edo or 76ed2), also called 76-tone equal temperament (76tet) or 76 equal temperament (76et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 76 equal parts of about 15.8 ¢ each. Each step represents a frequency ratio of 21/76, or the 76th root of 2.

## Theory

76edo's patent val is contorted in the 5-limit, reflecting the fact that 76 = 4 × 19. In the 7-limit it tempers out 2401/2400 in addition to 81/80, and so supports the squares temperament. In the 11-limit, it tempers out 245/242 and 385/384, and supports pombe, the 24 & 26 temperament. In the 13-limit, it tempers out 105/104, 144/143, 351/350 and 364/363. While the 44\76 = 11\19 fifth is already flat, the 43\76 fifth, even flatter, is an almost perfect approximation to the hornbostel temperament's POTE fifth, whereas its sharp fifth, 45\76, makes for an excellent superpyth fifth. Hence you can do hornbostel/mavila, squares/meantone, and superpyth all with the same equal division.

Using the 76dgh val, 76edo provides an excellent tuning for teff temperament, a low-complexity, medium-accuracy, and high-limit (17- or 19-limit) temperament.

### Odd harmonics

Approximation of odd harmonics in 76edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -7.22 -7.37 -5.67 +1.35 +1.31 -3.69 +1.20 +5.57 +2.49 +2.90 +3.30
Relative (%) -45.7 -46.7 -35.9 +8.6 +8.3 -23.3 +7.6 +35.3 +15.8 +18.4 +20.9
Steps
(reduced)
120
(44)
176
(24)
213
(61)
241
(13)
263
(35)
281
(53)
297
(69)
311
(7)
323
(19)
334
(30)
344
(40)

### Subsets and supersets

Since 76 factors into 22 × 19, 76edo has subset edos 2, 4, 19, and 38. 152edo, which doubles it, is a zeta peak edo.

## Intervals

Steps Cents Approximate Ratios Ups and Downs Notation
(Dual Flat Fifth 44\76)
Ups and Downs Notation
(Dual Sharp Fifth 45\76)
0 0 1/1 D D
1 15.789 ^D, v3E♭♭ ^D, vvE♭
2 31.579 49/48, 50/49, 56/55, 66/65 ^^D, vvE♭♭ ^^D, vE♭
3 47.368 33/32, 36/35, 40/39 ^3D, vE♭♭ ^3D, E♭
4 63.158 80/77 D♯, E♭♭ ^4D, v10E
5 78.947 21/20 ^D♯, v3E♭ ^5D, v9E
6 94.737 55/52 ^^D♯, vvE♭ ^6D, v8E
7 110.526 ^3D♯, vE♭ ^7D, v7E
8 126.316 14/13 D𝄪, E♭ ^8D, v6E
9 142.105 13/12 ^D𝄪, v3E ^9D, v5E
10 157.895 ^^D𝄪, vvE ^10D, v4E
11 173.684 72/65 ^3D𝄪, vE D♯, v3E
12 189.474 39/35 E ^D♯, vvE
13 205.263 55/49 ^E, v3F♭ ^^D♯, vE
14 221.053 ^^E, vvF♭ E
15 236.842 8/7, 55/48 ^3E, vF♭ ^E, vvF
16 252.632 65/56 E♯, F♭ ^^E, vF
17 268.421 7/6, 64/55 ^E♯, v3F F
18 284.211 13/11, 33/28 ^^E♯, vvF ^F, vvG♭
19 300 25/21 ^3E♯, vF ^^F, vG♭
20 315.789 6/5, 77/64 F ^3F, G♭
21 331.579 40/33 ^F, v3G♭♭ ^4F, v10G
22 347.368 49/40, 60/49 ^^F, vvG♭♭ ^5F, v9G
23 363.158 16/13 ^3F, vG♭♭ ^6F, v8G
24 378.947 F♯, G♭♭ ^7F, v7G
25 394.737 49/39, 63/50 ^F♯, v3G♭ ^8F, v6G
26 410.526 33/26 ^^F♯, vvG♭ ^9F, v5G
27 426.316 50/39 ^3F♯, vG♭ ^10F, v4G
28 442.105 F𝄪, G♭ F♯, v3G
29 457.895 13/10 ^F𝄪, v3G ^F♯, vvG
30 473.684 ^^F𝄪, vvG ^^F♯, vG
31 489.474 65/49 ^3F𝄪, vG G
32 505.263 G ^G, vvA♭
33 521.053 65/48, 66/49 ^G, v3A♭♭ ^^G, vA♭
34 536.842 49/36 ^^G, vvA♭♭ ^3G, A♭
35 552.632 11/8, 48/35 ^3G, vA♭♭ ^4G, v10A
36 568.421 25/18, 39/28 G♯, A♭♭ ^5G, v9A
37 584.211 7/5 ^G♯, v3A♭ ^6G, v8A
38 600 55/39, 78/55 ^^G♯, vvA♭ ^7G, v7A
39 615.789 10/7 ^3G♯, vA♭ ^8G, v6A
40 631.579 36/25, 56/39 G𝄪, A♭ ^9G, v5A
41 647.368 16/11, 35/24 ^G𝄪, v3A ^10G, v4A
42 663.158 72/49 ^^G𝄪, vvA G♯, v3A
43 678.947 49/33, 77/52 ^3G𝄪, vA ^G♯, vvA
44 694.737 A ^^G♯, vA
45 710.526 ^A, v3B♭♭ A
46 726.316 ^^A, vvB♭♭ ^A, vvB♭
47 742.105 20/13 ^3A, vB♭♭ ^^A, vB♭
48 757.895 65/42 A♯, B♭♭ ^3A, B♭
49 773.684 39/25 ^A♯, v3B♭ ^4A, v10B
50 789.474 52/33 ^^A♯, vvB♭ ^5A, v9B
51 805.263 78/49 ^3A♯, vB♭ ^6A, v8B
52 821.053 77/48 A𝄪, B♭ ^7A, v7B
53 836.842 13/8 ^A𝄪, v3B ^8A, v6B
54 852.632 49/30, 80/49 ^^A𝄪, vvB ^9A, v5B
55 868.421 33/20 ^3A𝄪, vB ^10A, v4B
56 884.211 5/3 B A♯, v3B
57 900 42/25 ^B, v3C♭ ^A♯, vvB
58 915.789 22/13, 56/33 ^^B, vvC♭ ^^A♯, vB
59 931.579 12/7, 55/32 ^3B, vC♭ B
60 947.368 B♯, C♭ ^B, vvC
61 963.158 7/4 ^B♯, v3C ^^B, vC
62 978.947 ^^B♯, vvC C
63 994.737 ^3B♯, vC ^C, vvD♭
64 1010.526 70/39 C ^^C, vD♭
65 1026.316 65/36 ^C, v3D♭♭ ^3C, D♭
66 1042.105 ^^C, vvD♭♭ ^4C, v10D
67 1057.895 24/13 ^3C, vD♭♭ ^5C, v9D
68 1073.684 13/7 C♯, D♭♭ ^6C, v8D
69 1089.474 ^C♯, v3D♭ ^7C, v7D
70 1105.263 ^^C♯, vvD♭ ^8C, v6D
71 1121.053 40/21 ^3C♯, vD♭ ^9C, v5D
72 1136.842 77/40 C𝄪, D♭ ^10C, v4D
73 1152.632 35/18, 39/20, 64/33 ^C𝄪, v3D C♯, v3D
74 1168.421 49/25, 55/28, 65/33 ^^C𝄪, vvD ^C♯, vvD
75 1184.211 ^3C𝄪, vD ^^C♯, vD
76 1200 2/1 D D