78edo

 ← 77edo 78edo 79edo →
Prime factorization 2 × 3 × 13
Step size 15.3846¢
Fifth 46\78 (707.692¢) (→23\39)
Semitones (A1:m2) 10:4 (153.8¢ : 61.54¢)
Dual sharp fifth 46\78 (707.692¢) (→23\39)
Dual flat fifth 45\78 (692.308¢) (→15\26)
Dual major 2nd 13\78 (200¢) (→1\6)
Consistency limit 7
Distinct consistency limit 7

78 equal divisions of the octave (abbreviated 78edo or 78ed2), also called 78-tone equal temperament (78tet) or 78 equal temperament (78et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 78 equal parts of about 15.4 ¢ each. Each step represents a frequency ratio of 21/78, or the 78th root of 2.

Theory

78edo is consistent to the 7-odd-limit, but the error of harmonic 3, inherited from 39edo, is quite large for the size of the system.

This tuning tempers out 2048/2025 in the 5-limit; 875/864 and 2401/2400 in the 7-limit; and 100/99, 385/384 and 1375/1372 in the 11-limit. It provides the optimal patent val for 11-limit keen temperament.

Much like 100bddd, the 78dd val can be used to construct an alternative to 22edo for pajara. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7 cents. The major third is 384.6 cents; less than two cents flat of just. The harmonic seventh is 984.6 cents, or about 15.8 cents sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16 cents} off. The 22-note 2mos generated in this way could be used to build straight-fretted guitars that would be tuned in tritones. The appeal of this scale is that it is less xenharmonic than 22edo is, for listeners accustomed to 12edo. In particular, the 163.6 cents "flat minor whole tone" of 22edo is now 169.2 cents, making it more clearly a whole tone (albeit noticeably flat), rather than a neutral second.

Odd harmonics

Approximation of odd harmonics in 78edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +5.74 -1.70 +0.40 -3.91 +2.53 +5.63 +4.04 +2.74 -5.21 +6.14 +2.49
Relative (%) +37.3 -11.0 +2.6 -25.4 +16.4 +36.6 +26.3 +17.8 -33.8 +39.9 +16.2
Steps
(reduced)
124
(46)
181
(25)
219
(63)
247
(13)
270
(36)
289
(55)
305
(71)
319
(7)
331
(19)
343
(31)
353
(41)

Subsets and supersets

Since 78 factors into 2 × 3 × 13, 78edo has subset edos 2, 3, 6, 13, 26, and 39. 156edo, which doubles it, is a notable tuning.

Intervals

Steps Cents Approximate Ratios Ups and Downs Notation
(Dual Flat Fifth 45\78)
Ups and Downs Notation
(Dual Sharp Fifth 46\78)
0 0 1/1 D D
1 15.385 ^D, vvE♭♭♭ ^D, v3E♭
2 30.769 49/48, 50/49, 56/55, 65/64, 66/65 ^^D, vE♭♭♭ ^^D, vvE♭
3 46.154 77/75 D♯, E♭♭♭ ^3D, vE♭
4 61.538 80/77 ^D♯, vvE♭♭ ^4D, E♭
5 76.923 22/21 ^^D♯, vE♭♭ ^5D, v9E
6 92.308 55/52 D𝄪, E♭♭ ^6D, v8E
7 107.692 16/15, 52/49 ^D𝄪, vvE♭ ^7D, v7E
8 123.077 14/13, 15/14 ^^D𝄪, vE♭ ^8D, v6E
9 138.462 13/12 D♯𝄪, E♭ ^9D, v5E
10 153.846 12/11, 35/32 ^D♯𝄪, vvE D♯, v4E
11 169.231 11/10 ^^D♯𝄪, vE ^D♯, v3E
12 184.615 49/44 E ^^D♯, vvE
13 200 28/25, 55/49 ^E, vvF♭♭ ^3D♯, vE
14 215.385 25/22 ^^E, vF♭♭ E
15 230.769 8/7, 55/48 E♯, F♭♭ ^E, v3F
16 246.154 15/13, 52/45 ^E♯, vvF♭ ^^E, vvF
17 261.538 7/6, 64/55, 65/56 ^^E♯, vF♭ ^3E, vF
18 276.923 75/64 E𝄪, F♭ F
19 292.308 13/11, 77/65 ^E𝄪, vvF ^F, v3G♭
20 307.692 ^^E𝄪, vF ^^F, vvG♭
21 323.077 77/64 F ^3F, vG♭
22 338.462 ^F, vvG♭♭♭ ^4F, G♭
23 353.846 16/13, 49/40, 60/49 ^^F, vG♭♭♭ ^5F, v9G
24 369.231 26/21 F♯, G♭♭♭ ^6F, v8G
25 384.615 5/4 ^F♯, vvG♭♭ ^7F, v7G
26 400 44/35 ^^F♯, vG♭♭ ^8F, v6G
27 415.385 14/11, 33/26 F𝄪, G♭♭ ^9F, v5G
28 430.769 32/25, 77/60 ^F𝄪, vvG♭ F♯, v4G
29 446.154 ^^F𝄪, vG♭ ^F♯, v3G
30 461.538 55/42, 64/49 F♯𝄪, G♭ ^^F♯, vvG
31 476.923 21/16 ^F♯𝄪, vvG ^3F♯, vG
32 492.308 4/3, 65/49 ^^F♯𝄪, vG G
33 507.692 75/56 G ^G, v3A♭
34 523.077 65/48 ^G, vvA♭♭♭ ^^G, vvA♭
35 538.462 15/11 ^^G, vA♭♭♭ ^3G, vA♭
36 553.846 11/8 G♯, A♭♭♭ ^4G, A♭
37 569.231 18/13 ^G♯, vvA♭♭ ^5G, v9A
38 584.615 7/5 ^^G♯, vA♭♭ ^6G, v8A
39 600 G𝄪, A♭♭ ^7G, v7A
40 615.385 10/7 ^G𝄪, vvA♭ ^8G, v6A
41 630.769 13/9, 75/52 ^^G𝄪, vA♭ ^9G, v5A
42 646.154 16/11 G♯𝄪, A♭ G♯, v4A
43 661.538 22/15 ^G♯𝄪, vvA ^G♯, v3A
44 676.923 65/44, 77/52 ^^G♯𝄪, vA ^^G♯, vvA
45 692.308 A ^3G♯, vA
46 707.692 3/2 ^A, vvB♭♭♭ A
47 723.077 32/21 ^^A, vB♭♭♭ ^A, v3B♭
48 738.462 49/32, 75/49 A♯, B♭♭♭ ^^A, vvB♭
49 753.846 65/42 ^A♯, vvB♭♭ ^3A, vB♭
50 769.231 25/16 ^^A♯, vB♭♭ ^4A, B♭
51 784.615 11/7, 52/33 A𝄪, B♭♭ ^5A, v9B
52 800 35/22 ^A𝄪, vvB♭ ^6A, v8B
53 815.385 8/5, 77/48 ^^A𝄪, vB♭ ^7A, v7B
54 830.769 21/13 A♯𝄪, B♭ ^8A, v6B
55 846.154 13/8, 49/30, 80/49 ^A♯𝄪, vvB ^9A, v5B
56 861.538 ^^A♯𝄪, vB A♯, v4B
57 876.923 B ^A♯, v3B
58 892.308 ^B, vvC♭♭ ^^A♯, vvB
59 907.692 22/13 ^^B, vC♭♭ ^3A♯, vB
60 923.077 75/44 B♯, C♭♭ B
61 938.462 12/7, 55/32 ^B♯, vvC♭ ^B, v3C
62 953.846 26/15, 45/26 ^^B♯, vC♭ ^^B, vvC
63 969.231 7/4 B𝄪, C♭ ^3B, vC
64 984.615 44/25 ^B𝄪, vvC C
65 1000 25/14 ^^B𝄪, vC ^C, v3D♭
66 1015.385 C ^^C, vvD♭
67 1030.769 20/11 ^C, vvD♭♭♭ ^3C, vD♭
68 1046.154 11/6, 64/35 ^^C, vD♭♭♭ ^4C, D♭
69 1061.538 24/13 C♯, D♭♭♭ ^5C, v9D
70 1076.923 13/7, 28/15 ^C♯, vvD♭♭ ^6C, v8D
71 1092.308 15/8, 49/26 ^^C♯, vD♭♭ ^7C, v7D
72 1107.692 C𝄪, D♭♭ ^8C, v6D
73 1123.077 21/11 ^C𝄪, vvD♭ ^9C, v5D
74 1138.462 77/40 ^^C𝄪, vD♭ C♯, v4D
75 1153.846 C♯𝄪, D♭ ^C♯, v3D
76 1169.231 49/25, 55/28, 65/33 ^C♯𝄪, vvD ^^C♯, vvD
77 1184.615 ^^C♯𝄪, vD ^3C♯, vD
78 1200 2/1 D D