78edo

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← 77edo78edo79edo →
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

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)

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 ¢. The major third is 384.6 ¢; less than two cents flat of just. The harmonic seventh is 984.6 ¢, or about 15.8 ¢ 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 ¢ 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 ¢ "flat minor whole tone" of 22edo is now 169.2 ¢, making it more clearly a whole tone (albeit noticeably flat), rather than a neutral second.

Intervals

Steps Cents Ups and Downs Notation
(Dual Flat Fifth 45\78)
Ups and Downs Notation
(Dual Sharp Fifth 46\78)
Approximate Ratios
0 0 D D 1/1
1 15.3846 ^D, vvE♭♭♭ ^D, v3E♭
2 30.7692 ^^D, vE♭♭♭ ^^D, vvE♭ 49/48, 50/49, 56/55, 65/64, 66/65
3 46.1538 D♯, E♭♭♭ ^3D, vE♭ 77/75
4 61.5385 ^D♯, vvE♭♭ ^4D, E♭ 80/77
5 76.9231 ^^D♯, vE♭♭ ^5D, v9E 22/21
6 92.3077 D𝄪, E♭♭ ^6D, v8E 55/52
7 107.692 ^D𝄪, vvE♭ ^7D, v7E 16/15, 52/49
8 123.077 ^^D𝄪, vE♭ ^8D, v6E 14/13, 15/14
9 138.462 D♯𝄪, E♭ ^9D, v5E 13/12
10 153.846 ^D♯𝄪, vvE D♯, v4E 12/11, 35/32
11 169.231 ^^D♯𝄪, vE ^D♯, v3E 11/10
12 184.615 E ^^D♯, vvE 49/44
13 200 ^E, vvF♭♭ ^3D♯, vE 28/25, 55/49
14 215.385 ^^E, vF♭♭ E 25/22
15 230.769 E♯, F♭♭ ^E, v3F 8/7, 55/48
16 246.154 ^E♯, vvF♭ ^^E, vvF 15/13, 52/45
17 261.538 ^^E♯, vF♭ ^3E, vF 7/6, 64/55, 65/56
18 276.923 E𝄪, F♭ F 75/64
19 292.308 ^E𝄪, vvF ^F, v3G♭ 13/11, 77/65
20 307.692 ^^E𝄪, vF ^^F, vvG♭
21 323.077 F ^3F, vG♭ 77/64
22 338.462 ^F, vvG♭♭♭ ^4F, G♭
23 353.846 ^^F, vG♭♭♭ ^5F, v9G 16/13, 49/40, 60/49
24 369.231 F♯, G♭♭♭ ^6F, v8G 26/21
25 384.615 ^F♯, vvG♭♭ ^7F, v7G 5/4
26 400 ^^F♯, vG♭♭ ^8F, v6G 44/35
27 415.385 F𝄪, G♭♭ ^9F, v5G 14/11, 33/26
28 430.769 ^F𝄪, vvG♭ F♯, v4G 32/25, 77/60
29 446.154 ^^F𝄪, vG♭ ^F♯, v3G
30 461.538 F♯𝄪, G♭ ^^F♯, vvG 55/42, 64/49
31 476.923 ^F♯𝄪, vvG ^3F♯, vG 21/16
32 492.308 ^^F♯𝄪, vG G 4/3, 65/49
33 507.692 G ^G, v3A♭ 75/56
34 523.077 ^G, vvA♭♭♭ ^^G, vvA♭ 65/48
35 538.462 ^^G, vA♭♭♭ ^3G, vA♭ 15/11
36 553.846 G♯, A♭♭♭ ^4G, A♭ 11/8
37 569.231 ^G♯, vvA♭♭ ^5G, v9A 18/13
38 584.615 ^^G♯, vA♭♭ ^6G, v8A 7/5
39 600 G𝄪, A♭♭ ^7G, v7A
40 615.385 ^G𝄪, vvA♭ ^8G, v6A 10/7
41 630.769 ^^G𝄪, vA♭ ^9G, v5A 13/9, 75/52
42 646.154 G♯𝄪, A♭ G♯, v4A 16/11
43 661.538 ^G♯𝄪, vvA ^G♯, v3A 22/15
44 676.923 ^^G♯𝄪, vA ^^G♯, vvA 65/44, 77/52
45 692.308 A ^3G♯, vA
46 707.692 ^A, vvB♭♭♭ A 3/2
47 723.077 ^^A, vB♭♭♭ ^A, v3B♭ 32/21
48 738.462 A♯, B♭♭♭ ^^A, vvB♭ 49/32, 75/49
49 753.846 ^A♯, vvB♭♭ ^3A, vB♭ 65/42
50 769.231 ^^A♯, vB♭♭ ^4A, B♭ 25/16
51 784.615 A𝄪, B♭♭ ^5A, v9B 11/7, 52/33
52 800 ^A𝄪, vvB♭ ^6A, v8B 35/22
53 815.385 ^^A𝄪, vB♭ ^7A, v7B 8/5, 77/48
54 830.769 A♯𝄪, B♭ ^8A, v6B 21/13
55 846.154 ^A♯𝄪, vvB ^9A, v5B 13/8, 49/30, 80/49
56 861.538 ^^A♯𝄪, vB A♯, v4B
57 876.923 B ^A♯, v3B
58 892.308 ^B, vvC♭♭ ^^A♯, vvB
59 907.692 ^^B, vC♭♭ ^3A♯, vB 22/13
60 923.077 B♯, C♭♭ B 75/44
61 938.462 ^B♯, vvC♭ ^B, v3C 12/7, 55/32
62 953.846 ^^B♯, vC♭ ^^B, vvC 26/15, 45/26
63 969.231 B𝄪, C♭ ^3B, vC 7/4
64 984.615 ^B𝄪, vvC C 44/25
65 1000 ^^B𝄪, vC ^C, v3D♭ 25/14
66 1015.38 C ^^C, vvD♭
67 1030.77 ^C, vvD♭♭♭ ^3C, vD♭ 20/11
68 1046.15 ^^C, vD♭♭♭ ^4C, D♭ 11/6, 64/35
69 1061.54 C♯, D♭♭♭ ^5C, v9D 24/13
70 1076.92 ^C♯, vvD♭♭ ^6C, v8D 13/7, 28/15
71 1092.31 ^^C♯, vD♭♭ ^7C, v7D 15/8, 49/26
72 1107.69 C𝄪, D♭♭ ^8C, v6D
73 1123.08 ^C𝄪, vvD♭ ^9C, v5D 21/11
74 1138.46 ^^C𝄪, vD♭ C♯, v4D 77/40
75 1153.85 C♯𝄪, D♭ ^C♯, v3D
76 1169.23 ^C♯𝄪, vvD ^^C♯, vvD 49/25, 55/28, 65/33
77 1184.62 ^^C♯𝄪, vD ^3C♯, vD
78 1200 D D 2/1