# 112edo

 ← 111edo 112edo 113edo →
Prime factorization 24 × 7
Step size 10.7143¢
Fifth 66\112 (707.143¢) (→33\56)
Semitones (A1:m2) 14:6 (150¢ : 64.29¢)
Dual sharp fifth 66\112 (707.143¢) (→33\56)
Dual flat fifth 65\112 (696.429¢)
Dual major 2nd 19\112 (203.571¢)
Consistency limit 3
Distinct consistency limit 3

112 equal divisions of the octave (abbreviated 112edo or 112ed2), also called 112-tone equal temperament (112tet) or 112 equal temperament (112et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 112 equal parts of about 10.7 ¢ each. Each step represents a frequency ratio of 21/112, or the 112th root of 2.

## Theory

112edo has two great perfect fifths, the lower of which approximates quarter-comma meantone (just a tad lower), and the upper of which – the patent fifth – is identical to the perfect fifth of 56edo, a great inverse gentle fifth where +5 fifths gives a near-just 28/27 while -8 fifths gives a near-just 39/32 (identical to 2 degrees of 7edo) and +9 fifths gives a close approximation to 21/17.

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from 17edo, but sharing a similar structure.

### Odd harmonics

Approximation of odd harmonics in 112edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +5.19 -0.60 -4.54 -0.34 -4.89 -4.81 +4.59 +2.19 +2.49 +0.65 +3.87
Relative (%) +48.4 -5.6 -42.4 -3.2 -45.6 -44.9 +42.8 +20.4 +23.2 +6.0 +36.1
Steps
(reduced)
178
(66)
260
(36)
314
(90)
355
(19)
387
(51)
414
(78)
438
(102)
458
(10)
476
(28)
492
(44)
507
(59)

### Subsets and supersets

Since 112 factors into 24 × 7, 112edo has subset edos 2, 4, 7, 8, 14, 16, 28, and 56. 224edo, which doubles it, is a strong 13-limit system.

## Intervals

Steps Cents Ups and Downs Notation
(Dual Flat Fifth 65\112)
Ups and Downs Notation
(Dual Sharp Fifth 66\112)
Approximate Ratios
0 0 D D 1/1
1 10.7143 ^D, v3E♭♭ ^D, v5E♭
2 21.4286 ^^D, vvE♭♭ ^^D, v4E♭
3 32.1429 ^3D, vE♭♭ ^3D, v3E♭ 56/55
4 42.8571 ^4D, E♭♭ ^4D, vvE♭ 40/39
5 53.5714 ^5D, v6E♭ ^5D, vE♭ 33/32
6 64.2857 ^6D, v5E♭ ^6D, E♭ 26/25
7 75 D♯, v4E♭ ^7D, v13E
8 85.7143 ^D♯, v3E♭ ^8D, v12E 21/20
9 96.4286 ^^D♯, vvE♭ ^9D, v11E 55/52
10 107.143 ^3D♯, vE♭ ^10D, v10E 52/49
11 117.857 ^4D♯, E♭ ^11D, v9E
12 128.571 ^5D♯, v6E ^12D, v8E 14/13
13 139.286 ^6D♯, v5E ^13D, v7E
14 150 D𝄪, v4E D♯, v6E
15 160.714 ^D𝄪, v3E ^D♯, v5E
16 171.429 ^^D𝄪, vvE ^^D♯, v4E
17 182.143 ^3D𝄪, vE ^3D♯, v3E 49/44
18 192.857 E ^4D♯, vvE 28/25
19 203.571 ^E, v3F♭ ^5D♯, vE 55/49
20 214.286 ^^E, vvF♭ E
21 225 ^3E, vF♭ ^E, v5F 25/22
22 235.714 ^4E, F♭ ^^E, v4F
23 246.429 ^5E, v6F ^3E, v3F
24 257.143 ^6E, v5F ^4E, vvF 65/56
25 267.857 E♯, v4F ^5E, vF
26 278.571 ^E♯, v3F F 75/64
27 289.286 ^^E♯, vvF ^F, v5G♭ 13/11, 77/65
28 300 ^3E♯, vF ^^F, v4G♭ 25/21
29 310.714 F ^3F, v3G♭
30 321.429 ^F, v3G♭♭ ^4F, vvG♭
31 332.143 ^^F, vvG♭♭ ^5F, vG♭ 40/33
32 342.857 ^3F, vG♭♭ ^6F, G♭ 39/32
33 353.571 ^4F, G♭♭ ^7F, v13G
34 364.286 ^5F, v6G♭ ^8F, v12G
35 375 ^6F, v5G♭ ^9F, v11G
36 385.714 F♯, v4G♭ ^10F, v10G 5/4
37 396.429 ^F♯, v3G♭ ^11F, v9G 44/35
38 407.143 ^^F♯, vvG♭ ^12F, v8G
39 417.857 ^3F♯, vG♭ ^13F, v7G 14/11
40 428.571 ^4F♯, G♭ F♯, v6G 32/25, 50/39
41 439.286 ^5F♯, v6G ^F♯, v5G
42 450 ^6F♯, v5G ^^F♯, v4G 13/10
43 460.714 F𝄪, v4G ^3F♯, v3G
44 471.429 ^F𝄪, v3G ^4F♯, vvG 21/16
45 482.143 ^^F𝄪, vvG ^5F♯, vG 33/25
46 492.857 ^3F𝄪, vG G 65/49
47 503.571 G ^G, v5A♭
48 514.286 ^G, v3A♭♭ ^^G, v4A♭ 35/26
49 525 ^^G, vvA♭♭ ^3G, v3A♭
50 535.714 ^3G, vA♭♭ ^4G, vvA♭
51 546.429 ^4G, A♭♭ ^5G, vA♭
52 557.143 ^5G, v6A♭ ^6G, A♭
53 567.857 ^6G, v5A♭ ^7G, v13A
54 578.571 G♯, v4A♭ ^8G, v12A 7/5
55 589.286 ^G♯, v3A♭ ^9G, v11A
56 600 ^^G♯, vvA♭ ^10G, v10A
57 610.714 ^3G♯, vA♭ ^11G, v9A
58 621.429 ^4G♯, A♭ ^12G, v8A 10/7
59 632.143 ^5G♯, v6A ^13G, v7A
60 642.857 ^6G♯, v5A G♯, v6A
61 653.571 G𝄪, v4A ^G♯, v5A
62 664.286 ^G𝄪, v3A ^^G♯, v4A
63 675 ^^G𝄪, vvA ^3G♯, v3A 65/44
64 685.714 ^3G𝄪, vA ^4G♯, vvA 52/35
65 696.429 A ^5G♯, vA
66 707.143 ^A, v3B♭♭ A
67 717.857 ^^A, vvB♭♭ ^A, v5B♭ 50/33
68 728.571 ^3A, vB♭♭ ^^A, v4B♭ 32/21
69 739.286 ^4A, B♭♭ ^3A, v3B♭
70 750 ^5A, v6B♭ ^4A, vvB♭ 20/13
71 760.714 ^6A, v5B♭ ^5A, vB♭
72 771.429 A♯, v4B♭ ^6A, B♭ 25/16, 39/25
73 782.143 ^A♯, v3B♭ ^7A, v13B 11/7
74 792.857 ^^A♯, vvB♭ ^8A, v12B
75 803.571 ^3A♯, vB♭ ^9A, v11B 35/22
76 814.286 ^4A♯, B♭ ^10A, v10B 8/5
77 825 ^5A♯, v6B ^11A, v9B
78 835.714 ^6A♯, v5B ^12A, v8B
79 846.429 A𝄪, v4B ^13A, v7B
80 857.143 ^A𝄪, v3B A♯, v6B 64/39
81 867.857 ^^A𝄪, vvB ^A♯, v5B 33/20
82 878.571 ^3A𝄪, vB ^^A♯, v4B
83 889.286 B ^3A♯, v3B
84 900 ^B, v3C♭ ^4A♯, vvB 42/25
85 910.714 ^^B, vvC♭ ^5A♯, vB 22/13
86 921.429 ^3B, vC♭ B
87 932.143 ^4B, C♭ ^B, v5C
88 942.857 ^5B, v6C ^^B, v4C
89 953.571 ^6B, v5C ^3B, v3C
90 964.286 B♯, v4C ^4B, vvC
91 975 ^B♯, v3C ^5B, vC 44/25
92 985.714 ^^B♯, vvC C
93 996.429 ^3B♯, vC ^C, v5D♭
94 1007.14 C ^^C, v4D♭ 25/14
95 1017.86 ^C, v3D♭♭ ^3C, v3D♭
96 1028.57 ^^C, vvD♭♭ ^4C, vvD♭
97 1039.29 ^3C, vD♭♭ ^5C, vD♭
98 1050 ^4C, D♭♭ ^6C, D♭
99 1060.71 ^5C, v6D♭ ^7C, v13D
100 1071.43 ^6C, v5D♭ ^8C, v12D 13/7
101 1082.14 C♯, v4D♭ ^9C, v11D
102 1092.86 ^C♯, v3D♭ ^10C, v10D 49/26
103 1103.57 ^^C♯, vvD♭ ^11C, v9D
104 1114.29 ^3C♯, vD♭ ^12C, v8D 40/21
105 1125 ^4C♯, D♭ ^13C, v7D
106 1135.71 ^5C♯, v6D C♯, v6D 25/13
107 1146.43 ^6C♯, v5D ^C♯, v5D 64/33
108 1157.14 C𝄪, v4D ^^C♯, v4D 39/20
109 1167.86 ^C𝄪, v3D ^3C♯, v3D 55/28
110 1178.57 ^^C𝄪, vvD ^4C♯, vvD
111 1189.29 ^3C𝄪, vD ^5C♯, vD
112 1200 D D 2/1

Cam Taylor