# 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 Approximate Ratios Ups and Downs Notation
(Dual Flat Fifth 65\112)
Ups and Downs Notation
(Dual Sharp Fifth 66\112)
0 0 1/1 D D
1 10.714 ^D, v3E♭♭ ^D, v5E♭
2 21.429 ^^D, vvE♭♭ ^^D, v4E♭
3 32.143 56/55 ^3D, vE♭♭ ^3D, v3E♭
4 42.857 40/39 ^4D, E♭♭ ^4D, vvE♭
5 53.571 33/32 ^5D, v6E♭ ^5D, vE♭
6 64.286 26/25 ^6D, v5E♭ ^6D, E♭
7 75 D♯, v4E♭ ^7D, v13E
8 85.714 21/20 ^D♯, v3E♭ ^8D, v12E
9 96.429 55/52 ^^D♯, vvE♭ ^9D, v11E
10 107.143 52/49 ^3D♯, vE♭ ^10D, v10E
11 117.857 ^4D♯, E♭ ^11D, v9E
12 128.571 14/13 ^5D♯, v6E ^12D, v8E
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 49/44 ^3D𝄪, vE ^3D♯, v3E
18 192.857 28/25 E ^4D♯, vvE
19 203.571 55/49 ^E, v3F♭ ^5D♯, vE
20 214.286 ^^E, vvF♭ E
21 225 25/22 ^3E, vF♭ ^E, v5F
22 235.714 ^4E, F♭ ^^E, v4F
23 246.429 ^5E, v6F ^3E, v3F
24 257.143 65/56 ^6E, v5F ^4E, vvF
25 267.857 E♯, v4F ^5E, vF
26 278.571 75/64 ^E♯, v3F F
27 289.286 13/11, 77/65 ^^E♯, vvF ^F, v5G♭
28 300 25/21 ^3E♯, vF ^^F, v4G♭
29 310.714 F ^3F, v3G♭
30 321.429 ^F, v3G♭♭ ^4F, vvG♭
31 332.143 40/33 ^^F, vvG♭♭ ^5F, vG♭
32 342.857 39/32 ^3F, vG♭♭ ^6F, G♭
33 353.571 ^4F, G♭♭ ^7F, v13G
34 364.286 ^5F, v6G♭ ^8F, v12G
35 375 ^6F, v5G♭ ^9F, v11G
36 385.714 5/4 F♯, v4G♭ ^10F, v10G
37 396.429 44/35 ^F♯, v3G♭ ^11F, v9G
38 407.143 ^^F♯, vvG♭ ^12F, v8G
39 417.857 14/11 ^3F♯, vG♭ ^13F, v7G
40 428.571 32/25, 50/39 ^4F♯, G♭ F♯, v6G
41 439.286 ^5F♯, v6G ^F♯, v5G
42 450 13/10 ^6F♯, v5G ^^F♯, v4G
43 460.714 F𝄪, v4G ^3F♯, v3G
44 471.429 21/16 ^F𝄪, v3G ^4F♯, vvG
45 482.143 33/25 ^^F𝄪, vvG ^5F♯, vG
46 492.857 65/49 ^3F𝄪, vG G
47 503.571 G ^G, v5A♭
48 514.286 35/26 ^G, v3A♭♭ ^^G, v4A♭
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 7/5 G♯, v4A♭ ^8G, v12A
55 589.286 ^G♯, v3A♭ ^9G, v11A
56 600 ^^G♯, vvA♭ ^10G, v10A
57 610.714 ^3G♯, vA♭ ^11G, v9A
58 621.429 10/7 ^4G♯, A♭ ^12G, v8A
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 65/44 ^^G𝄪, vvA ^3G♯, v3A
64 685.714 52/35 ^3G𝄪, vA ^4G♯, vvA
65 696.429 A ^5G♯, vA
66 707.143 ^A, v3B♭♭ A
67 717.857 50/33 ^^A, vvB♭♭ ^A, v5B♭
68 728.571 32/21 ^3A, vB♭♭ ^^A, v4B♭
69 739.286 ^4A, B♭♭ ^3A, v3B♭
70 750 20/13 ^5A, v6B♭ ^4A, vvB♭
71 760.714 ^6A, v5B♭ ^5A, vB♭
72 771.429 25/16, 39/25 A♯, v4B♭ ^6A, B♭
73 782.143 11/7 ^A♯, v3B♭ ^7A, v13B
74 792.857 ^^A♯, vvB♭ ^8A, v12B
75 803.571 35/22 ^3A♯, vB♭ ^9A, v11B
76 814.286 8/5 ^4A♯, B♭ ^10A, v10B
77 825 ^5A♯, v6B ^11A, v9B
78 835.714 ^6A♯, v5B ^12A, v8B
79 846.429 A𝄪, v4B ^13A, v7B
80 857.143 64/39 ^A𝄪, v3B A♯, v6B
81 867.857 33/20 ^^A𝄪, vvB ^A♯, v5B
82 878.571 ^3A𝄪, vB ^^A♯, v4B
83 889.286 B ^3A♯, v3B
84 900 42/25 ^B, v3C♭ ^4A♯, vvB
85 910.714 22/13 ^^B, vvC♭ ^5A♯, vB
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 44/25 ^B♯, v3C ^5B, vC
92 985.714 ^^B♯, vvC C
93 996.429 ^3B♯, vC ^C, v5D♭
94 1007.143 25/14 C ^^C, v4D♭
95 1017.857 ^C, v3D♭♭ ^3C, v3D♭
96 1028.571 ^^C, vvD♭♭ ^4C, vvD♭
97 1039.286 ^3C, vD♭♭ ^5C, vD♭
98 1050 ^4C, D♭♭ ^6C, D♭
99 1060.714 ^5C, v6D♭ ^7C, v13D
100 1071.429 13/7 ^6C, v5D♭ ^8C, v12D
101 1082.143 C♯, v4D♭ ^9C, v11D
102 1092.857 49/26 ^C♯, v3D♭ ^10C, v10D
103 1103.571 ^^C♯, vvD♭ ^11C, v9D
104 1114.286 40/21 ^3C♯, vD♭ ^12C, v8D
105 1125 ^4C♯, D♭ ^13C, v7D
106 1135.714 25/13 ^5C♯, v6D C♯, v6D
107 1146.429 64/33 ^6C♯, v5D ^C♯, v5D
108 1157.143 39/20 C𝄪, v4D ^^C♯, v4D
109 1167.857 55/28 ^C𝄪, v3D ^3C♯, v3D
110 1178.571 ^^C𝄪, vvD ^4C♯, vvD
111 1189.286 ^3C𝄪, vD ^5C♯, vD
112 1200 2/1 D D

Cam Taylor