112edo

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← 111edo112edo113edo →
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

Music

Cam Taylor
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