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 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

Music

Cam Taylor