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'''112EDO''' has two great perfect fifths, the lower of which approximates 1/4-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|28:27]] while -8 fifths gives a near-just [[39/32|32:39]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17|17:21]].
{{Infobox ET}}
{{ED intro}}
 
== Theory ==
112edo has two great [[3/2|perfect fifth]]s, 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.
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.


Since 112edo has a step of 10.714 cents, it also allows one to use its MOS scales as circulating temperaments.
=== Odd harmonics ===
{{Harmonics in equal|112|intervals=odd}}


{| class="wikitable"
=== Subsets and supersets ===
|-
Since 112 factors into {{factorization|112}}, 112edo has subset edos {{EDOs| 2, 4, 7, 8, 14, 16, 28, and 56 }}. [[224edo]], which doubles it, is a strong 13-limit system.
|+ Circulating temperaments in 112edo
|-
! Tones
! Pattern
! L:s
|-
| 5
| [[2L 3s]]
| 23:22
|-
| 6
| [[4L 2s]]
| 19:18
|-
| 7
| [[7edo]]
| rowspan="2" | equal
|-
| 8
| [[8edo]]
|-
| 9
| [[4L 5s]]
| 13:12
|-
| 10
| [[2L 8s]]
| 12:11
|-
| 11
| [[2L 9s]]
| 11:10
|-
| 12
| [[4L 8s]]
| 10:9
|-
| 13
| [[8L 5s]]
| 9:8
|-
| 14
| [[14edo]]
| equal
|-
| 15
| [[6L 9s]]
| 8:7
|-
| 16
| [[16edo]]
| equal
|-
| 17
| [[10L 7s]]
| rowspan="2" | 7:6
|-
| 18
| 4L 14s
|-
| 19
| [[17L 2s]]
| rowspan="4" | 6:5
|-
| 20
| 12L 8s
|-
| 21
| 7L 14s
|-
| 22
| 2L 20s
|-
| 23
| 20L 3s
| rowspan="5" | 5:4
|-
| 24
| 16L 8s
|-
| 25
| 12L 13s
|-
| 26
| 8L 18s
|-
| 27
| 4L 23s
|-
| 28
| [[28edo]]
| equal
|-
| 29
| 25L 4s
| rowspan="9" | 4:3
|-
| 30
| 22L 8s
|-
| 31
| 19L 12s
|-
| 32
| 16L 16s
|-
| 33
| 13L 20s
|-
| 34
| 10L 24s
|-
| 35
| 7L 28s
|-
| 36
| 4L 32s
|-
| 37
| 1L 36s
|-
| 38
| 36L 2s
| rowspan="18" | 3:2
|-
| 39
| 34L 5s
|-
| 40
| 32L 8s
|-
| 41
| 30L 11s
|-
| 42
| 28L 14s
|-
| 43
| 26L 17s
|-
| 44
| 24L 20s
|-
| 45
| 22L 23s
|-
| 46
| 20L 26s
|-
| 47
| 18L 29s
|-
| 48
| 16L 32s
|-
| 49
| 14L 35s
|-
| 50
| 12L 38s
|-
| 51
| 10L 41s
|-
| 52
| 8L 44s
|-
| 53
| 6L 47s
|-
| 54
| 4L 50s
|-
| 55
| 2L 53s
|-
| 56
| [[56edo]]
| equal
|-
| 57
| 55L 2s
| rowspan="33" | 2:1
|-
| 58
| 54L 4s
|-
| 59
| 53L 6s
|-
| 60
| 52L 8s
|-
| 61
| 51L 10s
|-
| 62
| 50L 12s
|-
| 63
| 49L 14s
|-
| 64
| 48L 16s
|-
| 65
| 47L 18s
|-
| 66
| 46L 20s
|-
| 67
| 45L 22s
|-
| 68
| 44L 24s
|-
| 69
| 43L 26s
|-
| 70
| 42L 28s
|-
| 71
| 41L 30s
|-
| 72
| 40L 32s
|-
| 73
| 39L 34s
|-
| 74
| 38L 36s
|-
| 75
| 37L 38s
|-
| 76
| 36L 40s
|-
| 77
| 35L 42s
|-
| 78
| 34L 44s
|-
| 79
| 33L 46s
|-
| 80
| 32L 48s
|-
| 81
| 31L 50s
|-
| 82
| 30L 52s
|-
| 83
| 29L 54s
|-
| 84
| 28L 56s
|-
| 85
| 27L 58s
|-
| 86
| 26L 60s
|-
| 87
| 25L 62s
|-
| 88
| 24L 64s
|-
| 89
| 23L 66s
|}


== Music in 112EDO ==
== Intervals ==
{{Interval table}}


* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv Circulating 2.3.7.11.13 Floaty Piano Improv] by [[Cam Taylor]]
== Music ==
; [[Cam Taylor]]
* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv ''Circulating 2.3.7.11.13 Floaty Piano Improv'']


[[Category:Equal divisions of the octave]]
== See also ==
[[Skip fretting system 112 9 11]]
[[Category:Listen]]

Latest revision as of 15:30, 23 April 2025

← 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 10.7 ^D, v3E♭♭ ^D, v5E♭
2 21.4 ^^D, vvE♭♭ ^^D, v4E♭
3 32.1 ^3D, vE♭♭ ^3D, v3E♭
4 42.9 39/38, 40/39, 41/40, 42/41, 43/42 v3D♯, E♭♭ ^4D, vvE♭
5 53.6 32/31, 33/32, 34/33 vvD♯, ^E♭♭ ^5D, vE♭
6 64.3 vD♯, ^^E♭♭ ^6D, E♭
7 75 24/23 D♯, ^3E♭♭ ^7D, ^E♭
8 85.7 21/20, 41/39 ^D♯, v3E♭ v6D♯, ^^E♭
9 96.4 37/35 ^^D♯, vvE♭ v5D♯, ^3E♭
10 107.1 33/31 ^3D♯, vE♭ v4D♯, ^4E♭
11 117.9 46/43 v3D𝄪, E♭ v3D♯, ^5E♭
12 128.6 14/13 vvD𝄪, ^E♭ vvD♯, ^6E♭
13 139.3 vD𝄪, ^^E♭ vD♯, v7E
14 150 D𝄪, ^3E♭ D♯, v6E
15 160.7 34/31 ^D𝄪, v3E ^D♯, v5E
16 171.4 21/19, 32/29 ^^D𝄪, vvE ^^D♯, v4E
17 182.1 ^3D𝄪, vE ^3D♯, v3E
18 192.9 19/17, 47/42 E ^4D♯, vvE
19 203.6 ^E, v3F♭ ^5D♯, vE
20 214.3 43/38 ^^E, vvF♭ E
21 225 33/29 ^3E, vF♭ ^E, v5F
22 235.7 39/34, 47/41 v3E♯, F♭ ^^E, v4F
23 246.4 vvE♯, ^F♭ ^3E, v3F
24 257.1 29/25 vE♯, ^^F♭ ^4E, vvF
25 267.9 E♯, ^3F♭ ^5E, vF
26 278.6 47/40 ^E♯, v3F F
27 289.3 13/11 ^^E♯, vvF ^F, v5G♭
28 300 25/21, 44/37 ^3E♯, vF ^^F, v4G♭
29 310.7 F ^3F, v3G♭
30 321.4 47/39 ^F, v3G♭♭ ^4F, vvG♭
31 332.1 23/19, 40/33 ^^F, vvG♭♭ ^5F, vG♭
32 342.9 39/32 ^3F, vG♭♭ ^6F, G♭
33 353.6 38/31 v3F♯, G♭♭ ^7F, ^G♭
34 364.3 21/17 vvF♯, ^G♭♭ v6F♯, ^^G♭
35 375 41/33 vF♯, ^^G♭♭ v5F♯, ^3G♭
36 385.7 5/4 F♯, ^3G♭♭ v4F♯, ^4G♭
37 396.4 39/31, 44/35 ^F♯, v3G♭ v3F♯, ^5G♭
38 407.1 19/15, 43/34 ^^F♯, vvG♭ vvF♯, ^6G♭
39 417.9 14/11 ^3F♯, vG♭ vF♯, v7G
40 428.6 32/25, 41/32 v3F𝄪, G♭ F♯, v6G
41 439.3 40/31 vvF𝄪, ^G♭ ^F♯, v5G
42 450 vF𝄪, ^^G♭ ^^F♯, v4G
43 460.7 30/23 F𝄪, ^3G♭ ^3F♯, v3G
44 471.4 21/16 ^F𝄪, v3G ^4F♯, vvG
45 482.1 33/25, 37/28, 41/31 ^^F𝄪, vvG ^5F♯, vG
46 492.9 ^3F𝄪, vG G
47 503.6 G ^G, v5A♭
48 514.3 35/26, 39/29 ^G, v3A♭♭ ^^G, v4A♭
49 525 23/17, 42/31 ^^G, vvA♭♭ ^3G, v3A♭
50 535.7 ^3G, vA♭♭ ^4G, vvA♭
51 546.4 v3G♯, A♭♭ ^5G, vA♭
52 557.1 29/21, 40/29 vvG♯, ^A♭♭ ^6G, A♭
53 567.9 43/31 vG♯, ^^A♭♭ ^7G, ^A♭
54 578.6 G♯, ^3A♭♭ v6G♯, ^^A♭
55 589.3 ^G♯, v3A♭ v5G♯, ^3A♭
56 600 41/29 ^^G♯, vvA♭ v4G♯, ^4A♭
57 610.7 37/26, 47/33 ^3G♯, vA♭ v3G♯, ^5A♭
58 621.4 43/30 v3G𝄪, A♭ vvG♯, ^6A♭
59 632.1 vvG𝄪, ^A♭ vG♯, v7A
60 642.9 29/20, 42/29 vG𝄪, ^^A♭ G♯, v6A
61 653.6 G𝄪, ^3A♭ ^G♯, v5A
62 664.3 47/32 ^G𝄪, v3A ^^G♯, v4A
63 675 31/21, 34/23 ^^G𝄪, vvA ^3G♯, v3A
64 685.7 ^3G𝄪, vA ^4G♯, vvA
65 696.4 A ^5G♯, vA
66 707.1 ^A, v3B♭♭ A
67 717.9 ^^A, vvB♭♭ ^A, v5B♭
68 728.6 32/21 ^3A, vB♭♭ ^^A, v4B♭
69 739.3 23/15 v3A♯, B♭♭ ^3A, v3B♭
70 750 vvA♯, ^B♭♭ ^4A, vvB♭
71 760.7 31/20 vA♯, ^^B♭♭ ^5A, vB♭
72 771.4 25/16, 39/25 A♯, ^3B♭♭ ^6A, B♭
73 782.1 11/7 ^A♯, v3B♭ ^7A, ^B♭
74 792.9 30/19 ^^A♯, vvB♭ v6A♯, ^^B♭
75 803.6 35/22 ^3A♯, vB♭ v5A♯, ^3B♭
76 814.3 8/5 v3A𝄪, B♭ v4A♯, ^4B♭
77 825 vvA𝄪, ^B♭ v3A♯, ^5B♭
78 835.7 34/21, 47/29 vA𝄪, ^^B♭ vvA♯, ^6B♭
79 846.4 31/19 A𝄪, ^3B♭ vA♯, v7B
80 857.1 41/25 ^A𝄪, v3B A♯, v6B
81 867.9 33/20, 38/23 ^^A𝄪, vvB ^A♯, v5B
82 878.6 ^3A𝄪, vB ^^A♯, v4B
83 889.3 B ^3A♯, v3B
84 900 37/22, 42/25 ^B, v3C♭ ^4A♯, vvB
85 910.7 22/13 ^^B, vvC♭ ^5A♯, vB
86 921.4 ^3B, vC♭ B
87 932.1 v3B♯, C♭ ^B, v5C
88 942.9 vvB♯, ^C♭ ^^B, v4C
89 953.6 vB♯, ^^C♭ ^3B, v3C
90 964.3 B♯, ^3C♭ ^4B, vvC
91 975 ^B♯, v3C ^5B, vC
92 985.7 ^^B♯, vvC C
93 996.4 ^3B♯, vC ^C, v5D♭
94 1007.1 34/19 C ^^C, v4D♭
95 1017.9 ^C, v3D♭♭ ^3C, v3D♭
96 1028.6 29/16, 38/21 ^^C, vvD♭♭ ^4C, vvD♭
97 1039.3 31/17 ^3C, vD♭♭ ^5C, vD♭
98 1050 v3C♯, D♭♭ ^6C, D♭
99 1060.7 vvC♯, ^D♭♭ ^7C, ^D♭
100 1071.4 13/7 vC♯, ^^D♭♭ v6C♯, ^^D♭
101 1082.1 43/23 C♯, ^3D♭♭ v5C♯, ^3D♭
102 1092.9 47/25 ^C♯, v3D♭ v4C♯, ^4D♭
103 1103.6 ^^C♯, vvD♭ v3C♯, ^5D♭
104 1114.3 40/21 ^3C♯, vD♭ vvC♯, ^6D♭
105 1125 23/12 v3C𝄪, D♭ vC♯, v7D
106 1135.7 vvC𝄪, ^D♭ C♯, v6D
107 1146.4 31/16, 33/17 vC𝄪, ^^D♭ ^C♯, v5D
108 1157.1 39/20, 41/21 C𝄪, ^3D♭ ^^C♯, v4D
109 1167.9 ^C𝄪, v3D ^3C♯, v3D
110 1178.6 ^^C𝄪, vvD ^4C♯, vvD
111 1189.3 ^3C𝄪, vD ^5C♯, vD
112 1200 2/1 D D

Music

Cam Taylor

See also

Skip fretting system 112 9 11

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