# 114edo

 ← 113edo 114edo 115edo →
Prime factorization 2 × 3 × 19
Step size 10.5263¢
Fifth 67\114 (705.263¢)
Semitones (A1:m2) 13:7 (136.8¢ : 73.68¢)
Consistency limit 7
Distinct consistency limit 7

114 equal divisions of the octave (abbreviated 114edo or 114ed2), also called 114-tone equal temperament (114tet) or 114 equal temperament (114et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 114 equal parts of about 10.5 ¢ each. Each step represents a frequency ratio of 21/114, or the 114th root of 2.

## Theory

In the 5-limit the equal temperament tempers out 2048/2025 (diaschisma), in the 7-limit 245/243, in the 11-limit 121/120, 176/175 and notably the quartisma, in the 13-limit 196/195 and 325/324, in the 17-limit 136/135 and 154/153, in the 19-limit 286/285 and 343/342. These commas make for 114edo being an excellent tuning for the shrutar temperament; it is in fact the optimal patent val for shrutar in the 11-, 13-, 17-, and 19-limit, as well as the rank-3 bisector temperament.

### Odd harmonics

Approximation of odd harmonics in 114edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +3.31 +3.16 -0.40 -3.91 -3.95 +1.58 -4.06 +0.31 -2.78 +2.90 +3.30
Relative (%) +31.4 +30.0 -3.8 -37.1 -37.5 +15.0 -38.6 +2.9 -26.4 +27.6 +31.4
Steps
(reduced)
181
(67)
265
(37)
320
(92)
361
(19)
394
(52)
422
(80)
445
(103)
466
(10)
484
(28)
501
(45)
516
(60)

### Subsets and supersets

Since 114 factors into 2 × 3 × 19, 114 edo has subset edos 2, 3, 6, 19, 38, and 57.

## Intervals

Steps Cents Approximate Ratios Ups and Downs Notation
0 0 1/1 D
1 10.526 ^D, v6E♭
2 21.053 ^^D, v5E♭
3 31.579 49/48, 56/55 ^3D, v4E♭
4 42.105 40/39 ^4D, v3E♭
5 52.632 33/32, 36/35, 65/63 ^5D, vvE♭
6 63.158 ^6D, vE♭
7 73.684 25/24 ^7D, E♭
8 84.211 21/20 ^8D, v12E
9 94.737 55/52 ^9D, v11E
10 105.263 35/33, 52/49 ^10D, v10E
11 115.789 ^11D, v9E
12 126.316 14/13 ^12D, v8E
13 136.842 13/12, 27/25 D♯, v7E
14 147.368 ^D♯, v6E
15 157.895 35/32 ^^D♯, v5E
16 168.421 ^3D♯, v4E
17 178.947 10/9, 72/65 ^4D♯, v3E
18 189.474 39/35, 49/44 ^5D♯, vvE
19 200 55/49 ^6D♯, vE
20 210.526 E
21 221.053 ^E, v6F
22 231.579 8/7, 55/48 ^^E, v5F
23 242.105 ^3E, v4F
24 252.632 ^4E, v3F
25 263.158 7/6, 64/55 ^5E, vvF
26 273.684 ^6E, vF
27 284.211 33/28 F
28 294.737 ^F, v6G♭
29 305.263 25/21 ^^F, v5G♭
30 315.789 6/5 ^3F, v4G♭
31 326.316 ^4F, v3G♭
32 336.842 40/33 ^5F, vvG♭
33 347.368 49/40 ^6F, vG♭
34 357.895 16/13 ^7F, G♭
35 368.421 26/21 ^8F, v12G
36 378.947 ^9F, v11G
37 389.474 5/4 ^10F, v10G
38 400 63/50 ^11F, v9G
39 410.526 33/26, 80/63 ^12F, v8G
40 421.053 14/11 F♯, v7G
41 431.579 50/39 ^F♯, v6G
42 442.105 ^^F♯, v5G
43 452.632 13/10 ^3F♯, v4G
44 463.158 55/42, 64/49 ^4F♯, v3G
45 473.684 21/16 ^5F♯, vvG
46 484.211 ^6F♯, vG
47 494.737 4/3 G
48 505.263 ^G, v6A♭
49 515.789 35/26, 66/49 ^^G, v5A♭
50 526.316 65/48 ^3G, v4A♭
51 536.842 ^4G, v3A♭
52 547.368 11/8, 48/35 ^5G, vvA♭
53 557.895 ^6G, vA♭
54 568.421 25/18 ^7G, A♭
55 578.947 7/5 ^8G, v12A
56 589.474 ^9G, v11A
57 600 ^10G, v10A
58 610.526 ^11G, v9A
59 621.053 10/7 ^12G, v8A
60 631.579 36/25 G♯, v7A
61 642.105 ^G♯, v6A
62 652.632 16/11, 35/24 ^^G♯, v5A
63 663.158 ^3G♯, v4A
64 673.684 ^4G♯, v3A
65 684.211 49/33, 52/35 ^5G♯, vvA
66 694.737 ^6G♯, vA
67 705.263 3/2 A
68 715.789 ^A, v6B♭
69 726.316 32/21 ^^A, v5B♭
70 736.842 49/32 ^3A, v4B♭
71 747.368 20/13 ^4A, v3B♭
72 757.895 65/42 ^5A, vvB♭
73 768.421 39/25 ^6A, vB♭
74 778.947 11/7 ^7A, B♭
75 789.474 52/33, 63/40 ^8A, v12B
76 800 ^9A, v11B
77 810.526 8/5 ^10A, v10B
78 821.053 ^11A, v9B
79 831.579 21/13 ^12A, v8B
80 842.105 13/8 A♯, v7B
81 852.632 80/49 ^A♯, v6B
82 863.158 33/20 ^^A♯, v5B
83 873.684 ^3A♯, v4B
84 884.211 5/3 ^4A♯, v3B
85 894.737 42/25 ^5A♯, vvB
86 905.263 ^6A♯, vB
87 915.789 56/33 B
88 926.316 ^B, v6C
89 936.842 12/7, 55/32 ^^B, v5C
90 947.368 ^3B, v4C
91 957.895 ^4B, v3C
92 968.421 7/4 ^5B, vvC
93 978.947 ^6B, vC
94 989.474 C
95 1000 ^C, v6D♭
96 1010.526 70/39 ^^C, v5D♭
97 1021.053 9/5, 65/36 ^3C, v4D♭
98 1031.579 ^4C, v3D♭
99 1042.105 64/35 ^5C, vvD♭
100 1052.632 ^6C, vD♭
101 1063.158 24/13, 50/27 ^7C, D♭
102 1073.684 13/7 ^8C, v12D
103 1084.211 ^9C, v11D
104 1094.737 49/26, 66/35 ^10C, v10D
105 1105.263 ^11C, v9D
106 1115.789 40/21 ^12C, v8D
107 1126.316 48/25 C♯, v7D
108 1136.842 ^C♯, v6D
109 1147.368 35/18, 64/33 ^^C♯, v5D
110 1157.895 39/20 ^3C♯, v4D
111 1168.421 55/28 ^4C♯, v3D
112 1178.947 ^5C♯, vvD
113 1189.474 ^6C♯, vD
114 1200 2/1 D