114edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 113edo114edo115edo →
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