107edo

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← 106edo107edo108edo →
Prime factorization 107 (prime)
Step size 11.215¢ 
Fifth 63\107 (706.542¢)
Semitones (A1:m2) 13:6 (145.8¢ : 67.29¢)
Dual sharp fifth 63\107 (706.542¢)
Dual flat fifth 62\107 (695.327¢)
Dual major 2nd 18\107 (201.869¢)
Consistency limit 3
Distinct consistency limit 3

107 equal divisions of the octave (abbreviated 107edo or 107ed2), also called 107-tone equal temperament (107tet) or 107 equal temperament (107et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 107 equal parts of about 11.2 ¢ each. Each step represents a frequency ratio of 21/107, or the 107th root of 2.

Theory

107edo is inconsistent to the 5-odd-limit and higher limits, and harmonics 3, 5, and 7 are about halfway between its steps. Either the 2.9.5.7 or 2.9.15.21 subgroup can be used. For the full 7-limit, it has four possible mappings: 107 170 248 300] (patent val), 107 169 248 300] (107b), 107 170 249 300] (107c), and 107 170 249 301] (107cd).

Using the patent val, it tempers out 3125/3072 (magic comma) and [28 -22 3 in the 5-limit; 1029/1024, 2240/2187, and 3125/3087 in the 7-limit; 100/99, 1232/1215, and 1331/1323 in the 11-limit.

Using the 107cd val, it tempers out 1728/1715, 4000/3969, and 28672/28125 in the 7-limit; 121/120, 896/891, 1375/1372, and 3168/3125 in the 11-limit.

Using the 107c val, it tempers out 1638400/1594323 (immunity comma) and 1990656/1953125 (valentine comma) in the 5-limit; 126/125, 1029/1024, and 307328/295245 in the 7-limit; 121/120, 176/175, 441/440, and 184877/177147 in the 11-limit.

Using the 107b val, it tempers out 81/80 (syntonic comma) and [-61 -1 27; in the 5-limit; 2401/2400, 2430/2401, and 234375/229376 in the 7-limit; 385/384, 1350/1331, 1375/1372, and 1944/1925 in the 11-limit.

Odd harmonics

Approximation of odd harmonics in 107edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +4.59 -5.01 -4.34 -2.04 -1.79 +0.59 -0.42 -4.02 +5.29 +0.25 -0.24
Relative (%) +40.9 -44.6 -38.7 -18.2 -15.9 +5.3 -3.7 -35.9 +47.2 +2.2 -2.1
Steps
(reduced)
170
(63)
248
(34)
300
(86)
339
(18)
370
(49)
396
(75)
418
(97)
437
(9)
455
(27)
470
(42)
484
(56)

Subsets and supersets

107edo is the 28th prime edo. 214edo, which doubles it, provides correction for the approximation to harmonics 3, 5, and 7.

Intervals

Steps Cents Ups and Downs Notation
(Dual Flat Fifth 62\107)
Ups and Downs Notation
(Dual Sharp Fifth 63\107)
Approximate Ratios
0 0 D D 1/1
1 11.215 ^D, v4E♭♭ ^D, v5E♭
2 22.4299 ^^D, v3E♭♭ ^^D, v4E♭ 65/64
3 33.6449 ^3D, vvE♭♭ ^3D, v3E♭ 50/49, 56/55
4 44.8598 ^4D, vE♭♭ ^4D, vvE♭ 77/75
5 56.0748 ^5D, E♭♭ ^5D, vE♭ 33/32
6 67.2897 D♯, v5E♭ ^6D, E♭ 80/77
7 78.5047 ^D♯, v4E♭ ^7D, v12E 22/21
8 89.7196 ^^D♯, v3E♭ ^8D, v11E
9 100.935 ^3D♯, vvE♭ ^9D, v10E
10 112.15 ^4D♯, vE♭ ^10D, v9E 16/15
11 123.364 ^5D♯, E♭ ^11D, v8E 15/14
12 134.579 D𝄪, v5E ^12D, v7E 13/12
13 145.794 ^D𝄪, v4E D♯, v6E
14 157.009 ^^D𝄪, v3E ^D♯, v5E
15 168.224 ^3D𝄪, vvE ^^D♯, v4E 11/10
16 179.439 ^4D𝄪, vE ^3D♯, v3E
17 190.654 E ^4D♯, vvE
18 201.869 ^E, v4F♭ ^5D♯, vE 55/49
19 213.084 ^^E, v3F♭ E
20 224.299 ^3E, vvF♭ ^E, v5F
21 235.514 ^4E, vF♭ ^^E, v4F 8/7
22 246.729 ^5E, F♭ ^3E, v3F 15/13, 52/45
23 257.944 E♯, v5F ^4E, vvF 65/56
24 269.159 ^E♯, v4F ^5E, vF
25 280.374 ^^E♯, v3F F
26 291.589 ^3E♯, vvF ^F, v5G♭ 13/11, 77/65
27 302.804 ^4E♯, vF ^^F, v4G♭
28 314.019 F ^3F, v3G♭
29 325.234 ^F, v4G♭♭ ^4F, vvG♭
30 336.449 ^^F, v3G♭♭ ^5F, vG♭ 63/52
31 347.664 ^3F, vvG♭♭ ^6F, G♭ 49/40
32 358.879 ^4F, vG♭♭ ^7F, v12G 16/13
33 370.093 ^5F, G♭♭ ^8F, v11G 26/21
34 381.308 F♯, v5G♭ ^9F, v10G
35 392.523 ^F♯, v4G♭ ^10F, v9G
36 403.738 ^^F♯, v3G♭ ^11F, v8G
37 414.953 ^3F♯, vvG♭ ^12F, v7G 14/11, 33/26
38 426.168 ^4F♯, vG♭ F♯, v6G
39 437.383 ^5F♯, G♭ ^F♯, v5G
40 448.598 F𝄪, v5G ^^F♯, v4G
41 459.813 ^F𝄪, v4G ^3F♯, v3G
42 471.028 ^^F𝄪, v3G ^4F♯, vvG 21/16
43 482.243 ^3F𝄪, vvG ^5F♯, vG
44 493.458 ^4F𝄪, vG G 65/49
45 504.673 G ^G, v5A♭ 75/56
46 515.888 ^G, v4A♭♭ ^^G, v4A♭
47 527.103 ^^G, v3A♭♭ ^3G, v3A♭
48 538.318 ^3G, vvA♭♭ ^4G, vvA♭ 15/11
49 549.533 ^4G, vA♭♭ ^5G, vA♭ 11/8
50 560.748 ^5G, A♭♭ ^6G, A♭
51 571.963 G♯, v5A♭ ^7G, v12A
52 583.178 ^G♯, v4A♭ ^8G, v11A 7/5
53 594.393 ^^G♯, v3A♭ ^9G, v10A 45/32
54 605.607 ^3G♯, vvA♭ ^10G, v9A 64/45
55 616.822 ^4G♯, vA♭ ^11G, v8A 10/7
56 628.037 ^5G♯, A♭ ^12G, v7A
57 639.252 G𝄪, v5A G♯, v6A
58 650.467 ^G𝄪, v4A ^G♯, v5A 16/11
59 661.682 ^^G𝄪, v3A ^^G♯, v4A 22/15
60 672.897 ^3G𝄪, vvA ^3G♯, v3A 65/44
61 684.112 ^4G𝄪, vA ^4G♯, vvA
62 695.327 A ^5G♯, vA
63 706.542 ^A, v4B♭♭ A
64 717.757 ^^A, v3B♭♭ ^A, v5B♭
65 728.972 ^3A, vvB♭♭ ^^A, v4B♭ 32/21
66 740.187 ^4A, vB♭♭ ^3A, v3B♭ 75/49
67 751.402 ^5A, B♭♭ ^4A, vvB♭ 77/50
68 762.617 A♯, v5B♭ ^5A, vB♭
69 773.832 ^A♯, v4B♭ ^6A, B♭
70 785.047 ^^A♯, v3B♭ ^7A, v12B 11/7, 52/33
71 796.262 ^3A♯, vvB♭ ^8A, v11B
72 807.477 ^4A♯, vB♭ ^9A, v10B
73 818.692 ^5A♯, B♭ ^10A, v9B
74 829.907 A𝄪, v5B ^11A, v8B 21/13
75 841.121 ^A𝄪, v4B ^12A, v7B 13/8
76 852.336 ^^A𝄪, v3B A♯, v6B 80/49
77 863.551 ^3A𝄪, vvB ^A♯, v5B
78 874.766 ^4A𝄪, vB ^^A♯, v4B
79 885.981 B ^3A♯, v3B
80 897.196 ^B, v4C♭ ^4A♯, vvB
81 908.411 ^^B, v3C♭ ^5A♯, vB 22/13
82 919.626 ^3B, vvC♭ B 75/44
83 930.841 ^4B, vC♭ ^B, v5C
84 942.056 ^5B, C♭ ^^B, v4C
85 953.271 B♯, v5C ^3B, v3C 26/15, 45/26
86 964.486 ^B♯, v4C ^4B, vvC 7/4
87 975.701 ^^B♯, v3C ^5B, vC
88 986.916 ^3B♯, vvC C
89 998.131 ^4B♯, vC ^C, v5D♭
90 1009.35 C ^^C, v4D♭
91 1020.56 ^C, v4D♭♭ ^3C, v3D♭
92 1031.78 ^^C, v3D♭♭ ^4C, vvD♭ 20/11
93 1042.99 ^3C, vvD♭♭ ^5C, vD♭
94 1054.21 ^4C, vD♭♭ ^6C, D♭
95 1065.42 ^5C, D♭♭ ^7C, v12D 24/13
96 1076.64 C♯, v5D♭ ^8C, v11D 28/15
97 1087.85 ^C♯, v4D♭ ^9C, v10D 15/8
98 1099.07 ^^C♯, v3D♭ ^10C, v9D
99 1110.28 ^3C♯, vvD♭ ^11C, v8D
100 1121.5 ^4C♯, vD♭ ^12C, v7D 21/11
101 1132.71 ^5C♯, D♭ C♯, v6D 77/40
102 1143.93 C𝄪, v5D ^C♯, v5D 64/33
103 1155.14 ^C𝄪, v4D ^^C♯, v4D
104 1166.36 ^^C𝄪, v3D ^3C♯, v3D 49/25, 55/28
105 1177.57 ^3C𝄪, vvD ^4C♯, vvD
106 1188.79 ^4C𝄪, vD ^5C♯, vD
107 1200 D D 2/1

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.9 [339 -107 [107 339]] +0.322 0.322 2.87
2.9.5 9765625/9565938, [-34 10 1 [107 339 248]] +0.933 0.904 8.06
2.9.5.7 225/224, 84035/82944, [14 -6 7 -4 [107 339 248 300]] +1.087 0.827 7.37
2.9.5.7.11 225/224, 441/440, 26411/26244, 161280/161051 [107 339 248 300 370]] +0.973 0.774 6.90
2.9.5.7.11.13 225/224, 325/324, 441/440, 847/845, 24500/24167 [107 339 248 300 370 396]] +0.783 0.823 7.33
2.9.5.7.11.13.17 170/169, 225/224, 325/324, 441/440, 847/845, 2000/1989 [107 339 248 300 370 396 437]] +0.812 0.765 6.82