# 107edo

 ← 106edo 107edo 108edo →
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, following 103edo and before 109edo. 214edo, which doubles it, provides correction for the approximation to harmonics 3, 5, and 7.

## Intervals

Steps Cents Approximate Ratios Ups and Downs Notation
(Dual Flat Fifth 62\107)
Ups and Downs Notation
(Dual Sharp Fifth 63\107)
0 0 1/1 D D
1 11.215 ^D, v4E♭♭ ^D, v5E♭
2 22.43 65/64 ^^D, v3E♭♭ ^^D, v4E♭
3 33.645 50/49, 56/55 ^3D, vvE♭♭ ^3D, v3E♭
4 44.86 77/75 ^4D, vE♭♭ ^4D, vvE♭
5 56.075 33/32 ^5D, E♭♭ ^5D, vE♭
6 67.29 80/77 D♯, v5E♭ ^6D, E♭
7 78.505 22/21 ^D♯, v4E♭ ^7D, v12E
8 89.72 ^^D♯, v3E♭ ^8D, v11E
9 100.935 ^3D♯, vvE♭ ^9D, v10E
10 112.15 16/15 ^4D♯, vE♭ ^10D, v9E
11 123.364 15/14 ^5D♯, E♭ ^11D, v8E
12 134.579 13/12 D𝄪, v5E ^12D, v7E
13 145.794 ^D𝄪, v4E D♯, v6E
14 157.009 ^^D𝄪, v3E ^D♯, v5E
15 168.224 11/10 ^3D𝄪, vvE ^^D♯, v4E
16 179.439 ^4D𝄪, vE ^3D♯, v3E
17 190.654 E ^4D♯, vvE
18 201.869 55/49 ^E, v4F♭ ^5D♯, vE
19 213.084 ^^E, v3F♭ E
20 224.299 ^3E, vvF♭ ^E, v5F
21 235.514 8/7 ^4E, vF♭ ^^E, v4F
22 246.729 15/13, 52/45 ^5E, F♭ ^3E, v3F
23 257.944 65/56 E♯, v5F ^4E, vvF
24 269.159 ^E♯, v4F ^5E, vF
25 280.374 ^^E♯, v3F F
26 291.589 13/11, 77/65 ^3E♯, vvF ^F, v5G♭
27 302.804 ^4E♯, vF ^^F, v4G♭
28 314.019 F ^3F, v3G♭
29 325.234 ^F, v4G♭♭ ^4F, vvG♭
30 336.449 63/52 ^^F, v3G♭♭ ^5F, vG♭
31 347.664 49/40 ^3F, vvG♭♭ ^6F, G♭
32 358.879 16/13 ^4F, vG♭♭ ^7F, v12G
33 370.093 26/21 ^5F, G♭♭ ^8F, v11G
34 381.308 F♯, v5G♭ ^9F, v10G
35 392.523 ^F♯, v4G♭ ^10F, v9G
36 403.738 ^^F♯, v3G♭ ^11F, v8G
37 414.953 14/11, 33/26 ^3F♯, vvG♭ ^12F, v7G
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 21/16 ^^F𝄪, v3G ^4F♯, vvG
43 482.243 ^3F𝄪, vvG ^5F♯, vG
44 493.458 65/49 ^4F𝄪, vG G
45 504.673 75/56 G ^G, v5A♭
46 515.888 ^G, v4A♭♭ ^^G, v4A♭
47 527.103 ^^G, v3A♭♭ ^3G, v3A♭
48 538.318 15/11 ^3G, vvA♭♭ ^4G, vvA♭
49 549.533 11/8 ^4G, vA♭♭ ^5G, vA♭
50 560.748 ^5G, A♭♭ ^6G, A♭
51 571.963 G♯, v5A♭ ^7G, v12A
52 583.178 7/5 ^G♯, v4A♭ ^8G, v11A
53 594.393 45/32 ^^G♯, v3A♭ ^9G, v10A
54 605.607 64/45 ^3G♯, vvA♭ ^10G, v9A
55 616.822 10/7 ^4G♯, vA♭ ^11G, v8A
56 628.037 ^5G♯, A♭ ^12G, v7A
57 639.252 G𝄪, v5A G♯, v6A
58 650.467 16/11 ^G𝄪, v4A ^G♯, v5A
59 661.682 22/15 ^^G𝄪, v3A ^^G♯, v4A
60 672.897 65/44 ^3G𝄪, vvA ^3G♯, v3A
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 32/21 ^3A, vvB♭♭ ^^A, v4B♭
66 740.187 75/49 ^4A, vB♭♭ ^3A, v3B♭
67 751.402 77/50 ^5A, B♭♭ ^4A, vvB♭
68 762.617 A♯, v5B♭ ^5A, vB♭
69 773.832 ^A♯, v4B♭ ^6A, B♭
70 785.047 11/7, 52/33 ^^A♯, v3B♭ ^7A, v12B
71 796.262 ^3A♯, vvB♭ ^8A, v11B
72 807.477 ^4A♯, vB♭ ^9A, v10B
73 818.692 ^5A♯, B♭ ^10A, v9B
74 829.907 21/13 A𝄪, v5B ^11A, v8B
75 841.121 13/8 ^A𝄪, v4B ^12A, v7B
76 852.336 80/49 ^^A𝄪, v3B A♯, v6B
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 22/13 ^^B, v3C♭ ^5A♯, vB
82 919.626 75/44 ^3B, vvC♭ B
83 930.841 ^4B, vC♭ ^B, v5C
84 942.056 ^5B, C♭ ^^B, v4C
85 953.271 26/15, 45/26 B♯, v5C ^3B, v3C
86 964.486 7/4 ^B♯, v4C ^4B, vvC
87 975.701 ^^B♯, v3C ^5B, vC
88 986.916 ^3B♯, vvC C
89 998.131 ^4B♯, vC ^C, v5D♭
90 1009.346 C ^^C, v4D♭
91 1020.561 ^C, v4D♭♭ ^3C, v3D♭
92 1031.776 20/11 ^^C, v3D♭♭ ^4C, vvD♭
93 1042.991 ^3C, vvD♭♭ ^5C, vD♭
94 1054.206 ^4C, vD♭♭ ^6C, D♭
95 1065.421 24/13 ^5C, D♭♭ ^7C, v12D
96 1076.636 28/15 C♯, v5D♭ ^8C, v11D
97 1087.85 15/8 ^C♯, v4D♭ ^9C, v10D
98 1099.065 ^^C♯, v3D♭ ^10C, v9D
99 1110.28 ^3C♯, vvD♭ ^11C, v8D
100 1121.495 21/11 ^4C♯, vD♭ ^12C, v7D
101 1132.71 77/40 ^5C♯, D♭ C♯, v6D
102 1143.925 64/33 C𝄪, v5D ^C♯, v5D
103 1155.14 ^C𝄪, v4D ^^C♯, v4D
104 1166.355 49/25, 55/28 ^^C𝄪, v3D ^3C♯, v3D
105 1177.57 ^3C𝄪, vvD ^4C♯, vvD
106 1188.785 ^4C𝄪, vD ^5C♯, vD
107 1200 2/1 D D

## 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