107edo

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← 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)

Octave stretch

107edo’s approximations of 3/1, 5/1, 7/1, 13/1, 17/1 and 19/1 are all improved by APS70tina, a stretched-octave version of 107edo. The trade-off is a slightly worse 2/1 and 11/1.

There are also several nearby Zeta peak index (ZPI) tunings which can be used for this same purpose: 622zpi, 623zpi, 624zpi, 625zpi, 626zpu, 627zpi, 628zpi and 629zpi.

The details of each of those ZPI tunings are visible in User:Contribution’s gallery of Zeta Peak Indexes (1 - 10 000). Warning: due to its length, that page may slow down your device while it is open. The effect will go away after you close the page.

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 ^^D, v3E♭♭ ^^D, v4E♭
3 33.645 ^3D, vvE♭♭ ^3D, v3E♭
4 44.86 39/38, 41/40 ^4D, vE♭♭ ^4D, vvE♭
5 56.075 31/30, 32/31 ^5D, E♭♭ ^5D, vE♭
6 67.29 D♯, v5E♭ ^6D, E♭
7 78.505 22/21, 23/22, 45/43 ^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 44/41 ^5D♯, E♭ ^11D, v8E
12 134.579 40/37 D𝄪, v5E ^12D, v7E
13 145.794 37/34 ^D𝄪, v4E D♯, v6E
14 157.009 23/21 ^^D𝄪, v3E ^D♯, v5E
15 168.224 32/29, 43/39 ^3D𝄪, vvE ^^D♯, v4E
16 179.439 41/37 ^4D𝄪, vE ^3D♯, v3E
17 190.654 29/26 E ^4D♯, vvE
18 201.869 ^E, v4F♭ ^5D♯, vE
19 213.084 26/23, 43/38 ^^E, v3F♭ E
20 224.299 33/29 ^3E, vvF♭ ^E, v5F
21 235.514 ^4E, vF♭ ^^E, v4F
22 246.729 15/13 ^5E, F♭ ^3E, v3F
23 257.944 E♯, v5F ^4E, vvF
24 269.159 ^E♯, v4F ^5E, vF
25 280.374 20/17 ^^E♯, v3F F
26 291.589 45/38 ^3E♯, vvF ^F, v5G♭
27 302.804 31/26 ^4E♯, vF ^^F, v4G♭
28 314.019 F ^3F, v3G♭
29 325.234 41/34 ^F, v4G♭♭ ^4F, vvG♭
30 336.449 17/14 ^^F, v3G♭♭ ^5F, vG♭
31 347.664 ^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 24/19 ^^F♯, v3G♭ ^11F, v8G
37 414.953 33/26 ^3F♯, vvG♭ ^12F, v7G
38 426.168 ^4F♯, vG♭ F♯, v6G
39 437.383 ^5F♯, G♭ ^F♯, v5G
40 448.598 22/17 F𝄪, v5G ^^F♯, v4G
41 459.813 30/23, 43/33 ^F𝄪, v4G ^3F♯, v3G
42 471.028 21/16 ^^F𝄪, v3G ^4F♯, vvG
43 482.243 37/28, 41/31 ^3F𝄪, vvG ^5F♯, vG
44 493.458 ^4F𝄪, vG G
45 504.673 G ^G, v5A♭
46 515.888 31/23 ^G, v4A♭♭ ^^G, v4A♭
47 527.103 42/31 ^^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 29/21 ^5G, A♭♭ ^6G, A♭
51 571.963 32/23 G♯, v5A♭ ^7G, v12A
52 583.178 7/5 ^G♯, v4A♭ ^8G, v11A
53 594.393 31/22 ^^G♯, v3A♭ ^9G, v10A
54 605.607 44/31 ^3G♯, vvA♭ ^10G, v9A
55 616.822 10/7 ^4G♯, vA♭ ^11G, v8A
56 628.037 23/16 ^5G♯, A♭ ^12G, v7A
57 639.252 42/29 G𝄪, v5A G♯, v6A
58 650.467 16/11 ^G𝄪, v4A ^G♯, v5A
59 661.682 22/15, 41/28 ^^G𝄪, v3A ^^G♯, v4A
60 672.897 31/21 ^3G𝄪, vvA ^3G♯, v3A
61 684.112 43/29, 46/31 ^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 23/15 ^4A, vB♭♭ ^3A, v3B♭
67 751.402 17/11 ^5A, B♭♭ ^4A, vvB♭
68 762.617 45/29 A♯, v5B♭ ^5A, vB♭
69 773.832 ^A♯, v4B♭ ^6A, B♭
70 785.047 ^^A♯, v3B♭ ^7A, v12B
71 796.262 19/12 ^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 ^^A𝄪, v3B A♯, v6B
77 863.551 28/17 ^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
82 919.626 17/10 ^3B, vvC♭ B
83 930.841 ^4B, vC♭ ^B, v5C
84 942.056 ^5B, C♭ ^^B, v4C
85 953.271 26/15 B♯, v5C ^3B, v3C
86 964.486 ^B♯, v4C ^4B, vvC
87 975.701 ^^B♯, v3C ^5B, vC
88 986.916 23/13 ^3B♯, vvC C
89 998.131 ^4B♯, vC ^C, v5D♭
90 1009.346 43/24 C ^^C, v4D♭
91 1020.561 ^C, v4D♭♭ ^3C, v3D♭
92 1031.776 29/16 ^^C, v3D♭♭ ^4C, vvD♭
93 1042.991 42/23 ^3C, vvD♭♭ ^5C, vD♭
94 1054.206 ^4C, vD♭♭ ^6C, D♭
95 1065.421 37/20 ^5C, D♭♭ ^7C, v12D
96 1076.636 41/22 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, 44/23 ^4C♯, vD♭ ^12C, v7D
101 1132.71 ^5C♯, D♭ C♯, v6D
102 1143.925 31/16 C𝄪, v5D ^C♯, v5D
103 1155.14 ^C𝄪, v4D ^^C♯, v4D
104 1166.355 ^^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
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