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 11.2 ^D, ^^E♭♭♭ ^D, v5E♭
2 22.4 ^^D, v3E♭♭ ^^D, v4E♭
3 33.6 ^3D, vvE♭♭ ^3D, v3E♭
4 44.9 39/38, 41/40 vvD♯, vE♭♭ ^4D, vvE♭
5 56.1 31/30, 32/31 vD♯, E♭♭ ^5D, vE♭
6 67.3 D♯, ^E♭♭ ^6D, E♭
7 78.5 22/21, 23/22, 45/43 ^D♯, ^^E♭♭ v6D♯, ^E♭
8 89.7 ^^D♯, v3E♭ v5D♯, ^^E♭
9 100.9 ^3D♯, vvE♭ v4D♯, ^3E♭
10 112.1 16/15 vvD𝄪, vE♭ v3D♯, ^4E♭
11 123.4 44/41 vD𝄪, E♭ vvD♯, ^5E♭
12 134.6 40/37 D𝄪, ^E♭ vD♯, ^6E♭
13 145.8 37/34 ^D𝄪, ^^E♭ D♯, v6E
14 157 23/21 ^^D𝄪, v3E ^D♯, v5E
15 168.2 32/29, 43/39 ^3D𝄪, vvE ^^D♯, v4E
16 179.4 41/37 vvD♯𝄪, vE ^3D♯, v3E
17 190.7 29/26 E ^4D♯, vvE
18 201.9 ^E, ^^F♭♭ ^5D♯, vE
19 213.1 26/23, 43/38 ^^E, v3F♭ E
20 224.3 33/29 ^3E, vvF♭ ^E, v5F
21 235.5 vvE♯, vF♭ ^^E, v4F
22 246.7 15/13 vE♯, F♭ ^3E, v3F
23 257.9 E♯, ^F♭ ^4E, vvF
24 269.2 ^E♯, ^^F♭ ^5E, vF
25 280.4 20/17 ^^E♯, v3F F
26 291.6 45/38 ^3E♯, vvF ^F, v5G♭
27 302.8 31/26 vvE𝄪, vF ^^F, v4G♭
28 314 F ^3F, v3G♭
29 325.2 41/34 ^F, ^^G♭♭♭ ^4F, vvG♭
30 336.4 17/14 ^^F, v3G♭♭ ^5F, vG♭
31 347.7 ^3F, vvG♭♭ ^6F, G♭
32 358.9 16/13 vvF♯, vG♭♭ v6F♯, ^G♭
33 370.1 26/21 vF♯, G♭♭ v5F♯, ^^G♭
34 381.3 F♯, ^G♭♭ v4F♯, ^3G♭
35 392.5 ^F♯, ^^G♭♭ v3F♯, ^4G♭
36 403.7 24/19 ^^F♯, v3G♭ vvF♯, ^5G♭
37 415 33/26 ^3F♯, vvG♭ vF♯, ^6G♭
38 426.2 vvF𝄪, vG♭ F♯, v6G
39 437.4 vF𝄪, G♭ ^F♯, v5G
40 448.6 22/17 F𝄪, ^G♭ ^^F♯, v4G
41 459.8 30/23, 43/33 ^F𝄪, ^^G♭ ^3F♯, v3G
42 471 21/16 ^^F𝄪, v3G ^4F♯, vvG
43 482.2 37/28, 41/31 ^3F𝄪, vvG ^5F♯, vG
44 493.5 vvF♯𝄪, vG G
45 504.7 G ^G, v5A♭
46 515.9 31/23 ^G, ^^A♭♭♭ ^^G, v4A♭
47 527.1 42/31 ^^G, v3A♭♭ ^3G, v3A♭
48 538.3 15/11 ^3G, vvA♭♭ ^4G, vvA♭
49 549.5 11/8 vvG♯, vA♭♭ ^5G, vA♭
50 560.7 29/21 vG♯, A♭♭ ^6G, A♭
51 572 32/23 G♯, ^A♭♭ v6G♯, ^A♭
52 583.2 7/5 ^G♯, ^^A♭♭ v5G♯, ^^A♭
53 594.4 31/22 ^^G♯, v3A♭ v4G♯, ^3A♭
54 605.6 44/31 ^3G♯, vvA♭ v3G♯, ^4A♭
55 616.8 10/7 vvG𝄪, vA♭ vvG♯, ^5A♭
56 628 23/16 vG𝄪, A♭ vG♯, ^6A♭
57 639.3 42/29 G𝄪, ^A♭ G♯, v6A
58 650.5 16/11 ^G𝄪, ^^A♭ ^G♯, v5A
59 661.7 22/15, 41/28 ^^G𝄪, v3A ^^G♯, v4A
60 672.9 31/21 ^3G𝄪, vvA ^3G♯, v3A
61 684.1 43/29, 46/31 vvG♯𝄪, vA ^4G♯, vvA
62 695.3 A ^5G♯, vA
63 706.5 ^A, ^^B♭♭♭ A
64 717.8 ^^A, v3B♭♭ ^A, v5B♭
65 729 32/21 ^3A, vvB♭♭ ^^A, v4B♭
66 740.2 23/15 vvA♯, vB♭♭ ^3A, v3B♭
67 751.4 17/11 vA♯, B♭♭ ^4A, vvB♭
68 762.6 45/29 A♯, ^B♭♭ ^5A, vB♭
69 773.8 ^A♯, ^^B♭♭ ^6A, B♭
70 785 ^^A♯, v3B♭ v6A♯, ^B♭
71 796.3 19/12 ^3A♯, vvB♭ v5A♯, ^^B♭
72 807.5 vvA𝄪, vB♭ v4A♯, ^3B♭
73 818.7 vA𝄪, B♭ v3A♯, ^4B♭
74 829.9 21/13 A𝄪, ^B♭ vvA♯, ^5B♭
75 841.1 13/8 ^A𝄪, ^^B♭ vA♯, ^6B♭
76 852.3 ^^A𝄪, v3B A♯, v6B
77 863.6 28/17 ^3A𝄪, vvB ^A♯, v5B
78 874.8 vvA♯𝄪, vB ^^A♯, v4B
79 886 B ^3A♯, v3B
80 897.2 ^B, ^^C♭♭ ^4A♯, vvB
81 908.4 ^^B, v3C♭ ^5A♯, vB
82 919.6 17/10 ^3B, vvC♭ B
83 930.8 vvB♯, vC♭ ^B, v5C
84 942.1 vB♯, C♭ ^^B, v4C
85 953.3 26/15 B♯, ^C♭ ^3B, v3C
86 964.5 ^B♯, ^^C♭ ^4B, vvC
87 975.7 ^^B♯, v3C ^5B, vC
88 986.9 23/13 ^3B♯, vvC C
89 998.1 vvB𝄪, vC ^C, v5D♭
90 1009.3 43/24 C ^^C, v4D♭
91 1020.6 ^C, ^^D♭♭♭ ^3C, v3D♭
92 1031.8 29/16 ^^C, v3D♭♭ ^4C, vvD♭
93 1043 42/23 ^3C, vvD♭♭ ^5C, vD♭
94 1054.2 vvC♯, vD♭♭ ^6C, D♭
95 1065.4 37/20 vC♯, D♭♭ v6C♯, ^D♭
96 1076.6 41/22 C♯, ^D♭♭ v5C♯, ^^D♭
97 1087.9 15/8 ^C♯, ^^D♭♭ v4C♯, ^3D♭
98 1099.1 ^^C♯, v3D♭ v3C♯, ^4D♭
99 1110.3 ^3C♯, vvD♭ vvC♯, ^5D♭
100 1121.5 21/11, 44/23 vvC𝄪, vD♭ vC♯, ^6D♭
101 1132.7 vC𝄪, D♭ C♯, v6D
102 1143.9 31/16 C𝄪, ^D♭ ^C♯, v5D
103 1155.1 ^C𝄪, ^^D♭ ^^C♯, v4D
104 1166.4 ^^C𝄪, v3D ^3C♯, v3D
105 1177.6 ^3C𝄪, vvD ^4C♯, vvD
106 1188.8 vvC♯𝄪, 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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