110edo

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← 109edo110edo111edo →
Prime factorization 2 × 5 × 11
Step size 10.9091¢
Fifth 64\110 (698.182¢) (→32\55)
Semitones (A1:m2) 8:10 (87.27¢ : 109.1¢)
Dual sharp fifth 65\110 (709.091¢) (→13\22)
Dual flat fifth 64\110 (698.182¢) (→32\55)
Dual major 2nd 19\110 (207.273¢)
Consistency limit 5
Distinct consistency limit 5

110 equal divisions of the octave (abbreviated 110edo or 110ed2), also called 110-tone equal temperament (110tet) or 110 equal temperament (110et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 110 equal parts of about 10.9 ¢ each. Each step represents a frequency ratio of 21/110, or the 110th root of 2.

Theory

It tempers out 15625/15552 and 3486784401/3355443200 in the 5-limit. Using the patent val, it tempers out 1728/1715, 3125/3087, and 3645/3584 in the 7-limit.


Approximation of odd harmonics in 110edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error absolute (¢) -3.77 -4.50 +2.08 +3.36 +5.05 -0.53 +2.64 +4.14 -2.97 -1.69 +4.45
relative (%) -35 -41 +19 +31 +46 -5 +24 +38 -27 -15 +41
Steps
(reduced) 		174
(64) 		255
(35) 		309
(89) 		349
(19) 		381
(51) 		407
(77) 		430
(100) 		450
(10) 		467
(27) 		483
(43) 		498
(58)

Intervals

"
Steps Cents Ups and downs notation
(dual flat fifth 64\110) 		Ups and downs notation
(dual sharp fifth 65\110) 		Approximate ratios
0 0 D D 1/1
1 10.9091 ^D, vE♭♭ ^D, v4E♭
2 21.8182 ^^D, E♭♭ ^^D, v3E♭
3 32.7273 ^3D, v7E♭ ^3D, vvE♭ 56/55
4 43.6364 ^4D, v6E♭ ^4D, vE♭ 40/39
5 54.5455 ^5D, v5E♭ ^5D, E♭ 33/32, 65/63
6 65.4545 ^6D, v4E♭ ^6D, v14E
7 76.3636 ^7D, v3E♭ ^7D, v13E
8 87.2727 D♯, vvE♭ ^8D, v12E 21/20
9 98.1818 ^D♯, vE♭ ^9D, v11E 35/33, 55/52
10 109.091 ^^D♯, E♭ ^10D, v10E
11 120 ^3D♯, v7E ^11D, v9E
12 130.909 ^4D♯, v6E ^12D, v8E 14/13, 27/25
13 141.818 ^5D♯, v5E ^13D, v7E 13/12
14 152.727 ^6D♯, v4E ^14D, v6E 35/32
15 163.636 ^7D♯, v3E D♯, v5E
16 174.545 D𝄪, vvE ^D♯, v4E 72/65
17 185.455 ^D𝄪, vE ^^D♯, v3E 10/9, 39/35, 49/44
18 196.364 E ^3D♯, vvE 55/49
19 207.273 ^E, vF♭ ^4D♯, vE
20 218.182 ^^E, F♭ E
21 229.091 ^3E, v7F ^E, v4F 8/7
22 240 ^4E, v6F ^^E, v3F 55/48
23 250.909 ^5E, v5F ^3E, vvF
24 261.818 ^6E, v4F ^4E, vF 64/55
25 272.727 ^7E, v3F F
26 283.636 E♯, vvF ^F, v4G♭ 33/28
27 294.545 ^E♯, vF ^^F, v3G♭
28 305.455 F ^3F, vvG♭
29 316.364 ^F, vG♭♭ ^4F, vG♭ 6/5
30 327.273 ^^F, G♭♭ ^5F, G♭
31 338.182 ^3F, v7G♭ ^6F, v14G 39/32
32 349.091 ^4F, v6G♭ ^7F, v13G
33 360 ^5F, v5G♭ ^8F, v12G 16/13
34 370.909 ^6F, v4G♭ ^9F, v11G 26/21
35 381.818 ^7F, v3G♭ ^10F, v10G
36 392.727 F♯, vvG♭ ^11F, v9G
37 403.636 ^F♯, vG♭ ^12F, v8G 63/50
38 414.545 ^^F♯, G♭ ^13F, v7G 14/11, 33/26, 80/63
39 425.455 ^3F♯, v7G ^14F, v6G
40 436.364 ^4F♯, v6G F♯, v5G
41 447.273 ^5F♯, v5G ^F♯, v4G
42 458.182 ^6F♯, v4G ^^F♯, v3G 13/10, 64/49
43 469.091 ^7F♯, v3G ^3F♯, vvG 21/16, 55/42
44 480 F𝄪, vvG ^4F♯, vG
45 490.909 ^F𝄪, vG G
46 501.818 G ^G, v4A♭ 4/3
47 512.727 ^G, vA♭♭ ^^G, v3A♭ 35/26, 66/49
48 523.636 ^^G, A♭♭ ^3G, vvA♭ 65/48
49 534.545 ^3G, v7A♭ ^4G, vA♭
50 545.455 ^4G, v6A♭ ^5G, A♭ 48/35
51 556.364 ^5G, v5A♭ ^6G, v14A
52 567.273 ^6G, v4A♭ ^7G, v13A 25/18
53 578.182 ^7G, v3A♭ ^8G, v12A
54 589.091 G♯, vvA♭ ^9G, v11A
55 600 ^G♯, vA♭ ^10G, v10A
56 610.909 ^^G♯, A♭ ^11G, v9A
57 621.818 ^3G♯, v7A ^12G, v8A
58 632.727 ^4G♯, v6A ^13G, v7A 36/25
59 643.636 ^5G♯, v5A ^14G, v6A
60 654.545 ^6G♯, v4A G♯, v5A 35/24
61 665.455 ^7G♯, v3A ^G♯, v4A
62 676.364 G𝄪, vvA ^^G♯, v3A
63 687.273 ^G𝄪, vA ^3G♯, vvA 49/33, 52/35
64 698.182 A ^4G♯, vA 3/2
65 709.091 ^A, vB♭♭ A
66 720 ^^A, B♭♭ ^A, v4B♭
67 730.909 ^3A, v7B♭ ^^A, v3B♭ 32/21
68 741.818 ^4A, v6B♭ ^3A, vvB♭ 20/13, 49/32
69 752.727 ^5A, v5B♭ ^4A, vB♭ 65/42
70 763.636 ^6A, v4B♭ ^5A, B♭
71 774.545 ^7A, v3B♭ ^6A, v14B
72 785.455 A♯, vvB♭ ^7A, v13B 11/7, 52/33, 63/40
73 796.364 ^A♯, vB♭ ^8A, v12B
74 807.273 ^^A♯, B♭ ^9A, v11B
75 818.182 ^3A♯, v7B ^10A, v10B
76 829.091 ^4A♯, v6B ^11A, v9B 21/13
77 840 ^5A♯, v5B ^12A, v8B 13/8
78 850.909 ^6A♯, v4B ^13A, v7B
79 861.818 ^7A♯, v3B ^14A, v6B 64/39
80 872.727 A𝄪, vvB A♯, v5B
81 883.636 ^A𝄪, vB ^A♯, v4B 5/3
82 894.545 B ^^A♯, v3B
83 905.455 ^B, vC♭ ^3A♯, vvB
84 916.364 ^^B, C♭ ^4A♯, vB 56/33
85 927.273 ^3B, v7C B
86 938.182 ^4B, v6C ^B, v4C 55/32
87 949.091 ^5B, v5C ^^B, v3C
88 960 ^6B, v4C ^3B, vvC
89 970.909 ^7B, v3C ^4B, vC 7/4
90 981.818 B♯, vvC C
91 992.727 ^B♯, vC ^C, v4D♭
92 1003.64 C ^^C, v3D♭
93 1014.55 ^C, vD♭♭ ^3C, vvD♭ 9/5, 70/39
94 1025.45 ^^C, D♭♭ ^4C, vD♭ 65/36
95 1036.36 ^3C, v7D♭ ^5C, D♭
96 1047.27 ^4C, v6D♭ ^6C, v14D 64/35
97 1058.18 ^5C, v5D♭ ^7C, v13D 24/13
98 1069.09 ^6C, v4D♭ ^8C, v12D 13/7, 50/27
99 1080 ^7C, v3D♭ ^9C, v11D
100 1090.91 C♯, vvD♭ ^10C, v10D
101 1101.82 ^C♯, vD♭ ^11C, v9D 66/35
102 1112.73 ^^C♯, D♭ ^12C, v8D 40/21
103 1123.64 ^3C♯, v7D ^13C, v7D
104 1134.55 ^4C♯, v6D ^14C, v6D
105 1145.45 ^5C♯, v5D C♯, v5D 64/33
106 1156.36 ^6C♯, v4D ^C♯, v4D 39/20
107 1167.27 ^7C♯, v3D ^^C♯, v3D 55/28
108 1178.18 C𝄪, vvD ^3C♯, vvD
109 1189.09 ^C𝄪, vD ^4C♯, vD
110 1200 D D 2/1

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