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 10.909 ¢ 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, vEbb ^D, v4Eb
2 21.8182 ^^D, Ebb ^^D, v3Eb
3 32.7273 ^3D, v7Eb ^3D, vvEb 56/55
4 43.6364 ^4D, v6Eb ^4D, vEb 40/39
5 54.5455 ^5D, v5Eb ^5D, Eb 33/32, 65/63
6 65.4545 ^6D, v4Eb ^6D, v14E
7 76.3636 ^7D, v3Eb ^7D, v13E
8 87.2727 D#, vvEb ^8D, v12E 21/20
9 98.1818 ^D#, vEb ^9D, v11E 35/33, 55/52
10 109.091 ^^D#, Eb ^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 Dx, vvE ^D#, v4E 72/65
17 185.455 ^Dx, vE ^^D#, v3E 10/9, 39/35, 49/44
18 196.364 E ^3D#, vvE 55/49
19 207.273 ^E, vFb ^4D#, vE
20 218.182 ^^E, Fb 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, v4Gb 33/28
27 294.545 ^E#, vF ^^F, v3Gb
28 305.455 F ^3F, vvGb
29 316.364 ^F, vGbb ^4F, vGb 6/5
30 327.273 ^^F, Gbb ^5F, Gb
31 338.182 ^3F, v7Gb ^6F, v14G 39/32
32 349.091 ^4F, v6Gb ^7F, v13G
33 360 ^5F, v5Gb ^8F, v12G 16/13
34 370.909 ^6F, v4Gb ^9F, v11G 26/21
35 381.818 ^7F, v3Gb ^10F, v10G
36 392.727 F#, vvGb ^11F, v9G
37 403.636 ^F#, vGb ^12F, v8G 63/50
38 414.545 ^^F#, Gb ^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 Fx, vvG ^4F#, vG
45 490.909 ^Fx, vG G
46 501.818 G ^G, v4Ab 4/3
47 512.727 ^G, vAbb ^^G, v3Ab 35/26, 66/49
48 523.636 ^^G, Abb ^3G, vvAb 65/48
49 534.545 ^3G, v7Ab ^4G, vAb
50 545.455 ^4G, v6Ab ^5G, Ab 48/35
51 556.364 ^5G, v5Ab ^6G, v14A
52 567.273 ^6G, v4Ab ^7G, v13A 25/18
53 578.182 ^7G, v3Ab ^8G, v12A
54 589.091 G#, vvAb ^9G, v11A
55 600 ^G#, vAb ^10G, v10A
56 610.909 ^^G#, Ab ^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 Gx, vvA ^^G#, v3A
63 687.273 ^Gx, vA ^3G#, vvA 49/33, 52/35
64 698.182 A ^4G#, vA 3/2
65 709.091 ^A, vBbb A
66 720 ^^A, Bbb ^A, v4Bb
67 730.909 ^3A, v7Bb ^^A, v3Bb 32/21
68 741.818 ^4A, v6Bb ^3A, vvBb 20/13, 49/32
69 752.727 ^5A, v5Bb ^4A, vBb 65/42
70 763.636 ^6A, v4Bb ^5A, Bb
71 774.545 ^7A, v3Bb ^6A, v14B
72 785.455 A#, vvBb ^7A, v13B 11/7, 52/33, 63/40
73 796.364 ^A#, vBb ^8A, v12B
74 807.273 ^^A#, Bb ^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 Ax, vvB A#, v5B
81 883.636 ^Ax, vB ^A#, v4B 5/3
82 894.545 B ^^A#, v3B
83 905.455 ^B, vCb ^3A#, vvB
84 916.364 ^^B, Cb ^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, v4Db
92 1003.64 C ^^C, v3Db
93 1014.55 ^C, vDbb ^3C, vvDb 9/5, 70/39
94 1025.45 ^^C, Dbb ^4C, vDb 65/36
95 1036.36 ^3C, v7Db ^5C, Db
96 1047.27 ^4C, v6Db ^6C, v14D 64/35
97 1058.18 ^5C, v5Db ^7C, v13D 24/13
98 1069.09 ^6C, v4Db ^8C, v12D 13/7, 50/27
99 1080 ^7C, v3Db ^9C, v11D
100 1090.91 C#, vvDb ^10C, v10D
101 1101.82 ^C#, vDb ^11C, v9D 66/35
102 1112.73 ^^C#, Db ^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 Cx, vvD ^3C#, vvD
109 1189.09 ^Cx, vD ^4C#, vD
110 1200 D D 2/1

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