110edo
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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
← 109edo | 110edo | 111edo → |
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.
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, ↓E♭♭ | ↑D, ↓4E♭ | |
2 | 21.8182 | ↑↑D, E♭♭ | ↑↑D, ↓3E♭ | |
3 | 32.7273 | ↑3D, ↓7E♭ | ↑3D, ↓↓E♭ | 56/55 |
4 | 43.6364 | ↑4D, ↓6E♭ | ↑4D, ↓E♭ | 40/39 |
5 | 54.5455 | ↑5D, ↓5E♭ | ↑5D, E♭ | 33/32, 65/63 |
6 | 65.4545 | ↑6D, ↓4E♭ | ↑6D, ↓14E | |
7 | 76.3636 | ↑7D, ↓3E♭ | ↑7D, ↓13E | |
8 | 87.2727 | D♯, ↓↓E♭ | ↑8D, ↓12E | 21/20 |
9 | 98.1818 | ↑D♯, ↓E♭ | ↑9D, ↓11E | 35/33, 55/52 |
10 | 109.091 | ↑↑D♯, E♭ | ↑10D, ↓10E | |
11 | 120 | ↑3D♯, ↓7E | ↑11D, ↓9E | |
12 | 130.909 | ↑4D♯, ↓6E | ↑12D, ↓8E | 14/13, 27/25 |
13 | 141.818 | ↑5D♯, ↓5E | ↑13D, ↓7E | 13/12 |
14 | 152.727 | ↑6D♯, ↓4E | ↑14D, ↓6E | 35/32 |
15 | 163.636 | ↑7D♯, ↓3E | D♯, ↓5E | |
16 | 174.545 | D𝄪, ↓↓E | ↑D♯, ↓4E | 72/65 |
17 | 185.455 | ↑D𝄪, ↓E | ↑↑D♯, ↓3E | 10/9, 39/35, 49/44 |
18 | 196.364 | E | ↑3D♯, ↓↓E | 55/49 |
19 | 207.273 | ↑E, ↓F♭ | ↑4D♯, ↓E | |
20 | 218.182 | ↑↑E, F♭ | E | |
21 | 229.091 | ↑3E, ↓7F | ↑E, ↓4F | 8/7 |
22 | 240 | ↑4E, ↓6F | ↑↑E, ↓3F | 55/48 |
23 | 250.909 | ↑5E, ↓5F | ↑3E, ↓↓F | |
24 | 261.818 | ↑6E, ↓4F | ↑4E, ↓F | 64/55 |
25 | 272.727 | ↑7E, ↓3F | F | |
26 | 283.636 | E♯, ↓↓F | ↑F, ↓4G♭ | 33/28 |
27 | 294.545 | ↑E♯, ↓F | ↑↑F, ↓3G♭ | |
28 | 305.455 | F | ↑3F, ↓↓G♭ | |
29 | 316.364 | ↑F, ↓G♭♭ | ↑4F, ↓G♭ | 6/5 |
30 | 327.273 | ↑↑F, G♭♭ | ↑5F, G♭ | |
31 | 338.182 | ↑3F, ↓7G♭ | ↑6F, ↓14G | 39/32 |
32 | 349.091 | ↑4F, ↓6G♭ | ↑7F, ↓13G | |
33 | 360 | ↑5F, ↓5G♭ | ↑8F, ↓12G | 16/13 |
34 | 370.909 | ↑6F, ↓4G♭ | ↑9F, ↓11G | 26/21 |
35 | 381.818 | ↑7F, ↓3G♭ | ↑10F, ↓10G | |
36 | 392.727 | F♯, ↓↓G♭ | ↑11F, ↓9G | |
37 | 403.636 | ↑F♯, ↓G♭ | ↑12F, ↓8G | 63/50 |
38 | 414.545 | ↑↑F♯, G♭ | ↑13F, ↓7G | 14/11, 33/26, 80/63 |
39 | 425.455 | ↑3F♯, ↓7G | ↑14F, ↓6G | |
40 | 436.364 | ↑4F♯, ↓6G | F♯, ↓5G | |
41 | 447.273 | ↑5F♯, ↓5G | ↑F♯, ↓4G | |
42 | 458.182 | ↑6F♯, ↓4G | ↑↑F♯, ↓3G | 13/10, 64/49 |
43 | 469.091 | ↑7F♯, ↓3G | ↑3F♯, ↓↓G | 21/16, 55/42 |
44 | 480 | F𝄪, ↓↓G | ↑4F♯, ↓G | |
45 | 490.909 | ↑F𝄪, ↓G | G | |
46 | 501.818 | G | ↑G, ↓4A♭ | 4/3 |
47 | 512.727 | ↑G, ↓A♭♭ | ↑↑G, ↓3A♭ | 35/26, 66/49 |
48 | 523.636 | ↑↑G, A♭♭ | ↑3G, ↓↓A♭ | 65/48 |
49 | 534.545 | ↑3G, ↓7A♭ | ↑4G, ↓A♭ | |
50 | 545.455 | ↑4G, ↓6A♭ | ↑5G, A♭ | 48/35 |
51 | 556.364 | ↑5G, ↓5A♭ | ↑6G, ↓14A | |
52 | 567.273 | ↑6G, ↓4A♭ | ↑7G, ↓13A | 25/18 |
53 | 578.182 | ↑7G, ↓3A♭ | ↑8G, ↓12A | |
54 | 589.091 | G♯, ↓↓A♭ | ↑9G, ↓11A | |
55 | 600 | ↑G♯, ↓A♭ | ↑10G, ↓10A | |
56 | 610.909 | ↑↑G♯, A♭ | ↑11G, ↓9A | |
57 | 621.818 | ↑3G♯, ↓7A | ↑12G, ↓8A | |
58 | 632.727 | ↑4G♯, ↓6A | ↑13G, ↓7A | 36/25 |
59 | 643.636 | ↑5G♯, ↓5A | ↑14G, ↓6A | |
60 | 654.545 | ↑6G♯, ↓4A | G♯, ↓5A | 35/24 |
61 | 665.455 | ↑7G♯, ↓3A | ↑G♯, ↓4A | |
62 | 676.364 | G𝄪, ↓↓A | ↑↑G♯, ↓3A | |
63 | 687.273 | ↑G𝄪, ↓A | ↑3G♯, ↓↓A | 49/33, 52/35 |
64 | 698.182 | A | ↑4G♯, ↓A | 3/2 |
65 | 709.091 | ↑A, ↓B♭♭ | A | |
66 | 720 | ↑↑A, B♭♭ | ↑A, ↓4B♭ | |
67 | 730.909 | ↑3A, ↓7B♭ | ↑↑A, ↓3B♭ | 32/21 |
68 | 741.818 | ↑4A, ↓6B♭ | ↑3A, ↓↓B♭ | 20/13, 49/32 |
69 | 752.727 | ↑5A, ↓5B♭ | ↑4A, ↓B♭ | 65/42 |
70 | 763.636 | ↑6A, ↓4B♭ | ↑5A, B♭ | |
71 | 774.545 | ↑7A, ↓3B♭ | ↑6A, ↓14B | |
72 | 785.455 | A♯, ↓↓B♭ | ↑7A, ↓13B | 11/7, 52/33, 63/40 |
73 | 796.364 | ↑A♯, ↓B♭ | ↑8A, ↓12B | |
74 | 807.273 | ↑↑A♯, B♭ | ↑9A, ↓11B | |
75 | 818.182 | ↑3A♯, ↓7B | ↑10A, ↓10B | |
76 | 829.091 | ↑4A♯, ↓6B | ↑11A, ↓9B | 21/13 |
77 | 840 | ↑5A♯, ↓5B | ↑12A, ↓8B | 13/8 |
78 | 850.909 | ↑6A♯, ↓4B | ↑13A, ↓7B | |
79 | 861.818 | ↑7A♯, ↓3B | ↑14A, ↓6B | 64/39 |
80 | 872.727 | A𝄪, ↓↓B | A♯, ↓5B | |
81 | 883.636 | ↑A𝄪, ↓B | ↑A♯, ↓4B | 5/3 |
82 | 894.545 | B | ↑↑A♯, ↓3B | |
83 | 905.455 | ↑B, ↓C♭ | ↑3A♯, ↓↓B | |
84 | 916.364 | ↑↑B, C♭ | ↑4A♯, ↓B | 56/33 |
85 | 927.273 | ↑3B, ↓7C | B | |
86 | 938.182 | ↑4B, ↓6C | ↑B, ↓4C | 55/32 |
87 | 949.091 | ↑5B, ↓5C | ↑↑B, ↓3C | |
88 | 960 | ↑6B, ↓4C | ↑3B, ↓↓C | |
89 | 970.909 | ↑7B, ↓3C | ↑4B, ↓C | 7/4 |
90 | 981.818 | B♯, ↓↓C | C | |
91 | 992.727 | ↑B♯, ↓C | ↑C, ↓4D♭ | |
92 | 1003.64 | C | ↑↑C, ↓3D♭ | |
93 | 1014.55 | ↑C, ↓D♭♭ | ↑3C, ↓↓D♭ | 9/5, 70/39 |
94 | 1025.45 | ↑↑C, D♭♭ | ↑4C, ↓D♭ | 65/36 |
95 | 1036.36 | ↑3C, ↓7D♭ | ↑5C, D♭ | |
96 | 1047.27 | ↑4C, ↓6D♭ | ↑6C, ↓14D | 64/35 |
97 | 1058.18 | ↑5C, ↓5D♭ | ↑7C, ↓13D | 24/13 |
98 | 1069.09 | ↑6C, ↓4D♭ | ↑8C, ↓12D | 13/7, 50/27 |
99 | 1080 | ↑7C, ↓3D♭ | ↑9C, ↓11D | |
100 | 1090.91 | C♯, ↓↓D♭ | ↑10C, ↓10D | |
101 | 1101.82 | ↑C♯, ↓D♭ | ↑11C, ↓9D | 66/35 |
102 | 1112.73 | ↑↑C♯, D♭ | ↑12C, ↓8D | 40/21 |
103 | 1123.64 | ↑3C♯, ↓7D | ↑13C, ↓7D | |
104 | 1134.55 | ↑4C♯, ↓6D | ↑14C, ↓6D | |
105 | 1145.45 | ↑5C♯, ↓5D | C♯, ↓5D | 64/33 |
106 | 1156.36 | ↑6C♯, ↓4D | ↑C♯, ↓4D | 39/20 |
107 | 1167.27 | ↑7C♯, ↓3D | ↑↑C♯, ↓3D | 55/28 |
108 | 1178.18 | C𝄪, ↓↓D | ↑3C♯, ↓↓D | |
109 | 1189.09 | ↑C𝄪, ↓D | ↑4C♯, ↓D | |
110 | 1200 | D | D | 2/1 |
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