113edo

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← 112edo113edo114edo →
Prime factorization 113 (prime)
Step size 10.6195¢ 
Fifth 66\113 (700.885¢)
Semitones (A1:m2) 10:9 (106.2¢ : 95.58¢)
Consistency limit 13
Distinct consistency limit 13

113 equal divisions of the octave (abbreviated 113edo or 113ed2), also called 113-tone equal temperament (113tet) or 113 equal temperament (113et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 113 equal parts of about 10.6 ¢ each. Each step represents a frequency ratio of 21/113, or the 113th root of 2.

Theory

113edo is distinctly consistent in the 13-odd-limit with a flat tendency. As an equal temperament, it tempers out the amity comma and the ampersand in the 5-limit; 225/224, 1029/1024 and 1071875/1062882 in the 7-limit; 243/242, 385/384, 441/440 and 540/539 in the 11-limit; 325/324, 364/363, 729/728, and 1625/1617 in the 13-limit. It notably supports the 5-limit amity temperament, 7-limit amicable temperament, 7- and 11-limit miracle temperament, and 13-limit manna temperament.

Prime harmonics

Approximation of prime harmonics in 113edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -1.07 -4.01 -2.45 +0.89 -1.59 +1.24 -0.17 -1.73 +0.51 +1.87
Relative (%) +0.0 -10.1 -37.8 -23.1 +8.4 -15.0 +11.7 -1.6 -16.3 +4.8 +17.6
Steps
(reduced) 		113
(0) 		179
(66) 		262
(36) 		317
(91) 		391
(52) 		418
(79) 		462
(10) 		480
(28) 		511
(59) 		549
(97) 		560
(108)

Subsets and supersets

113edo is the 30th prime edo, following 109edo and before 127edo.

Intervals

Steps Cents Approximate Ratios Ups and Downs Notation
0 0 1/1 D
1 10.619 ^D, v8E♭
2 21.239 ^^D, v7E♭
3 31.858 ^3D, v6E♭
4 42.478 40/39, 41/40, 42/41, 43/42 ^4D, v5E♭
5 53.097 32/31, 33/32, 34/33 ^5D, v4E♭
6 63.717 27/26, 28/27 ^6D, v3E♭
7 74.336 24/23 ^7D, vvE♭
8 84.956 21/20, 41/39 ^8D, vE♭
9 95.575 19/18 ^9D, E♭
10 106.195 17/16, 33/31 D♯, v9E
11 116.814 31/29, 46/43 ^D♯, v8E
12 127.434 14/13 ^^D♯, v7E
13 138.053 13/12 ^3D♯, v6E
14 148.673 12/11 ^4D♯, v5E
15 159.292 23/21, 34/31, 45/41 ^5D♯, v4E
16 169.912 32/29, 43/39 ^6D♯, v3E
17 180.531 10/9 ^7D♯, vvE
18 191.15 19/17, 29/26, 48/43 ^8D♯, vE
19 201.77 9/8 E
20 212.389 26/23, 43/38 ^E, v8F
21 223.009 33/29, 41/36 ^^E, v7F
22 233.628 ^3E, v6F
23 244.248 38/33 ^4E, v5F
24 254.867 22/19 ^5E, v4F
25 265.487 7/6 ^6E, v3F
26 276.106 27/23, 34/29 ^7E, vvF
27 286.726 46/39 ^8E, vF
28 297.345 19/16 F
29 307.965 37/31, 43/36 ^F, v8G♭
30 318.584 ^^F, v7G♭
31 329.204 23/19, 29/24 ^3F, v6G♭
32 339.823 28/23 ^4F, v5G♭
33 350.442 38/31 ^5F, v4G♭
34 361.062 16/13 ^6F, v3G♭
35 371.681 26/21 ^7F, vvG♭
36 382.301 ^8F, vG♭
37 392.92 ^9F, G♭
38 403.54 24/19 F♯, v9G
39 414.159 33/26, 47/37 ^F♯, v8G
40 424.779 23/18 ^^F♯, v7G
41 435.398 9/7 ^3F♯, v6G
42 446.018 22/17 ^4F♯, v5G
43 456.637 43/33 ^5F♯, v4G
44 467.257 38/29 ^6F♯, v3G
45 477.876 29/22 ^7F♯, vvG
46 488.496 ^8F♯, vG
47 499.115 4/3 G
48 509.735 43/32 ^G, v8A♭
49 520.354 27/20 ^^G, v7A♭
50 530.973 ^3G, v6A♭
51 541.593 26/19, 41/30 ^4G, v5A♭
52 552.212 11/8 ^5G, v4A♭
53 562.832 18/13 ^6G, v3A♭
54 573.451 32/23, 39/28, 46/33 ^7G, vvA♭
55 584.071 7/5 ^8G, vA♭
56 594.69 31/22 ^9G, A♭
57 605.31 44/31 G♯, v9A
58 615.929 10/7 ^G♯, v8A
59 626.549 23/16, 33/23 ^^G♯, v7A
60 637.168 13/9 ^3G♯, v6A
61 647.788 16/11 ^4G♯, v5A
62 658.407 19/13, 41/28 ^5G♯, v4A
63 669.027 ^6G♯, v3A
64 679.646 40/27 ^7G♯, vvA
65 690.265 ^8G♯, vA
66 700.885 3/2 A
67 711.504 ^A, v8B♭
68 722.124 41/27, 44/29, 47/31 ^^A, v7B♭
69 732.743 29/19 ^3A, v6B♭
70 743.363 43/28 ^4A, v5B♭
71 753.982 17/11 ^5A, v4B♭
72 764.602 14/9 ^6A, v3B♭
73 775.221 36/23 ^7A, vvB♭
74 785.841 ^8A, vB♭
75 796.46 19/12 ^9A, B♭
76 807.08 43/27 A♯, v9B
77 817.699 ^A♯, v8B
78 828.319 21/13 ^^A♯, v7B
79 838.938 13/8 ^3A♯, v6B
80 849.558 31/19 ^4A♯, v5B
81 860.177 23/14 ^5A♯, v4B
82 870.796 38/23, 43/26, 48/29 ^6A♯, v3B
83 881.416 ^7A♯, vvB
84 892.035 ^8A♯, vB
85 902.655 32/19 B
86 913.274 39/23 ^B, v8C
87 923.894 29/17, 46/27 ^^B, v7C
88 934.513 12/7 ^3B, v6C
89 945.133 19/11 ^4B, v5C
90 955.752 33/19 ^5B, v4C
91 966.372 ^6B, v3C
92 976.991 ^7B, vvC
93 987.611 23/13 ^8B, vC
94 998.23 16/9 C
95 1008.85 34/19, 43/24 ^C, v8D♭
96 1019.469 9/5 ^^C, v7D♭
97 1030.088 29/16 ^3C, v6D♭
98 1040.708 31/17, 42/23 ^4C, v5D♭
99 1051.327 11/6 ^5C, v4D♭
100 1061.947 24/13 ^6C, v3D♭
101 1072.566 13/7 ^7C, vvD♭
102 1083.186 43/23 ^8C, vD♭
103 1093.805 32/17 ^9C, D♭
104 1104.425 36/19 C♯, v9D
105 1115.044 40/21 ^C♯, v8D
106 1125.664 23/12 ^^C♯, v7D
107 1136.283 27/14 ^3C♯, v6D
108 1146.903 31/16, 33/17 ^4C♯, v5D
109 1157.522 39/20, 41/21 ^5C♯, v4D
110 1168.142 ^6C♯, v3D
111 1178.761 ^7C♯, vvD
112 1189.381 ^8C♯, vD
113 1200 2/1 D

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-179 113⟩ [⟨113 179]] +0.338 0.338 3.18
2.3.5 1600000/1594323, 34171875/33554432 [⟨113 179 262]] +0.801 0.712 6.70
2.3.5.7 225/224, 1029/1024, 1071875/1062882 [⟨113 179 262 317]] +0.820 0.617 5.81
2.3.5.7.11 225/224, 243/242, 385/384, 980000/970299 [⟨113 179 262 317 391]] +0.604 0.700 6.59
2.3.5.7.11.13 225/224, 243/242, 325/324, 385/384, 1875/1859 [⟨113 179 262 317 391 418]] +0.575 0.643 6.05

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 4\113 42.48 40/39 Humorous
1 6\113 63.72 28/27 Sycamore / betic
1 8\113 84.96 21/20 Amicable / pseudoamical / pseudoamorous
1 11\113 116.81 15/14~16/15 Miracle / manna
1 13\113 138.05 27/25 Quartemka
1 22\113 233.63 8/7 Slendric
1 27\113 286.73 13/11 Gamity
1 29\113 307.96 3200/2673 Familia
1 32\113 339.82 243/200 Houborizic
1 34\113 360.06 16/13 Phicordial
1 37\113 392.92 2744/2187 Emmthird
1 47\113 499.12 4/3 Gracecordial
1 56\113 594.69 55/39 Gaster

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

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