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 a 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 -10 -38 -23 +8 -15 +12 -2 -16 +5 +18
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.

Intervals

Steps Cents Ups and downs notation Approximate ratios
0 0 D 1/1
1 10.6195 ↑D, ↓8E♭
2 21.2389 ↑↑D, ↓7E♭ 78/77, 81/80
3 31.8584 ↑3D, ↓6E♭ 49/48, 50/49, 55/54, 56/55
4 42.4779 ↑4D, ↓5E♭ 40/39
5 53.0973 ↑5D, ↓4E♭ 33/32, 65/63
6 63.7168 ↑6D, ↓3E♭ 27/26, 28/27, 80/77
7 74.3363 ↑7D, ↓↓E♭
8 84.9558 ↑8D, ↓E♭ 21/20, 81/77
9 95.5752 ↑9D, E♭ 55/52
10 106.195 D♯, ↓9E 52/49
11 116.814 ↑D♯, ↓8E 15/14, 77/72
12 127.434 ↑↑D♯, ↓7E 14/13
13 138.053 ↑3D♯, ↓6E 13/12
14 148.673 ↑4D♯, ↓5E 12/11, 49/45
15 159.292 ↑5D♯, ↓4E
16 169.912 ↑6D♯, ↓3E 54/49
17 180.531 ↑7D♯, ↓↓E 10/9, 72/65
18 191.15 ↑8D♯, ↓E 39/35
19 201.77 E 9/8, 55/49
20 212.389 ↑E, ↓8F 44/39
21 223.009 ↑↑E, ↓7F
22 233.628 ↑3E, ↓6F 8/7, 55/48, 63/55
23 244.248 ↑4E, ↓5F 15/13
24 254.867 ↑5E, ↓4F 65/56, 81/70
25 265.487 ↑6E, ↓3F 7/6, 64/55
26 276.106 ↑7E, ↓↓F
27 286.726 ↑8E, ↓F 13/11, 33/28
28 297.345 F 32/27, 77/65
29 307.965 ↑F, ↓8G♭
30 318.584 ↑↑F, ↓7G♭ 6/5, 65/54, 77/64
31 329.204 ↑3F, ↓6G♭ 40/33, 63/52
32 339.823 ↑4F, ↓5G♭ 39/32
33 350.442 ↑5F, ↓4G♭ 11/9, 27/22, 49/40, 60/49
34 361.062 ↑6F, ↓3G♭ 16/13
35 371.681 ↑7F, ↓↓G♭ 26/21
36 382.301 ↑8F, ↓G♭ 5/4, 56/45, 81/65
37 392.92 ↑9F, G♭ 49/39
38 403.54 F♯, ↓9G 63/50
39 414.159 ↑F♯, ↓8G 14/11, 33/26, 80/63
40 424.779 ↑↑F♯, ↓7G
41 435.398 ↑3F♯, ↓6G 9/7, 77/60
42 446.018 ↑4F♯, ↓5G 35/27
43 456.637 ↑5F♯, ↓4G 13/10
44 467.257 ↑6F♯, ↓3G 21/16, 55/42, 72/55
45 477.876 ↑7F♯, ↓↓G
46 488.496 ↑8F♯, ↓G 65/49
47 499.115 G 4/3
48 509.735 ↑G, ↓8A♭
49 520.354 ↑↑G, ↓7A♭ 27/20
50 530.973 ↑3G, ↓6A♭ 49/36
51 541.593 ↑4G, ↓5A♭
52 552.212 ↑5G, ↓4A♭ 11/8
53 562.832 ↑6G, ↓3A♭ 18/13
54 573.451 ↑7G, ↓↓A♭ 39/28
55 584.071 ↑8G, ↓A♭ 7/5
56 594.69 ↑9G, A♭ 55/39
57 605.31 G♯, ↓9A 78/55
58 615.929 ↑G♯, ↓8A 10/7, 77/54
59 626.549 ↑↑G♯, ↓7A 56/39
60 637.168 ↑3G♯, ↓6A 13/9, 81/56
61 647.788 ↑4G♯, ↓5A 16/11
62 658.407 ↑5G♯, ↓4A
63 669.027 ↑6G♯, ↓3A 72/49, 81/55
64 679.646 ↑7G♯, ↓↓A 40/27, 77/52
65 690.265 ↑8G♯, ↓A
66 700.885 A 3/2
67 711.504 ↑A, ↓8B♭
68 722.124 ↑↑A, ↓7B♭
69 732.743 ↑3A, ↓6B♭ 32/21, 55/36, 75/49
70 743.363 ↑4A, ↓5B♭ 20/13
71 753.982 ↑5A, ↓4B♭ 54/35, 65/42
72 764.602 ↑6A, ↓3B♭ 14/9, 81/52
73 775.221 ↑7A, ↓↓B♭
74 785.841 ↑8A, ↓B♭ 11/7, 52/33, 63/40
75 796.46 ↑9A, B♭
76 807.08 A♯, ↓9B 78/49
77 817.699 ↑A♯, ↓8B 8/5, 45/28, 77/48
78 828.319 ↑↑A♯, ↓7B 21/13
79 838.938 ↑3A♯, ↓6B 13/8, 81/50
80 849.558 ↑4A♯, ↓5B 18/11, 44/27, 49/30, 80/49
81 860.177 ↑5A♯, ↓4B 64/39
82 870.796 ↑6A♯, ↓3B 33/20, 81/49
83 881.416 ↑7A♯, ↓↓B 5/3
84 892.035 ↑8A♯, ↓B
85 902.655 B 27/16
86 913.274 ↑B, ↓8C 22/13, 56/33
87 923.894 ↑↑B, ↓7C
88 934.513 ↑3B, ↓6C 12/7, 55/32
89 945.133 ↑4B, ↓5C
90 955.752 ↑5B, ↓4C 26/15
91 966.372 ↑6B, ↓3C 7/4
92 976.991 ↑7B, ↓↓C
93 987.611 ↑8B, ↓C 39/22
94 998.23 C 16/9
95 1008.85 ↑C, ↓8D♭ 70/39
96 1019.47 ↑↑C, ↓7D♭ 9/5, 65/36
97 1030.09 ↑3C, ↓6D♭ 49/27
98 1040.71 ↑4C, ↓5D♭
99 1051.33 ↑5C, ↓4D♭ 11/6
100 1061.95 ↑6C, ↓3D♭ 24/13
101 1072.57 ↑7C, ↓↓D♭ 13/7
102 1083.19 ↑8C, ↓D♭ 28/15
103 1093.81 ↑9C, D♭ 49/26
104 1104.42 C♯, ↓9D
105 1115.04 ↑C♯, ↓8D 40/21
106 1125.66 ↑↑C♯, ↓7D
107 1136.28 ↑3C♯, ↓6D 27/14, 52/27, 77/40
108 1146.9 ↑4C♯, ↓5D 64/33
109 1157.52 ↑5C♯, ↓4D 39/20
110 1168.14 ↑6C♯, ↓3D 49/25, 55/28
111 1178.76 ↑7C♯, ↓↓D 77/39
112 1189.38 ↑8C♯, ↓D
113 1200 D 2/1

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
(Reduced) 		Cents
(Reduced) 		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
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