113edo

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← 112edo 113edo 114edo →
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 comma 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 10.6 ^D, ^^E♭♭
2 21.2 ^^D, ^3E♭♭
3 31.9 ^3D, ^4E♭♭
4 42.5 40/39, 41/40, 42/41, 43/42 ^4D, v5E♭
5 53.1 32/31, 33/32, 34/33 ^5D, v4E♭
6 63.7 27/26, 28/27 v4D♯, v3E♭
7 74.3 24/23 v3D♯, vvE♭
8 85 21/20, 41/39 vvD♯, vE♭
9 95.6 19/18 vD♯, E♭
10 106.2 17/16, 33/31 D♯, ^E♭
11 116.8 31/29, 46/43 ^D♯, ^^E♭
12 127.4 14/13 ^^D♯, ^3E♭
13 138.1 13/12 ^3D♯, ^4E♭
14 148.7 12/11 ^4D♯, v5E
15 159.3 23/21, 34/31, 45/41 ^5D♯, v4E
16 169.9 32/29, 43/39 v4D𝄪, v3E
17 180.5 10/9 v3D𝄪, vvE
18 191.2 19/17, 29/26, 48/43 vvD𝄪, vE
19 201.8 9/8 E
20 212.4 26/23, 43/38 ^E, ^^F♭
21 223 33/29, 41/36 ^^E, ^3F♭
22 233.6 ^3E, ^4F♭
23 244.2 38/33 ^4E, v5F
24 254.9 22/19 ^5E, v4F
25 265.5 7/6 v4E♯, v3F
26 276.1 27/23, 34/29 v3E♯, vvF
27 286.7 46/39 vvE♯, vF
28 297.3 19/16 F
29 308 37/31, 43/36 ^F, ^^G♭♭
30 318.6 ^^F, ^3G♭♭
31 329.2 23/19, 29/24 ^3F, ^4G♭♭
32 339.8 28/23 ^4F, v5G♭
33 350.4 38/31 ^5F, v4G♭
34 361.1 16/13 v4F♯, v3G♭
35 371.7 26/21 v3F♯, vvG♭
36 382.3 vvF♯, vG♭
37 392.9 vF♯, G♭
38 403.5 24/19 F♯, ^G♭
39 414.2 33/26, 47/37 ^F♯, ^^G♭
40 424.8 23/18 ^^F♯, ^3G♭
41 435.4 9/7 ^3F♯, ^4G♭
42 446 22/17 ^4F♯, v5G
43 456.6 43/33 ^5F♯, v4G
44 467.3 38/29 v4F𝄪, v3G
45 477.9 29/22 v3F𝄪, vvG
46 488.5 vvF𝄪, vG
47 499.1 4/3 G
48 509.7 43/32 ^G, ^^A♭♭
49 520.4 27/20 ^^G, ^3A♭♭
50 531 ^3G, ^4A♭♭
51 541.6 26/19, 41/30 ^4G, v5A♭
52 552.2 11/8 ^5G, v4A♭
53 562.8 18/13 v4G♯, v3A♭
54 573.5 32/23, 39/28, 46/33 v3G♯, vvA♭
55 584.1 7/5 vvG♯, vA♭
56 594.7 31/22 vG♯, A♭
57 605.3 44/31 G♯, ^A♭
58 615.9 10/7 ^G♯, ^^A♭
59 626.5 23/16, 33/23 ^^G♯, ^3A♭
60 637.2 13/9 ^3G♯, ^4A♭
61 647.8 16/11 ^4G♯, v5A
62 658.4 19/13, 41/28 ^5G♯, v4A
63 669 v4G𝄪, v3A
64 679.6 40/27 v3G𝄪, vvA
65 690.3 vvG𝄪, vA
66 700.9 3/2 A
67 711.5 ^A, ^^B♭♭
68 722.1 41/27, 44/29, 47/31 ^^A, ^3B♭♭
69 732.7 29/19 ^3A, ^4B♭♭
70 743.4 43/28 ^4A, v5B♭
71 754 17/11 ^5A, v4B♭
72 764.6 14/9 v4A♯, v3B♭
73 775.2 36/23 v3A♯, vvB♭
74 785.8 vvA♯, vB♭
75 796.5 19/12 vA♯, B♭
76 807.1 43/27 A♯, ^B♭
77 817.7 ^A♯, ^^B♭
78 828.3 21/13 ^^A♯, ^3B♭
79 838.9 13/8 ^3A♯, ^4B♭
80 849.6 31/19 ^4A♯, v5B
81 860.2 23/14 ^5A♯, v4B
82 870.8 38/23, 43/26, 48/29 v4A𝄪, v3B
83 881.4 v3A𝄪, vvB
84 892 vvA𝄪, vB
85 902.7 32/19 B
86 913.3 39/23 ^B, ^^C♭
87 923.9 29/17, 46/27 ^^B, ^3C♭
88 934.5 12/7 ^3B, ^4C♭
89 945.1 19/11 ^4B, v5C
90 955.8 33/19 ^5B, v4C
91 966.4 v4B♯, v3C
92 977 v3B♯, vvC
93 987.6 23/13 vvB♯, vC
94 998.2 16/9 C
95 1008.8 34/19, 43/24 ^C, ^^D♭♭
96 1019.5 9/5 ^^C, ^3D♭♭
97 1030.1 29/16 ^3C, ^4D♭♭
98 1040.7 31/17, 42/23 ^4C, v5D♭
99 1051.3 11/6 ^5C, v4D♭
100 1061.9 24/13 v4C♯, v3D♭
101 1072.6 13/7 v3C♯, vvD♭
102 1083.2 43/23 vvC♯, vD♭
103 1093.8 32/17 vC♯, D♭
104 1104.4 36/19 C♯, ^D♭
105 1115 40/21 ^C♯, ^^D♭
106 1125.7 23/12 ^^C♯, ^3D♭
107 1136.3 27/14 ^3C♯, ^4D♭
108 1146.9 31/16, 33/17 ^4C♯, v5D
109 1157.5 39/20, 41/21 ^5C♯, v4D
110 1168.1 v4C𝄪, v3D
111 1178.8 v3C𝄪, vvD
112 1189.4 vvC𝄪, 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 distinct

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