113edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
Line 59: Line 51:
| 0.643
| 0.643
| 6.05
| 6.05
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
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| 55/39
| 55/39
| [[Gaster temperament|Gaster]]
| [[Gaster temperament|Gaster]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}

Revision as of 04:09, 16 November 2024

← 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

Template:EDO intro

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 10.6 ^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

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 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 Template:Rank-2 end Template:Orf

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