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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 113 (prime)
{{ED intro}}
| Step size = 10.619¢
| Fifth = 66\113 (700.885¢)
| Major 2nd = 19\113 (202¢)
| Semitones = 10:9 (106¢ : 96¢)
| Consistency = 13
}}
The '''113 equal divisions of the octave''' ('''113edo'''), or the '''113(-tone) equal temperament''' ('''113tet''', '''113et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 113 parts of about 10.6 [[cent]]s each.


== Theory ==
== 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.
113edo is [[consistency|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency. As an equal temperament, it [[tempering out|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 [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.


113edo is the 30th [[prime EDO]].
113edo might be notable as a no-fives system, where it is consistent in the [[29-odd-limit]] and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17.29 extension of slendric harmonies known as [[euslendric]].


=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|113}}
{{Harmonics in equal|113}}
 
=== Subsets and supersets ===
113edo is the 30th [[prime edo]], following [[109edo]] and before [[127edo]].
 
== Intervals ==
{{Interval table}}


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
Line 30: Line 30:
| 2.3
| 2.3
| {{monzo| -179 113 }}
| {{monzo| -179 113 }}
| [{{val| 113 179 }}]
| {{mapping| 113 179 }}
| +0.338
| +0.338
| 0.338
| 0.338
Line 37: Line 37:
| 2.3.5
| 2.3.5
| 1600000/1594323, 34171875/33554432
| 1600000/1594323, 34171875/33554432
| [{{val| 113 179 262 }}]
| {{mapping| 113 179 262 }}
| +0.801
| +0.801
| 0.712
| 0.712
Line 44: Line 44:
| 2.3.5.7
| 2.3.5.7
| 225/224, 1029/1024, 1071875/1062882
| 225/224, 1029/1024, 1071875/1062882
| [{{val| 113 179 262 317 }}]
| {{mapping| 113 179 262 317 }}
| +0.820
| +0.820
| 0.617
| 0.617
Line 51: Line 51:
| 2.3.5.7.11
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 980000/970299
| 225/224, 243/242, 385/384, 980000/970299
| [{{val| 113 179 262 317 391 }}]
| {{mapping| 113 179 262 317 391 }}
| +0.604
| +0.604
| 0.700
| 0.700
Line 58: Line 58:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 225/224, 243/242, 325/324, 385/384, 1875/1859
| 225/224, 243/242, 325/324, 385/384, 1875/1859
| [{{val| 113 179 262 317 391 418 }}]
| {{mapping| 113 179 262 317 391 418 }}
| +0.575
| +0.575
| 0.643
| 0.643
Line 66: Line 66:
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per octave
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
Line 125: Line 126:
| 339.82
| 339.82
| 243/200
| 243/200
| [[Amity]] / [[houborizic]]
| [[Houborizic]]
|-
|-
| 1
| 1
Line 144: Line 145:
| 4/3
| 4/3
| [[Gracecordial]]
| [[Gracecordial]]
|}
== Scales ==
Since 113edo has a step of 10.6195 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}. It is the first edo which allows one to use an MOS scale of 90 tones or more as a circulating temperament.
{| class="wikitable"
|+Circulating temperaments in 113edo
!Tones
!Pattern
!L:s
|-
|-
|5
| 1
|[[3L 2s]]
| 56\113
|23:22
| 594.69
|-
| 55/39
|6
| [[Gaster temperament|Gaster]]
|[[5L 1s]]
|19:18
|-
|7
|[[1L 6s]]
|17:16
|-
|8
|[[1L 7s]]
|15:14
|-
|9
|[[5L 4s]]
|13:12
|-
|10
|[[3L 7s]]
|12:11
|-
|11
|[[3L 8s]]
|11:10
|-
|12
|[[5L 7s]]
|10:9
|-
|13
|[[9L 4s]]
| rowspan="2" |9:8
|-
|14
|[[1L 13s]]
|-
|15
|[[7L 8s]]
| rowspan="2" |8:7
|-
|16
|1L 15s
|-
|17
|[[11L 6s]]
| rowspan="2" |7:6
|-
|18
|5L 13s
|-
|19
|18L 1s
| rowspan="4" |6:5
|-
|20
|[[13L 7s]]
|-
|21
|[[8L 13s]]
|-
|22
|[[3L 19s]]
|-
|23
|21L 2s
| rowspan="6" |5:4
|-
|24
|[[17L 7s]]
|-
|25
|13L 12s
|-
|26
|9L 17s
|-
|27
|[[5L 22s]]
|-
|28
|1L 27s
|-
|29
|26L 3s
| rowspan="9" |4:3
|-
|30
|23L 7s
|-
|31
|20L 11s
|-
|32
|17L 15s
|-
|33
|14L 19s
|-
|34
|11L 23s
|-
|35
|8L 27s
|-
|36
|5L 31s
|-
|37
|2L 35s
|-
|38
|37L 1s
| rowspan="19" |3:2
|-
|39
|35L 4s
|-
|40
|33L 7s
|-
|41
|31L 10s
|-
|42
|29L 13s
|-
|43
|27L 16s
|-
|44
|25L 19s
|-
|45
|23L 22s
|-
|46
|21L 25s
|-
|47
|19L 28s
|-
|48
|17L 31s
|-
|49
|15L 34s
|-
|50
|13L 37s
|-
|51
|11L 40s
|-
|52
|9L  43s
|-
|53
|7L 46s
|-
|54
|5L 49s
|-
|55
|3L 52s
|-
|56
|1L 55s
|-
|57
|56L 1s
| rowspan="34" |2:1
|-
|58
|55L 3s
|-
|59
|54L 5s
|-
|60
|53L 7s
|-
|61
|52L 9s
|-
|62
|51L 11s
|-
|63
|50L 13s
|-
|64
|49L 15s
|-
|65
|48L 17s
|-
|66
|47L 19s
|-
|67
|46L 21s
|-
|68
|45L 23s
|-
|69
|44L 25s
|-
|70
|43L 27s
|-
|71
|42L 29s
|-
|72
|41L 31s
|-
|73
|40L 33s
|-
|74
|39L 35s
|-
|75
|38L 37s
|-
|76
|37L 39s
|-
|77
|36L 41s
|-
|78
|35L 43s
|-
|79
|34L 45s
|-
|80
|33L 47s
|-
|81
|32L 49s
|-
|82
|31L 51s
|-
|83
|30L 53s
|-
|84
|29L 55s
|-
|85
|28L 57s
|-
|86
|27L 59s
|-
|87
|26L 61s
|-
|88
|25L 63s
|-
|89
|24L 65s
|-
|90
|23L 67s
|}
|}
 
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
[[Category:Equal divisions of the octave]]
[[Category:Prime EDO]]
[[Category:Theory]]

Latest revision as of 00:21, 10 July 2025

← 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.

113edo might be notable as a no-fives system, where it is consistent in the 29-odd-limit and serves as a nearly optimal tuning for slendric, in particular a 2.3.7.13.17.29 extension of slendric harmonies known as euslendric.

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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