116edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
'''116edo''' is the [[EDO|equal division of the octave]] into 116 parts of 10.3448 cents each. It tempers out 20000/19683 (tetracot comma) and 2197265625/2147483648 (wizard comma) in the 5-limit. Using the patent val, it tempers out 225/224, 15625/15309, and 51200/50421 in the 7-limit; 385/384, 540/539, 4000/3993, and 6655/6561 in the 11-limit; 169/168, 275/273, 352/351, and 640/637 in the 13-limit. 116edo provides the optimal patent val for [[Marvel temperaments|submajor temperament]].
{{ED intro}}


Since 116edo has a step of 10.3448 cents, it also allows one to use its MOS scales as circulating temperaments.
116edo is only [[consistent]] to the [[5-odd-limit]], and is not quite accurate for its size. It can be viewed as splitting [[58edo]]'s step in two, and the [[enfactoring|enfactored]] 116cef [[val]] comes out on top accuracy in the 7-, 11-, and 13-limit. In the 5-limit, however, the [[patent val]] {{val| 116 184 '''269''' }} beats the enfactored 116c val {{val| 116 184 '''270''' }} by a thin margin, and it [[Tempering out|tempers out]] 20000/19683 ([[tetracot comma]]) and 2197265625/2147483648 (wizard comma).  


{| class="wikitable"
In the 7-, 11- and 13-limit, the patent val {{val| 116 184 '''269''' 326 '''401''' '''429''' }} comes in second best after the enfactored 116cef val {{val| 116 184 '''270''' 326 '''402''' '''430''' }} , and it tempers out [[225/224]], 15625/15309, and 51200/50421 in the 7-limit; [[385/384]], [[540/539]], [[4000/3993]], and 6655/6561 in the 11-limit; [[169/168]], [[275/273]], [[352/351]], and [[640/637]] in the 13-limit. 116edo provides the [[optimal patent val]] for the [[submajor (temperament)|submajor]] temperament in the 11- and 13-limit.
|+Circulating temperaments in 116edo
!Tones
!Pattern
!L:s
|-
|5
|[[1L 4s]]
|24:23
|-
|6
|[[2L 4s]]
|20:19
|-
|7
|[[4L 3s]]
|17:16
|-
|8
|[[4L 4s]]
|15:14
|-
|9
|[[8L 1s]]
|13:12
|-
|10
|[[6L 4s]]
|12:11
|-
|11
|[[6L 5s]]
|11:10
|-
|12
|[[8L 4s]]
|10:9
|-
|13
|[[12L 1s]]
| rowspan="2" |9:8
|-
|14
|[[4L 10s]]
|-
|15
|[[11L 4s]]
| rowspan="2" |8:7
|-
|16
|4L 12s
|-
|17
|[[14L 3s]]
| rowspan="3" |7:6
|-
|18
|8L 10s
|-
|19
|[[2L 17s]]
|-
|20
|16L 4s
| rowspan="4" |6:5
|-
|21
|11L 10s
|-
|22
|[[6L 16s]]
|-
|23
|1L 22s
|-
|24
|20L 4s
| rowspan="5" |5:4
|-
|25
|16L 9s
|-
|26
|12L 14s
|-
|27
|8L 19s
|-
|28
|4L 24s
|-
|29
|[[29edo]]
|equal
|-
|30
|26L 4s
| rowspan="9" |4:3
|-
|31
|23L 8s
|-
|32
|20L 12s
|-
|33
|17L 16s
|-
|34
|14L 20s
|-
|35
|11L 24s
|-
|36
|8L 28s
|-
|37
|5L 32s
|-
|38
|2L 36s
|-
|39
|38L 1s
| rowspan="19" |3:2
|-
|40
|36L 4s
|-
|41
|34L 7s
|-
|42
|32L 10s
|-
|43
|30L 13s
|-
|44
|28L 16s
|-
|45
|26L 19s
|-
|46
|24L 22s
|-
|47
|22L 25s
|-
|48
|20L 28s
|-
|49
|18L 31s
|-
|50
|16L 34s
|-
|51
|14L 37s
|-
|52
|12L 40s
|-
|53
|10L 43s
|-
|54
|8L 46s
|-
|55
|6L 49s
|-
|56
|4L 52s
|-
|57
|2L 55s
|-
|58
|[[58edo]]
|equal
|-
|59
|57L 2s
| rowspan="34" |2:1
|-
|60
|56L 4s
|-
|61
|55L 6s
|-
|62
|54L 8s
|-
|63
|53L 10s
|-
|64
|52L 12s
|-
|65
|51L 14s
|-
|66
|50L 16s
|-
|67
|49L 18s
|-
|68
|48L 20s
|-
|69
|47L 22s
|-
|70
|46L 24s
|-
|71
|45L 26s
|-
|72
|44L 28s
|-
|73
|43L 30s
|-
|74
|42L 32s
|-
|75
|41L 34s
|-
|76
|40L 36s
|-
|77
|39L 38s
|-
|78
|38L 40s
|-
|79
|37L 42s
|-
|80
|36L 44s
|-
|81
|35L 46s
|-
|82
|34L 48s
|-
|83
|33L 50s
|-
|84
|32L 52s
|-
|85
|31L 54s
|-
|86
|30L 56s
|-
|87
|29L 58s
|-
|88
|28L 60s
|-
|89
|27L 62s
|-
|90
|26L 64s
|-
|91
|25L 66s
|-
|92
|24L 68s
|}


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
=== Prime harmonics ===
{{Harmonics in equal|116}}
 
=== Subsets and supersets ===
Since 116 factors into {{factorisation|116}}, 116edo has subset edos {{EDOs| 2, 4, 29, and 58 }}. [[232edo]], which doubles it, is a notable tuning.
 
== Intervals ==
{{Interval table}}
 
[[Category:Submajor (temperament)]]

Latest revision as of 19:24, 20 February 2025

← 115edo 116edo 117edo →
Prime factorization 22 × 29
Step size 10.3448 ¢ 
Fifth 68\116 (703.448 ¢) (→ 17\29)
Semitones (A1:m2) 12:8 (124.1 ¢ : 82.76 ¢)
Consistency limit 5
Distinct consistency limit 5

116 equal divisions of the octave (abbreviated 116edo or 116ed2), also called 116-tone equal temperament (116tet) or 116 equal temperament (116et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 116 equal parts of about 10.3 ¢ each. Each step represents a frequency ratio of 21/116, or the 116th root of 2.

116edo is only consistent to the 5-odd-limit, and is not quite accurate for its size. It can be viewed as splitting 58edo's step in two, and the enfactored 116cef val comes out on top accuracy in the 7-, 11-, and 13-limit. In the 5-limit, however, the patent val ⟨116 184 269] beats the enfactored 116c val ⟨116 184 270] by a thin margin, and it tempers out 20000/19683 (tetracot comma) and 2197265625/2147483648 (wizard comma).

In the 7-, 11- and 13-limit, the patent val ⟨116 184 269 326 401 429] comes in second best after the enfactored 116cef val ⟨116 184 270 326 402 430] , and it tempers out 225/224, 15625/15309, and 51200/50421 in the 7-limit; 385/384, 540/539, 4000/3993, and 6655/6561 in the 11-limit; 169/168, 275/273, 352/351, and 640/637 in the 13-limit. 116edo provides the optimal patent val for the submajor temperament in the 11- and 13-limit.

Prime harmonics

Approximation of prime harmonics in 116edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +1.49 -3.56 +3.59 -3.04 -2.60 -1.51 +2.49 +2.76 +4.91 +3.24
Relative (%) +0.0 +14.4 -34.4 +34.7 -29.4 -25.1 -14.6 +24.0 +26.7 +47.4 +31.3
Steps
(reduced) 		116
(0) 		184
(68) 		269
(37) 		326
(94) 		401
(53) 		429
(81) 		474
(10) 		493
(29) 		525
(61) 		564
(100) 		575
(111)

Subsets and supersets

Since 116 factors into 22 × 29, 116edo has subset edos 2, 4, 29, and 58. 232edo, which doubles it, is a notable tuning.

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 10.3 ^D, ^5E♭♭
2 20.7 ^^D, v6E♭
3 31 ^3D, v5E♭
4 41.4 41/40, 44/43 ^4D, v4E♭
5 51.7 33/32, 34/33, 35/34 ^5D, v3E♭
6 62.1 28/27, 29/28 ^6D, vvE♭
7 72.4 24/23 v5D♯, vE♭
8 82.8 43/41 v4D♯, E♭
9 93.1 19/18, 39/37 v3D♯, ^E♭
10 103.4 17/16, 35/33 vvD♯, ^^E♭
11 113.8 16/15, 31/29, 47/44 vD♯, ^3E♭
12 124.1 29/27, 43/40, 44/41 D♯, ^4E♭
13 134.5 40/37 ^D♯, ^5E♭
14 144.8 37/34 ^^D♯, v6E
15 155.2 35/32, 47/43 ^3D♯, v5E
16 165.5 11/10 ^4D♯, v4E
17 175.9 31/28, 41/37 ^5D♯, v3E
18 186.2 39/35 ^6D♯, vvE
19 196.6 37/33 v5D𝄪, vE
20 206.9 44/39 E
21 217.2 17/15 ^E, ^5F♭
22 227.6 ^^E, v6F
23 237.9 31/27, 39/34, 47/41 ^3E, v5F
24 248.3 15/13 ^4E, v4F
25 258.6 36/31, 43/37 ^5E, v3F
26 269 ^6E, vvF
27 279.3 20/17, 27/23, 47/40 v5E♯, vF
28 289.7 13/11 F
29 300 44/37 ^F, ^5G♭♭
30 310.3 ^^F, v6G♭
31 320.7 ^3F, v5G♭
32 331 23/19, 40/33 ^4F, v4G♭
33 341.4 28/23, 39/32 ^5F, v3G♭
34 351.7 38/31 ^6F, vvG♭
35 362.1 37/30 v5F♯, vG♭
36 372.4 36/29 v4F♯, G♭
37 382.8 v3F♯, ^G♭
38 393.1 vvF♯, ^^G♭
39 403.4 24/19 vF♯, ^3G♭
40 413.8 33/26, 47/37 F♯, ^4G♭
41 424.1 23/18 ^F♯, ^5G♭
42 434.5 9/7 ^^F♯, v6G
43 444.8 22/17, 31/24 ^3F♯, v5G
44 455.2 13/10 ^4F♯, v4G
45 465.5 17/13 ^5F♯, v3G
46 475.9 ^6F♯, vvG
47 486.2 45/34 v5F𝄪, vG
48 496.6 4/3 G
49 506.9 ^G, ^5A♭♭
50 517.2 31/23 ^^G, v6A♭
51 527.6 19/14, 42/31 ^3G, v5A♭
52 537.9 15/11 ^4G, v4A♭
53 548.3 48/35 ^5G, v3A♭
54 558.6 29/21, 47/34 ^6G, vvA♭
55 569 v5G♯, vA♭
56 579.3 v4G♯, A♭
57 589.7 38/27, 45/32 v3G♯, ^A♭
58 600 vvG♯, ^^A♭
59 610.3 27/19, 37/26, 47/33 vG♯, ^3A♭
60 620.7 G♯, ^4A♭
61 631 ^G♯, ^5A♭
62 641.4 42/29 ^^G♯, v6A
63 651.7 35/24 ^3G♯, v5A
64 662.1 22/15 ^4G♯, v4A
65 672.4 28/19, 31/21 ^5G♯, v3A
66 682.8 46/31 ^6G♯, vvA
67 693.1 v5G𝄪, vA
68 703.4 3/2 A
69 713.8 ^A, ^5B♭♭
70 724.1 ^^A, v6B♭
71 734.5 26/17 ^3A, v5B♭
72 744.8 20/13 ^4A, v4B♭
73 755.2 17/11, 48/31 ^5A, v3B♭
74 765.5 14/9 ^6A, vvB♭
75 775.9 36/23, 47/30 v5A♯, vB♭
76 786.2 v4A♯, B♭
77 796.6 19/12 v3A♯, ^B♭
78 806.9 vvA♯, ^^B♭
79 817.2 vA♯, ^3B♭
80 827.6 29/18 A♯, ^4B♭
81 837.9 ^A♯, ^5B♭
82 848.3 31/19 ^^A♯, v6B
83 858.6 23/14 ^3A♯, v5B
84 869 33/20, 38/23, 43/26 ^4A♯, v4B
85 879.3 ^5A♯, v3B
86 889.7 ^6A♯, vvB
87 900 37/22 v5A𝄪, vB
88 910.3 22/13 B
89 920.7 17/10, 46/27 ^B, ^5C♭
90 931 ^^B, v6C
91 941.4 31/18 ^3B, v5C
92 951.7 26/15, 45/26 ^4B, v4C
93 962.1 ^5B, v3C
94 972.4 ^6B, vvC
95 982.8 30/17 v5B♯, vC
96 993.1 39/22 C
97 1003.4 ^C, ^5D♭♭
98 1013.8 ^^C, v6D♭
99 1024.1 47/26 ^3C, v5D♭
100 1034.5 20/11 ^4C, v4D♭
101 1044.8 ^5C, v3D♭
102 1055.2 ^6C, vvD♭
103 1065.5 37/20 v5C♯, vD♭
104 1075.9 41/22 v4C♯, D♭
105 1086.2 15/8 v3C♯, ^D♭
106 1096.6 32/17 vvC♯, ^^D♭
107 1106.9 36/19 vC♯, ^3D♭
108 1117.2 C♯, ^4D♭
109 1127.6 23/12 ^C♯, ^5D♭
110 1137.9 27/14 ^^C♯, v6D
111 1148.3 33/17 ^3C♯, v5D
112 1158.6 43/22 ^4C♯, v4D
113 1169 ^5C♯, v3D
114 1179.3 ^6C♯, vvD
115 1189.7 v5C𝄪, vD
116 1200 2/1 D
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