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{{Infobox ET}}
{{Infobox ET}}
'''114edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 114 parts, each of 10.52632 [[cent]]s. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] 2048/2025, in the [[7-limit|7-limit]] 245/243, in the [[11-limit|11-limit]] 121/120, 176/175 and [[Quartisma|117440512/117406179]], in the [[13-limit|13-limit]] 196/195 and 325/324, in the [[17-limit|17-limit]] 136/135 and 154/153, in the [[19-limit|19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic_family|shrutar temperament]]; it is in fact the [[Optimal_patent_val|optimal patent val]] for [[Shrutar|shrutar]] in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.
{{ED intro}}


===Period of 19-limit Shrutar===
== Theory ==
In the [[5-limit]] the equal temperament [[tempering out|tempers out]] 2048/2025 ([[diaschisma]]), in the [[7-limit]] [[245/243]], in the [[11-limit]] [[121/120]], [[176/175]] and notably the [[quartisma]], in the [[13-limit]] [[196/195]] and [[325/324]], in the [[17-limit]] [[136/135]] and [[154/153]], in the [[19-limit]] [[286/285]] and [[343/342]]. These commas make for 114edo being an excellent tuning for the [[shrutar]] temperament; it is in fact the [[optimal patent val]] for shrutar in the 11-, 13-, 17-, and 19-limit, as well as the rank-3 [[bisector]] temperament.


{| class="wikitable"
=== Odd harmonics ===
|-
{{Harmonics in equal|114}}
! |Degree
 
! |Cents
=== Subsets and supersets ===
!Difference from 68edo
Since 114 factors into {{factorization|114}}, 114 edo has subset edos {{EDOs| 2, 3, 6, 19, 38, and 57 }}.
|-
 
| |2
== Intervals ==
| |21.05263
{{Interval table}}
|3.40557¢
|-
| |3
| |31.57895
| -3.71517¢
|-
| | 5
| | 52.63158
| -0.3096¢
|-
| |7
| |73.68421
|3.096¢
|-
| |8
| |84.21053
| -4.02477¢
|-
| |10
| |105.26316
|  -0.619195¢
|-
| |12
| | 126.31579
|2.78638¢
|-
| |13
| |136.842105
|  -4.334365¢
|-
| |15
| |157.89474
| -0.9288¢
|-
| | 17
| |178.94737
|2.47678¢
|-
| |18
| | 189.47369
| -4.644¢
|-
| |20
| |210.52632
| -1.23839¢
|-
| |22
| |231.57895
|2.16718¢
|-
| |23
| |242.10526
|  -4.953560372
|-
| |25
| |263.157895
| -1.548¢
|-
| |27
| |284.21053
|1.857585¢
|-
| |29
| |305.26316
|5.26316¢
|-
| |30
| |315.78947
| -1.857585¢
|-
| |32
| |336.842105
|1.548¢
|-
| |34
| |357.89474
|4.95356¢
|-
| |35
| | 368.42105
| -2.16718¢
|-
| |37
| |389.47368
|1.23839¢
|-
| |39
| |410.52632
|4.64396¢
|-
| |40
| |421.05263
| -2.47678¢
|-
| |42
| |442.10526
|0.92879¢
|-
| |44
| |463.157895
|4.334365¢
|-
| |45
| |473.68421
| -2.78638¢
|-
| |47
| |494.73684
|0.619195¢
|-
| |49
| |515.78947
|4.02477¢
|-
| |50
| |526.31579
| -3.095975¢
|-
| |52
| |547.36842
|0.3096¢
|-
| |54
| |568.42105
|3.71517¢
|-
| |55
| |578.94737
| -3.40557¢
|}
Since 114edo has a step of 10.52632 cents, it also allows one to use its MOS scales as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 114edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|23:22
|-
|6
|[[6edo]]
|equal
|-
|7
|[[2L 5s]]
|17:16
|-
|8
|[[2L 6s]]
|15:14
|-
|9
|[[6L 3s]]
|13:12
|-
|10
|[[4L 6s]]
|12:11
|-
|11
|[[4L 7s]]
|11:10
|-
|12
|[[6L 6s]]
|10:9
|-
|13
|[[10L 3s]]
| rowspan="2" |9:8
|-
|14
|[[2L 12s]]
|-
|15
|[[9L 6s]]
| rowspan="2" |8:7
|-
|16
|2L 14s
|-
|17
|[[12L 5s]]
| rowspan="2" |7:6
|-
|18
|6L 12s
|-
|19
|[[19edo]]
|equal
|-
|20
|14L 6s
| rowspan="3" |6:5
|-
|21
|9L 12s
|-
|22
|4L 18s
|-
|23
|22L 1s
| rowspan="6" |5:4
|-
|24
|18L 6s
|-
|25
|14L 11s
|-
|26
|10L 16s
|-
|27
|6L 21s
|-
|28
|2L 26s
|-
|29
|27L 2s
| rowspan="9" |4:3
|-
|30
|24L 6s
|-
|31
|21L 10s
|-
|32
|18L 14s
|-
|33
|15L 18s
|-
|34
|12L 22s
|-
|35
|9L 26s
|-
|36
|6L 30s
|-
|37
|3L 34s
|-
|38
|[[38edo]]
|equal
|-
|39
|36L 3s
| rowspan="18" |3:2
|-
|40
|34L 6s
|-
|41
|32L 9s
|-
|42
|30L 12s
|-
|43
|28L 15s
|-
|44
|26L 18s
|-
|45
|24L 21s
|-
|46
|22L 24s
|-
|47
|20L 27s
|-
|48
|18L 30L
|-
|49
|16L 33s
|-
|50
|14L 36s
|-
|51
|12L 39s
|-
|52
|10L 42s
|-
|53
|8L 45s
|-
|54
|6L 48s
|-
|55
|4L 52s
|-
|56
|2L 54s
|-
|57
|[[57edo]]
|equal
|-
|58
|56L 2s
| rowspan="34" |2:1
|-
|59
|55L 4s
|-
|60
|54L 6s
|-
|61
|53L 8s
|-
|62
|52L 10s
|-
|63
|51L 12s
|-
|64
|50L 14s
|-
|65
|49L 16s
|-
|66
|48L 18s
|-
|67
|47L 20s
|-
|68
|46L 22s
|-
|69
|45L 24s
|-
|70
|44L 26s
|-
|71
|43L 28s
|-
|72
|42L 30s
|-
|73
|41L 32s
|-
|74
|40L 34s
|-
|75
|39L 36s
|-
|76
|38L 38s
|-
|77
|37L 40s
|-
|78
|36L 42s
|-
|79
|35L 44s
|-
|80
|34L 46s
|-
|81
|33L 48s
|-
|82
|32L 50s
|-
|83
|31L 52s
|-
|84
|30L 54s
|-
|85
|29L 56s
|-
|86
|28L 58s
|-
|87
|27L 60s
|-
|88
|26L 62s
|-
|89
|25L 64s
|-
|90
|24L 66s
|-
|91
|23L 68s
|}


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Shrutar]]
[[Category:Shrutar]]
[[Category:Bisector]]

Latest revision as of 16:51, 18 February 2025

← 113edo 114edo 115edo →
Prime factorization 2 × 3 × 19
Step size 10.5263 ¢ 
Fifth 67\114 (705.263 ¢)
Semitones (A1:m2) 13:7 (136.8 ¢ : 73.68 ¢)
Consistency limit 7
Distinct consistency limit 7

114 equal divisions of the octave (abbreviated 114edo or 114ed2), also called 114-tone equal temperament (114tet) or 114 equal temperament (114et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 114 equal parts of about 10.5 ¢ each. Each step represents a frequency ratio of 21/114, or the 114th root of 2.

Theory

In the 5-limit the equal temperament tempers out 2048/2025 (diaschisma), in the 7-limit 245/243, in the 11-limit 121/120, 176/175 and notably the quartisma, in the 13-limit 196/195 and 325/324, in the 17-limit 136/135 and 154/153, in the 19-limit 286/285 and 343/342. These commas make for 114edo being an excellent tuning for the shrutar temperament; it is in fact the optimal patent val for shrutar in the 11-, 13-, 17-, and 19-limit, as well as the rank-3 bisector temperament.

Odd harmonics

Approximation of odd harmonics in 114edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +3.31 +3.16 -0.40 -3.91 -3.95 +1.58 -4.06 +0.31 -2.78 +2.90 +3.30
Relative (%) +31.4 +30.0 -3.8 -37.1 -37.5 +15.0 -38.6 +2.9 -26.4 +27.6 +31.4
Steps
(reduced) 		181
(67) 		265
(37) 		320
(92) 		361
(19) 		394
(52) 		422
(80) 		445
(103) 		466
(10) 		484
(28) 		501
(45) 		516
(60)

Subsets and supersets

Since 114 factors into 2 × 3 × 19, 114 edo has subset edos 2, 3, 6, 19, 38, and 57.

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 10.5 ^D, v6E♭
2 21.1 ^^D, v5E♭
3 31.6 ^3D, v4E♭
4 42.1 40/39, 41/40, 42/41, 43/42 ^4D, v3E♭
5 52.6 33/32, 34/33 ^5D, vvE♭
6 63.2 ^6D, vE♭
7 73.7 24/23 v6D♯, E♭
8 84.2 21/20, 43/41 v5D♯, ^E♭
9 94.7 37/35 v4D♯, ^^E♭
10 105.3 17/16 v3D♯, ^3E♭
11 115.8 31/29, 46/43, 47/44 vvD♯, ^4E♭
12 126.3 14/13, 43/40 vD♯, ^5E♭
13 136.8 13/12, 40/37 D♯, ^6E♭
14 147.4 37/34 ^D♯, v6E
15 157.9 23/21, 34/31 ^^D♯, v5E
16 168.4 32/29, 43/39 ^3D♯, v4E
17 178.9 41/37 ^4D♯, v3E
18 189.5 29/26, 48/43 ^5D♯, vvE
19 200 37/33, 46/41 ^6D♯, vE
20 210.5 26/23, 35/31 E
21 221.1 42/37 ^E, v6F
22 231.6 8/7 ^^E, v5F
23 242.1 23/20 ^3E, v4F
24 252.6 22/19, 37/32 ^4E, v3F
25 263.2 ^5E, vvF
26 273.7 34/29, 41/35, 48/41 ^6E, vF
27 284.2 33/28, 46/39 F
28 294.7 ^F, v6G♭
29 305.3 31/26, 37/31 ^^F, v5G♭
30 315.8 6/5 ^3F, v4G♭
31 326.3 29/24, 35/29 ^4F, v3G♭
32 336.8 17/14 ^5F, vvG♭
33 347.4 ^6F, vG♭
34 357.9 16/13, 43/35 v6F♯, G♭
35 368.4 26/21, 47/38 v5F♯, ^G♭
36 378.9 46/37 v4F♯, ^^G♭
37 389.5 v3F♯, ^3G♭
38 400 29/23 vvF♯, ^4G♭
39 410.5 vF♯, ^5G♭
40 421.1 37/29 F♯, ^6G♭
41 431.6 ^F♯, v6G
42 442.1 31/24, 40/31 ^^F♯, v5G
43 452.6 13/10, 48/37 ^3F♯, v4G
44 463.2 17/13 ^4F♯, v3G
45 473.7 46/35 ^5F♯, vvG
46 484.2 37/28, 41/31 ^6F♯, vG
47 494.7 G
48 505.3 ^G, v6A♭
49 515.8 31/23, 35/26 ^^G, v5A♭
50 526.3 42/31 ^3G, v4A♭
51 536.8 ^4G, v3A♭
52 547.4 48/35 ^5G, vvA♭
53 557.9 29/21, 40/29 ^6G, vA♭
54 568.4 25/18, 43/31 v6G♯, A♭
55 578.9 v5G♯, ^A♭
56 589.5 v4G♯, ^^A♭
57 600 41/29 v3G♯, ^3A♭
58 610.5 37/26, 47/33 vvG♯, ^4A♭
59 621.1 vG♯, ^5A♭
60 631.6 36/25 G♯, ^6A♭
61 642.1 29/20, 42/29 ^G♯, v6A
62 652.6 35/24 ^^G♯, v5A
63 663.2 ^3G♯, v4A
64 673.7 31/21 ^4G♯, v3A
65 684.2 46/31 ^5G♯, vvA
66 694.7 ^6G♯, vA
67 705.3 A
68 715.8 ^A, v6B♭
69 726.3 35/23 ^^A, v5B♭
70 736.8 26/17 ^3A, v4B♭
71 747.4 20/13, 37/24 ^4A, v3B♭
72 757.9 31/20, 48/31 ^5A, vvB♭
73 768.4 39/25 ^6A, vB♭
74 778.9 v6A♯, B♭
75 789.5 41/26 v5A♯, ^B♭
76 800 46/29 v4A♯, ^^B♭
77 810.5 v3A♯, ^3B♭
78 821.1 37/23 vvA♯, ^4B♭
79 831.6 21/13 vA♯, ^5B♭
80 842.1 13/8 A♯, ^6B♭
81 852.6 ^A♯, v6B
82 863.2 28/17 ^^A♯, v5B
83 873.7 48/29 ^3A♯, v4B
84 884.2 5/3 ^4A♯, v3B
85 894.7 47/28 ^5A♯, vvB
86 905.3 ^6A♯, vB
87 915.8 39/23 B
88 926.3 29/17, 41/24 ^B, v6C
89 936.8 43/25 ^^B, v5C
90 947.4 19/11 ^3B, v4C
91 957.9 40/23 ^4B, v3C
92 968.4 7/4 ^5B, vvC
93 978.9 37/21 ^6B, vC
94 989.5 23/13 C
95 1000 41/23 ^C, v6D♭
96 1010.5 43/24 ^^C, v5D♭
97 1021.1 ^3C, v4D♭
98 1031.6 29/16 ^4C, v3D♭
99 1042.1 31/17, 42/23 ^5C, vvD♭
100 1052.6 ^6C, vD♭
101 1063.2 24/13, 37/20 v6C♯, D♭
102 1073.7 13/7 v5C♯, ^D♭
103 1084.2 43/23 v4C♯, ^^D♭
104 1094.7 32/17 v3C♯, ^3D♭
105 1105.3 vvC♯, ^4D♭
106 1115.8 40/21 vC♯, ^5D♭
107 1126.3 23/12 C♯, ^6D♭
108 1136.8 ^C♯, v6D
109 1147.4 33/17 ^^C♯, v5D
110 1157.9 39/20, 41/21 ^3C♯, v4D
111 1168.4 ^4C♯, v3D
112 1178.9 ^5C♯, vvD
113 1189.5 ^6C♯, vD
114 1200 2/1 D
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