110edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
'''110edo''' is the [[EDO|equal division of the octave]] into 110 parts of 10.9090909091 cents each. It tempers out 15625/15552 and 3486784401/3355443200 in the 5-limit. Using the patent val, it tempers out 1728/1715, 3125/3087, and 3645/3584 in the 7-limit.
{{ED intro}}


== Theory ==
== Theory ==
The equal temperament [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]) and 3486784401/3355443200 in the 5-limit. Using the [[patent val]], it tempers out [[1728/1715]], [[3125/3087]], and 3645/3584 in the 7-limit.
=== Odd harmonics ===
{{Harmonics in equal|110}}
{{Harmonics in equal|110}}
Since 110edo has a step of 10.9090909091 cents, it also allows one to use its MOS scales as circulating temperaments. As 10*[[11edo]], it is the first edo which allows one to use consecutive smaller edos as [[circulating temperament]]s.
{| class="wikitable"
|+Circulating temperaments in 110edo
!Tones
!Pattern
!L:s
|-
|5
|[[5edo]]
|equal
|-
|6
|[[2L 4s]]
|19:18
|-
|7
|[[5L 2s]]
|16:15
|-
|8
|[[6L 2s]]
|14:13
|-
|9
|[[2L 7s]]
|13:12
|-
|10
|[[10edo]]
| rowspan="2" |equal
|-
|11
|[[11edo]]
|-
|12
|[[2L 10s]]
|10:9
|-
|13
|[[5L 8s]]
|9:8
|-
|14
|[[12L 2s]]
| rowspan="2" |8:7
|-
|15
|[[5L 10s]]
|-
|16
|14L 2s
| rowspan="3" |7:6
|-
|17
|[[8L 9s]]
|-
|18
|2L 16s
|-
|19
|[[15L 4s]]
| rowspan="3" |6:5
|-
|20
|10L 10s
|-
|21
|[[5L 16s]]
|-
|22
|[[22edo]]
|equal
|-
|23
|18L 5s
| rowspan="5" |5:4
|-
|24
|14L 10s
|-
|25
|10L 15s
|-
|26
|6L 20s
|-
|27
|2L 25s
|-
|28
|26L 2s
| rowspan="9" |4:3
|-
|29
|23L 6s
|-
|30
|20L 10s
|-
|31
|17L 14s
|-
|32
|14L 18s
|-
|33
|11L 22s
|-
|34
|8L 26s
|-
|35
|5L 30s
|-
|36
|2L 34s
|-
|37
|36L 1s
| rowspan="18" |3:2
|-
|38
|34L 4s
|-
|39
|32L 7s
|-
|40
|30L 10s
|-
|41
|28L 13s
|-
|42
|26L 16s
|-
|43
|24L 19s
|-
|44
|22L 22s
|-
|45
|20L 25s
|-
|46
|18L 28s
|-
|47
|16L 31s
|-
|48
|14L 34s
|-
|49
|12L 37s
|-
|50
|10L 40s
|-
|51
|8L 43s
|-
|52
|6L 46s
|-
|53
|4L 49s
|-
|54
|2L 52s
|-
|55
|[[55edo]]
|equal
|-
|56
|54L 2s
| rowspan="32" |2:1
|-
|57
|53L 4s
|-
|58
|52L 6s
|-
|59
|51L 8s
|-
|60
|50L 10s
|-
|61
|49L 12s
|-
|62
|48L 14s
|-
|63
|47L 16s
|-
|64
|46L 18s
|-
|65
|45L 20s
|-
|66
|44L 22s
|-
|67
|43L 24s
|-
|68
|42L 26s
|-
|69
|41L 28s
|-
|70
|40L 30s
|-
|71
|39L 32s
|-
|72
|38L 34s
|-
|73
|37L 36s
|-
|74
|36L 38s
|-
|75
|35L 40s
|-
|76
|34L 42s
|-
|77
|33L 44s
|-
|78
|32L 46s
|-
|79
|31L 48s
|-
|80
|30L 50s
|-
|81
|29L 52s
|-
|82
|28L 54s
|-
|83
|27L 56s
|-
|84
|26L 58s
|-
|85
|25L 60s
|-
|86
|24L 62s
|-
|87
|23L 64s
|}


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
=== Subsets and supersets ===
Since 110 factors into {{factorization|110}}, 110edo has subset edos {{EDOs| 2, 5, 10, 11, 22, and 55 }}.
 
== Intervals ==
{{Interval table}}
 
== Instruments ==
* [[Lumatone mapping for 110edo]]
 
{{stub}}

Latest revision as of 03:27, 3 August 2025

← 109edo 110edo 111edo →
Prime factorization 2 × 5 × 11
Step size 10.9091 ¢ 
Fifth 64\110 (698.182 ¢) (→ 32\55)
Semitones (A1:m2) 8:10 (87.27 ¢ : 109.1 ¢)
Dual sharp fifth 65\110 (709.091 ¢) (→ 13\22)
Dual flat fifth 64\110 (698.182 ¢) (→ 32\55)
Dual major 2nd 19\110 (207.273 ¢)
Consistency limit 5
Distinct consistency limit 5

110 equal divisions of the octave (abbreviated 110edo or 110ed2), also called 110-tone equal temperament (110tet) or 110 equal temperament (110et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 110 equal parts of about 10.9 ¢ each. Each step represents a frequency ratio of 21/110, or the 110th root of 2.

Theory

The equal temperament tempers out 15625/15552 (kleisma) and 3486784401/3355443200 in the 5-limit. Using the patent val, it tempers out 1728/1715, 3125/3087, and 3645/3584 in the 7-limit.

Odd harmonics

Approximation of odd harmonics in 110edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -3.77 -4.50 +2.08 +3.36 +5.05 -0.53 +2.64 +4.14 -2.97 -1.69 +4.45
Relative (%) -34.6 -41.2 +19.1 +30.8 +46.3 -4.8 +24.2 +37.9 -27.2 -15.5 +40.8
Steps
(reduced) 		174
(64) 		255
(35) 		309
(89) 		349
(19) 		381
(51) 		407
(77) 		430
(100) 		450
(10) 		467
(27) 		483
(43) 		498
(58)

Subsets and supersets

Since 110 factors into 2 × 5 × 11, 110edo has subset edos 2, 5, 10, 11, 22, and 55.

Intervals

Steps Cents Approximate ratios Ups and downs notation
(Dual flat fifth 64\110) 		Ups and downs notation
(Dual sharp fifth 65\110)
0 0 1/1 D D
1 10.9 ^D, vE♭♭ ^D, v4E♭
2 21.8 ^^D, E♭♭ ^^D, v3E♭
3 32.7 ^3D, ^E♭♭ ^3D, vvE♭
4 43.6 39/38, 40/39, 41/40, 42/41 ^4D, ^^E♭♭ ^4D, vE♭
5 54.5 32/31, 33/32 v3D♯, ^3E♭♭ ^5D, E♭
6 65.5 vvD♯, v4E♭ ^6D, ^E♭
7 76.4 23/22 vD♯, v3E♭ ^7D, ^^E♭
8 87.3 20/19, 41/39 D♯, vvE♭ v7D♯, ^3E♭
9 98.2 37/35 ^D♯, vE♭ v6D♯, ^4E♭
10 109.1 33/31 ^^D♯, E♭ v5D♯, ^5E♭
11 120 ^3D♯, ^E♭ v4D♯, ^6E♭
12 130.9 41/38 ^4D♯, ^^E♭ v3D♯, ^7E♭
13 141.8 38/35 v3D𝄪, ^3E♭ vvD♯, v7E
14 152.7 47/43 vvD𝄪, v4E vD♯, v6E
15 163.6 vD𝄪, v3E D♯, v5E
16 174.5 21/19, 31/28 D𝄪, vvE ^D♯, v4E
17 185.5 39/35 ^D𝄪, vE ^^D♯, v3E
18 196.4 37/33, 47/42 E ^3D♯, vvE
19 207.3 ^E, vF♭ ^4D♯, vE
20 218.2 42/37 ^^E, F♭ E
21 229.1 8/7 ^3E, ^F♭ ^E, v4F
22 240 ^4E, ^^F♭ ^^E, v3F
23 250.9 37/32 v3E♯, ^3F♭ ^3E, vvF
24 261.8 43/37 vvE♯, v4F ^4E, vF
25 272.7 41/35 vE♯, v3F F
26 283.6 33/28 E♯, vvF ^F, v4G♭
27 294.5 ^E♯, vF ^^F, v3G♭
28 305.5 31/26, 37/31 F ^3F, vvG♭
29 316.4 6/5 ^F, vG♭♭ ^4F, vG♭
30 327.3 29/24, 35/29 ^^F, G♭♭ ^5F, G♭
31 338.2 17/14 ^3F, ^G♭♭ ^6F, ^G♭
32 349.1 ^4F, ^^G♭♭ ^7F, ^^G♭
33 360 16/13 v3F♯, ^3G♭♭ v7F♯, ^3G♭
34 370.9 26/21 vvF♯, v4G♭ v6F♯, ^4G♭
35 381.8 vF♯, v3G♭ v5F♯, ^5G♭
36 392.7 F♯, vvG♭ v4F♯, ^6G♭
37 403.6 24/19 ^F♯, vG♭ v3F♯, ^7G♭
38 414.5 33/26, 47/37 ^^F♯, G♭ vvF♯, v7G
39 425.5 ^3F♯, ^G♭ vF♯, v6G
40 436.4 ^4F♯, ^^G♭ F♯, v5G
41 447.3 22/17 v3F𝄪, ^3G♭ ^F♯, v4G
42 458.2 43/33 vvF𝄪, v4G ^^F♯, v3G
43 469.1 21/16, 38/29 vF𝄪, v3G ^3F♯, vvG
44 480 F𝄪, vvG ^4F♯, vG
45 490.9 ^F𝄪, vG G
46 501.8 G ^G, v4A♭
47 512.7 35/26, 39/29, 43/32 ^G, vA♭♭ ^^G, v3A♭
48 523.6 23/17, 42/31 ^^G, A♭♭ ^3G, vvA♭
49 534.5 ^3G, ^A♭♭ ^4G, vA♭
50 545.5 ^4G, ^^A♭♭ ^5G, A♭
51 556.4 40/29 v3G♯, ^3A♭♭ ^6G, ^A♭
52 567.3 25/18, 43/31 vvG♯, v4A♭ ^7G, ^^A♭
53 578.2 vG♯, v3A♭ v7G♯, ^3A♭
54 589.1 G♯, vvA♭ v6G♯, ^4A♭
55 600 41/29 ^G♯, vA♭ v5G♯, ^5A♭
56 610.9 37/26, 47/33 ^^G♯, A♭ v4G♯, ^6A♭
57 621.8 ^3G♯, ^A♭ v3G♯, ^7A♭
58 632.7 36/25 ^4G♯, ^^A♭ vvG♯, v7A
59 643.6 29/20 v3G𝄪, ^3A♭ vG♯, v6A
60 654.5 35/24 vvG𝄪, v4A G♯, v5A
61 665.5 47/32 vG𝄪, v3A ^G♯, v4A
62 676.4 31/21, 34/23 G𝄪, vvA ^^G♯, v3A
63 687.3 ^G𝄪, vA ^3G♯, vvA
64 698.2 A ^4G♯, vA
65 709.1 ^A, vB♭♭ A
66 720 47/31 ^^A, B♭♭ ^A, v4B♭
67 730.9 29/19, 32/21 ^3A, ^B♭♭ ^^A, v3B♭
68 741.8 43/28 ^4A, ^^B♭♭ ^3A, vvB♭
69 752.7 17/11 v3A♯, ^3B♭♭ ^4A, vB♭
70 763.6 vvA♯, v4B♭ ^5A, B♭
71 774.5 vA♯, v3B♭ ^6A, ^B♭
72 785.5 A♯, vvB♭ ^7A, ^^B♭
73 796.4 19/12 ^A♯, vB♭ v7A♯, ^3B♭
74 807.3 ^^A♯, B♭ v6A♯, ^4B♭
75 818.2 ^3A♯, ^B♭ v5A♯, ^5B♭
76 829.1 21/13 ^4A♯, ^^B♭ v4A♯, ^6B♭
77 840 13/8 v3A𝄪, ^3B♭ v3A♯, ^7B♭
78 850.9 vvA𝄪, v4B vvA♯, v7B
79 861.8 28/17 vA𝄪, v3B vA♯, v6B
80 872.7 43/26 A𝄪, vvB A♯, v5B
81 883.6 5/3 ^A𝄪, vB ^A♯, v4B
82 894.5 47/28 B ^^A♯, v3B
83 905.5 ^B, vC♭ ^3A♯, vvB
84 916.4 ^^B, C♭ ^4A♯, vB
85 927.3 41/24 ^3B, ^C♭ B
86 938.2 ^4B, ^^C♭ ^B, v4C
87 949.1 v3B♯, ^3C♭ ^^B, v3C
88 960 vvB♯, v4C ^3B, vvC
89 970.9 7/4 vB♯, v3C ^4B, vC
90 981.8 37/21 B♯, vvC C
91 992.7 ^B♯, vC ^C, v4D♭
92 1003.6 C ^^C, v3D♭
93 1014.5 ^C, vD♭♭ ^3C, vvD♭
94 1025.5 38/21, 47/26 ^^C, D♭♭ ^4C, vD♭
95 1036.4 ^3C, ^D♭♭ ^5C, D♭
96 1047.3 ^4C, ^^D♭♭ ^6C, ^D♭
97 1058.2 35/19 v3C♯, ^3D♭♭ ^7C, ^^D♭
98 1069.1 vvC♯, v4D♭ v7C♯, ^3D♭
99 1080 vC♯, v3D♭ v6C♯, ^4D♭
100 1090.9 C♯, vvD♭ v5C♯, ^5D♭
101 1101.8 ^C♯, vD♭ v4C♯, ^6D♭
102 1112.7 19/10 ^^C♯, D♭ v3C♯, ^7D♭
103 1123.6 44/23 ^3C♯, ^D♭ vvC♯, v7D
104 1134.5 ^4C♯, ^^D♭ vC♯, v6D
105 1145.5 31/16 v3C𝄪, ^3D♭ C♯, v5D
106 1156.4 39/20, 41/21 vvC𝄪, v4D ^C♯, v4D
107 1167.3 vC𝄪, v3D ^^C♯, v3D
108 1178.2 C𝄪, vvD ^3C♯, vvD
109 1189.1 ^C𝄪, vD ^4C♯, vD
110 1200 2/1 D D

Instruments

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