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{{Infobox ET}}
{{Infobox ET}}
'''109edo''' is the [[equal division of the octave]] into 109 parts of 11.009 [[cent]]s each. It [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; [[245/243]], 2401/2400 and 65625/65536 in the [[7-limit]]; [[385/384]], 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11 and 13 limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.
{{ED intro}}


109edo is the 29th [[prime EDO]].
== Theory ==
109edo [[tempering out|tempers out]] 20000/19683 ([[tetracot comma]]) in the [[5-limit]]; [[245/243]], [[2401/2400]] and [[65625/65536]] in the [[7-limit]]; [[385/384]], [[1375/1372]], and [[4000/3993]] in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11- and 13-limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.


Since 109edo has a step of 11.009 cents, it also allows one to use its MOS scales as circulating temperaments. It is also the first edo which also allows one to use an MOS scale one octave of which fills a standard piano keyboard as a circulating temperament.
109edo has an excellent [[7/1|7th harmonic]], being a denominator of [[semiconvergent]] to log<sub>2</sub>7, and it is overall a strong 2.5.7.11.19.23.31.41 [[subgroup]] tuning, with errors of less than 10% on all harmonics. Some commas it tempers out in this subgroup are 575/574, 1331/1330, 1375/1372, 2255/2254, 2300/2299, 6860/6859, 10241/10240.
{| class="wikitable"
|+Circulating temperaments in 109edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|22:21
|-
|6
|[[1L 5s]]
|19:18
|-
|7
|[[4L 3s]]
|16:15
|-
|8
|[[5L 3s]]
|14:13
|-
|9
|[[1L 8s]]
|13:12
|-
|10
|[[9L 1s]]
|11:10
|-
|11
|[[10L 1s]]
| rowspan="2" |10:9
|-
|12
|[[1L 11s]]
|-
|13
|[[4L 9s]]
|9:8
|-
|14
|[[11L 3s]]
| rowspan="2" |8:7
|-
|15
|[[4L 11s]]
|-
|16
|13L 3s
| rowspan="3" |7:6
|-
|17
|[[7L 10s]]
|-
|18
|1L 17s
|-
|19
|14L 5s
| rowspan="3" |6:5
|-
|20
|9L 11s
|-
|21
|4L 17s
|-
|22
|21L 1s
| rowspan="6" |5:4
|-
|23
|17L 6s
|-
|24
|13L 11s
|-
|25
|9L 16s
|-
|26
|5L 21s
|-
|27
|1L 26s
|-
|28
|25L 3s
| rowspan="9" |4:3
|-
|29
|22L 7s
|-
|30
|19L 11s
|-
|31
|16L 15s
|-
|32
|13L 19s
|-
|33
|10L 23s
|-
|34
|7L 27s
|-
|35
|4L 31s
|-
|36
|1L 35s
|-
|37
|35L 2s
| rowspan="18" |3:2
|-
|38
|33L 5s
|-
|39
|31L 8s
|-
|40
|29L 11s
|-
|41
|27L 14s
|-
|42
|25L 17s
|-
|43
|23L 20s
|-
|44
|21L 23s
|-
|45
|19L 26s
|-
|46
|17L 29s
|-
|47
|15L 32s
|-
|48
|13L 35s
|-
|49
|11L 38L
|-
|50
|9L 41s
|-
|51
|7L 44s
|-
|52
|5L 47s
|-
|53
|3L 50s
|-
|54
|1L 53s
|-
|55
|54L 1s
| rowspan="33" |2:1
|-
|56
|53L 3s
|-
|57
|52L 5s
|-
|58
|51L 7s
|-
|59
|50L 9s
|-
|60
|49L 11s
|-
|61
|48L 13s
|-
|62
|47L 15s
|-
|63
|46L 17s
|-
|64
|45L 19s
|-
|65
|44L 21s
|-
|66
|43L 23s
|-
|67
|42L 25s
|-
|68
|41L 27s
|-
|69
|40L 29s
|-
|70
|39L 31s
|-
|71
|38L 33s
|-
|72
|37L 35s
|-
|73
|36L 37s
|-
|74
|35L 39s
|-
|75
|34L 41s
|-
|76
|33L 43s
|-
|77
|32L 45s
|-
|78
|31L 47s
|-
|79
|30L 49s
|-
|80
|29L 51s
|-
|81
|28L 53s
|-
|82
|27L 55s
|-
|83
|26L 57s
|-
|84
|25L 59s
|-
|85
|24L 61s
|-
|86
|23L 63s
|-
|87
|22L 65s
|}


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
=== Prime harmonics ===
[[Category:Prime EDO]]
{{Harmonics in equal|109|columns=16}}
 
=== Subsets and supersets ===
109edo is the 29th [[prime edo]], following [[107edo]] and before [[113edo]]. [[436edo]], which slices each step of 109edo in four, provides correction for the approximation to harmonic 3.
 
=== Nonoctave temperaments ===
Taking every 8 degree of 109edo produces a scale extremely close to [[88cET]].
 
== Intervals ==
{{Interval table}}
 
== Music ==
; [[Francium]]
* "Teenagerges" from ''TOTMC September to December 2024'' (2024) – [https://open.spotify.com/track/4oQglJSEyp6CsL5RNWuiBy Spotify] | [https://francium223.bandcamp.com/track/teenagerges Bandcamp] | [https://www.youtube.com/watch?v=v_J71U392_k YouTube] – in Tetracot[13], 109edo tuning
* "Catbabel" from ''Microtonal Six-Dimensional Cats'' (2025) – [https://open.spotify.com/track/0T7nW3ziEFvjV8c7v1EaMB Spotify] | [https://francium223.bandcamp.com/track/catbabel Bandcamp] | [https://www.youtube.com/watch?v=gtnTdPqiTDQ YouTube]
 
== See also ==
* [[109-7-comma]]

Latest revision as of 03:56, 19 October 2025

← 108edo 109edo 110edo →
Prime factorization 109 (prime)
Step size 11.0092 ¢ 
Fifth 64\109 (704.587 ¢)
Semitones (A1:m2) 12:7 (132.1 ¢ : 77.06 ¢)
Consistency limit 7
Distinct consistency limit 7

109 equal divisions of the octave (abbreviated 109edo or 109ed2), also called 109-tone equal temperament (109tet) or 109 equal temperament (109et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 109 equal parts of about 11 ¢ each. Each step represents a frequency ratio of 21/109, or the 109th root of 2.

Theory

109edo tempers out 20000/19683 (tetracot comma) in the 5-limit; 245/243, 2401/2400 and 65625/65536 in the 7-limit; 385/384, 1375/1372, and 4000/3993 in the 11-limit. It provides the optimal patent val for 7-limit octacot temperament, and 11- and 13-limit leapweek; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo has an excellent 7th harmonic, being a denominator of semiconvergent to log27, and it is overall a strong 2.5.7.11.19.23.31.41 subgroup tuning, with errors of less than 10% on all harmonics. Some commas it tempers out in this subgroup are 575/574, 1331/1330, 1375/1372, 2255/2254, 2300/2299, 6860/6859, 10241/10240.

Prime harmonics

Approximation of prime harmonics in 109edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53
Error Absolute (¢) +0.00 +2.63 -0.99 -0.02 -0.86 -3.83 +5.14 -0.27 -0.75 +5.29 -0.08 +1.87 +0.30 -5.10 -4.96 -3.78
Relative (%) +0.0 +23.9 -9.0 -0.2 -7.8 -34.8 +46.7 -2.4 -6.8 +48.0 -0.7 +17.0 +2.7 -46.3 -45.0 -34.3
Steps
(reduced) 		109
(0) 		173
(64) 		253
(35) 		306
(88) 		377
(50) 		403
(76) 		446
(10) 		463
(27) 		493
(57) 		530
(94) 		540
(104) 		568
(23) 		584
(39) 		591
(46) 		605
(60) 		624
(79)

Subsets and supersets

109edo is the 29th prime edo, following 107edo and before 113edo. 436edo, which slices each step of 109edo in four, provides correction for the approximation to harmonic 3.

Nonoctave temperaments

Taking every 8 degree of 109edo produces a scale extremely close to 88cET.

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 11 ^D, v6E♭
2 22 ^^D, v5E♭
3 33 ^3D, v4E♭
4 44 38/37, 39/38, 40/39, 41/40 ^4D, v3E♭
5 55 31/30, 32/31, 33/32 ^5D, vvE♭
6 66.1 26/25 ^6D, vE♭
7 77.1 23/22 v5D♯, E♭
8 88.1 20/19, 41/39 v4D♯, ^E♭
9 99.1 18/17 v3D♯, ^^E♭
10 110.1 16/15, 33/31 vvD♯, ^3E♭
11 121.1 15/14, 44/41 vD♯, ^4E♭
12 132.1 41/38 D♯, ^5E♭
13 143.1 25/23, 38/35 ^D♯, v6E
14 154.1 35/32, 47/43 ^^D♯, v5E
15 165.1 11/10 ^3D♯, v4E
16 176.1 31/28, 41/37 ^4D♯, v3E
17 187.2 39/35 ^5D♯, vvE
18 198.2 28/25, 37/33, 46/41 ^6D♯, vE
19 209.2 35/31, 44/39 E
20 220.2 25/22, 42/37 ^E, v6F
21 231.2 8/7 ^^E, v5F
22 242.2 23/20, 38/33 ^3E, v4F
23 253.2 22/19, 37/32 ^4E, v3F
24 264.2 ^5E, vvF
25 275.2 34/29, 41/35 ^6E, vF
26 286.2 33/28, 46/39 F
27 297.2 19/16 ^F, v6G♭
28 308.3 37/31 ^^F, v5G♭
29 319.3 ^3F, v4G♭
30 330.3 23/19 ^4F, v3G♭
31 341.3 28/23, 39/32 ^5F, vvG♭
32 352.3 38/31 ^6F, vG♭
33 363.3 37/30 v5F♯, G♭
34 374.3 31/25, 36/29, 41/33 v4F♯, ^G♭
35 385.3 5/4 v3F♯, ^^G♭
36 396.3 39/31, 44/35 vvF♯, ^3G♭
37 407.3 19/15 vF♯, ^4G♭
38 418.3 14/11 F♯, ^5G♭
39 429.4 32/25, 41/32 ^F♯, v6G
40 440.4 40/31 ^^F♯, v5G
41 451.4 ^3F♯, v4G
42 462.4 ^4F♯, v3G
43 473.4 25/19, 46/35 ^5F♯, vvG
44 484.4 37/28, 41/31, 45/34 ^6F♯, vG
45 495.4 G
46 506.4 ^G, v6A♭
47 517.4 31/23 ^^G, v5A♭
48 528.4 19/14 ^3G, v4A♭
49 539.4 41/30 ^4G, v3A♭
50 550.5 11/8 ^5G, vvA♭
51 561.5 ^6G, vA♭
52 572.5 32/23, 39/28 v5G♯, A♭
53 583.5 7/5 v4G♯, ^A♭
54 594.5 31/22 v3G♯, ^^A♭
55 605.5 44/31 vvG♯, ^3A♭
56 616.5 10/7 vG♯, ^4A♭
57 627.5 23/16 G♯, ^5A♭
58 638.5 ^G♯, v6A
59 649.5 16/11 ^^G♯, v5A
60 660.6 41/28 ^3G♯, v4A
61 671.6 28/19 ^4G♯, v3A
62 682.6 46/31 ^5G♯, vvA
63 693.6 ^6G♯, vA
64 704.6 A
65 715.6 ^A, v6B♭
66 726.6 35/23, 38/25 ^^A, v5B♭
67 737.6 ^3A, v4B♭
68 748.6 37/24 ^4A, v3B♭
69 759.6 31/20, 45/29 ^5A, vvB♭
70 770.6 25/16, 39/25 ^6A, vB♭
71 781.7 11/7 v5A♯, B♭
72 792.7 30/19 v4A♯, ^B♭
73 803.7 35/22 v3A♯, ^^B♭
74 814.7 8/5 vvA♯, ^3B♭
75 825.7 29/18 vA♯, ^4B♭
76 836.7 A♯, ^5B♭
77 847.7 31/19 ^A♯, v6B
78 858.7 23/14 ^^A♯, v5B
79 869.7 38/23, 43/26 ^3A♯, v4B
80 880.7 ^4A♯, v3B
81 891.7 ^5A♯, vvB
82 902.8 32/19 ^6A♯, vB
83 913.8 39/23 B
84 924.8 29/17 ^B, v6C
85 935.8 ^^B, v5C
86 946.8 19/11 ^3B, v4C
87 957.8 33/19, 40/23 ^4B, v3C
88 968.8 7/4 ^5B, vvC
89 979.8 37/21, 44/25 ^6B, vC
90 990.8 39/22 C
91 1001.8 25/14, 41/23 ^C, v6D♭
92 1012.8 ^^C, v5D♭
93 1023.9 47/26 ^3C, v4D♭
94 1034.9 20/11 ^4C, v3D♭
95 1045.9 ^5C, vvD♭
96 1056.9 35/19, 46/25 ^6C, vD♭
97 1067.9 v5C♯, D♭
98 1078.9 28/15, 41/22 v4C♯, ^D♭
99 1089.9 15/8 v3C♯, ^^D♭
100 1100.9 17/9 vvC♯, ^3D♭
101 1111.9 19/10 vC♯, ^4D♭
102 1122.9 44/23 C♯, ^5D♭
103 1133.9 25/13 ^C♯, v6D
104 1145 31/16 ^^C♯, v5D
105 1156 37/19, 39/20 ^3C♯, v4D
106 1167 ^4C♯, v3D
107 1178 ^5C♯, vvD
108 1189 ^6C♯, vD
109 1200 2/1 D

Music

Francium

See also

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