# 109edo

 ← 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, 2255/2244, 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
Error Absolute (¢) +0.00 +2.63 -0.99 -0.02 -0.86 -3.83 +5.14 -0.27 -0.75 +5.29 -0.08
Relative (%) +0.0 +23.9 -9.0 -0.2 -7.8 -34.8 +46.7 -2.4 -6.8 +48.0 -0.7
Steps
(reduced)
109
(0)
173
(64)
253
(35)
306
(88)
377
(50)
403
(76)
446
(10)
463
(27)
493
(57)
530
(94)
540
(104)

### 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 11.009 ^D, v6E♭
2 22.018 78/77 ^^D, v5E♭
3 33.028 49/48, 50/49, 56/55 ^3D, v4E♭
4 44.037 40/39, 77/75 ^4D, v3E♭
5 55.046 33/32 ^5D, vvE♭
6 66.055 26/25, 80/77 ^6D, vE♭
7 77.064 22/21 ^7D, E♭
8 88.073 21/20 ^8D, v11E
9 99.083 35/33, 52/49, 55/52 ^9D, v10E
10 110.092 16/15 ^10D, v9E
11 121.101 15/14 ^11D, v8E
12 132.11 14/13 D♯, v7E
13 143.119 49/45 ^D♯, v6E
14 154.128 12/11, 35/32 ^^D♯, v5E
15 165.138 11/10 ^3D♯, v4E
16 176.147 ^4D♯, v3E
17 187.156 39/35, 49/44 ^5D♯, vvE
18 198.165 28/25, 55/49 ^6D♯, vE
19 209.174 44/39 E
20 220.183 25/22 ^E, v6F
21 231.193 8/7 ^^E, v5F
22 242.202 ^3E, v4F
23 253.211 ^4E, v3F
24 264.22 7/6, 64/55 ^5E, vvF
25 275.229 75/64 ^6E, vF
26 286.239 13/11, 33/28 F
27 297.248 77/65 ^F, v6G♭
28 308.257 ^^F, v5G♭
29 319.266 6/5, 77/64 ^3F, v4G♭
30 330.275 40/33 ^4F, v3G♭
31 341.284 39/32 ^5F, vvG♭
32 352.294 49/40, 60/49 ^6F, vG♭
33 363.303 16/13 ^7F, G♭
34 374.312 56/45 ^8F, v11G
35 385.321 5/4 ^9F, v10G
36 396.33 44/35, 49/39 ^10F, v9G
37 407.339 ^11F, v8G
38 418.349 14/11 F♯, v7G
39 429.358 32/25, 50/39, 77/60 ^F♯, v6G
40 440.367 ^^F♯, v5G
41 451.376 13/10 ^3F♯, v4G
42 462.385 64/49 ^4F♯, v3G
43 473.394 21/16 ^5F♯, vvG
44 484.404 33/25 ^6F♯, vG
45 495.413 4/3 G
46 506.422 75/56 ^G, v6A♭
47 517.431 35/26, 66/49 ^^G, v5A♭
48 528.44 ^3G, v4A♭
49 539.45 15/11 ^4G, v3A♭
50 550.459 11/8, 48/35 ^5G, vvA♭
51 561.468 ^6G, vA♭
52 572.477 39/28 ^7G, A♭
53 583.486 7/5 ^8G, v11A
54 594.495 45/32, 55/39 ^9G, v10A
55 605.505 64/45, 78/55 ^10G, v9A
56 616.514 10/7 ^11G, v8A
57 627.523 56/39 G♯, v7A
58 638.532 ^G♯, v6A
59 649.541 16/11, 35/24 ^^G♯, v5A
60 660.55 22/15 ^3G♯, v4A
61 671.56 65/44 ^4G♯, v3A
62 682.569 49/33, 52/35, 77/52 ^5G♯, vvA
63 693.578 ^6G♯, vA
64 704.587 3/2 A
65 715.596 50/33 ^A, v6B♭
66 726.606 32/21 ^^A, v5B♭
67 737.615 49/32, 75/49 ^3A, v4B♭
68 748.624 20/13, 77/50 ^4A, v3B♭
69 759.633 ^5A, vvB♭
70 770.642 25/16, 39/25 ^6A, vB♭
71 781.651 11/7 ^7A, B♭
72 792.661 ^8A, v11B
73 803.67 35/22, 78/49 ^9A, v10B
74 814.679 8/5, 77/48 ^10A, v9B
75 825.688 45/28 ^11A, v8B
76 836.697 13/8 A♯, v7B
77 847.706 49/30, 80/49 ^A♯, v6B
78 858.716 64/39 ^^A♯, v5B
79 869.725 33/20 ^3A♯, v4B
80 880.734 5/3 ^4A♯, v3B
81 891.743 ^5A♯, vvB
82 902.752 ^6A♯, vB
83 913.761 22/13, 56/33 B
84 924.771 75/44 ^B, v6C
85 935.78 12/7, 55/32 ^^B, v5C
86 946.789 ^3B, v4C
87 957.798 ^4B, v3C
88 968.807 7/4 ^5B, vvC
89 979.817 44/25 ^6B, vC
90 990.826 39/22 C
91 1001.835 25/14 ^C, v6D♭
92 1012.844 70/39 ^^C, v5D♭
93 1023.853 ^3C, v4D♭
94 1034.862 20/11 ^4C, v3D♭
95 1045.872 11/6, 64/35 ^5C, vvD♭
96 1056.881 ^6C, vD♭
97 1067.89 13/7 ^7C, D♭
98 1078.899 28/15 ^8C, v11D
99 1089.908 15/8 ^9C, v10D
100 1100.917 49/26, 66/35 ^10C, v9D
101 1111.927 40/21 ^11C, v8D
102 1122.936 21/11 C♯, v7D
103 1133.945 25/13, 77/40 ^C♯, v6D
104 1144.954 64/33 ^^C♯, v5D
105 1155.963 39/20 ^3C♯, v4D
106 1166.972 49/25, 55/28 ^4C♯, v3D
107 1177.982 77/39 ^5C♯, vvD
108 1188.991 ^6C♯, vD
109 1200 2/1 D