# 108edo

 ← 107edo 108edo 109edo →
Prime factorization 22 × 33
Step size 11.1111¢
Fifth 63\108 (700¢) (→7\12)
Semitones (A1:m2) 9:9 (100¢ : 100¢)
Consistency limit 7
Distinct consistency limit 7

108 equal divisions of the octave (abbreviated 108edo or 108ed2), also called 108-tone equal temperament (108tet) or 108 equal temperament (108et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 108 equal parts of about 11.1 ¢ each. Each step represents a frequency ratio of 21/108, or the 108th root of 2.

## Theory

108et tempers out the Pythagorean comma in the 3-limit and 1990656/1953125 (valentine comma) in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting valentine, and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and 1344/1331, and the 108e val tempers out 121/120, 176/175, 385/384, and 441/440, supporting 11-limit valentine for which it is again a good tuning.

108edo is the smallest 12n-edo which offers a decent alternative fifth to 12edo's fifth, that is 27edo's superpyth fifth.

### Prime harmonics

Approximation of prime harmonics in 108edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -1.96 +2.58 -2.16 +4.24 +3.92 -4.96 +2.49 +5.06 +3.76 -0.59
Relative (%) +0.0 -17.6 +23.2 -19.4 +38.1 +35.3 -44.6 +22.4 +45.5 +33.8 -5.3
Steps
(reduced)
108
(0)
171
(63)
251
(35)
303
(87)
374
(50)
400
(76)
441
(9)
459
(27)
489
(57)
525
(93)
535
(103)

### Subsets and supersets

Since 108 factors into 22 × 33, 108edo has subset edos 2, 3, 4, 6, 9, 12, 18, 27, 36, and 54.

## Intervals

Steps Cents Approximate Ratios Ups and Downs Notation
0 0 1/1 D
1 11.111 ^D, v8E♭
2 22.222 66/65, 78/77 ^^D, v7E♭
3 33.333 49/48 ^3D, v6E♭
4 44.444 36/35, 40/39, 77/75 ^4D, v5E♭
5 55.556 33/32 ^5D, v4E♭
6 66.667 26/25, 28/27, 80/77 ^6D, v3E♭
7 77.778 ^7D, vvE♭
8 88.889 ^8D, vE♭
9 100 35/33, 55/52 D♯, E♭
10 111.111 16/15 ^D♯, v8E
11 122.222 15/14 ^^D♯, v7E
12 133.333 ^3D♯, v6E
13 144.444 49/45 ^4D♯, v5E
14 155.556 35/32 ^5D♯, v4E
15 166.667 11/10, 54/49 ^6D♯, v3E
16 177.778 ^7D♯, vvE
17 188.889 39/35 ^8D♯, vE
18 200 9/8 E
19 211.111 44/39 ^E, v8F
20 222.222 25/22 ^^E, v7F
21 233.333 8/7 ^3E, v6F
22 244.444 15/13 ^4E, v5F
23 255.556 ^5E, v4F
24 266.667 7/6 ^6E, v3F
25 277.778 75/64 ^7E, vvF
26 288.889 13/11, 33/28, 77/65 ^8E, vF
27 300 F
28 311.111 ^F, v8G♭
29 322.222 77/64 ^^F, v7G♭
30 333.333 40/33 ^3F, v6G♭
31 344.444 39/32 ^4F, v5G♭
32 355.556 16/13 ^5F, v4G♭
33 366.667 ^6F, v3G♭
34 377.778 56/45 ^7F, vvG♭
35 388.889 5/4 ^8F, vG♭
36 400 44/35 F♯, G♭
37 411.111 33/26 ^F♯, v8G
38 422.222 ^^F♯, v7G
39 433.333 9/7, 50/39, 77/60 ^3F♯, v6G
40 444.444 ^4F♯, v5G
41 455.556 13/10 ^5F♯, v4G
42 466.667 21/16, 64/49 ^6F♯, v3G
43 477.778 33/25 ^7F♯, vvG
44 488.889 ^8F♯, vG
45 500 4/3 G
46 511.111 35/26 ^G, v8A♭
47 522.222 ^^G, v7A♭
48 533.333 15/11, 49/36 ^3G, v6A♭
49 544.444 48/35 ^4G, v5A♭
50 555.556 11/8 ^5G, v4A♭
51 566.667 ^6G, v3A♭
52 577.778 39/28 ^7G, vvA♭
53 588.889 45/32 ^8G, vA♭
54 600 G♯, A♭
55 611.111 64/45 ^G♯, v8A
56 622.222 56/39 ^^G♯, v7A
57 633.333 75/52 ^3G♯, v6A
58 644.444 16/11 ^4G♯, v5A
59 655.556 35/24 ^5G♯, v4A
60 666.667 22/15, 72/49 ^6G♯, v3A
61 677.778 65/44, 77/52 ^7G♯, vvA
62 688.889 52/35 ^8G♯, vA
63 700 3/2 A
64 711.111 ^A, v8B♭
65 722.222 50/33 ^^A, v7B♭
66 733.333 32/21, 49/32 ^3A, v6B♭
67 744.444 20/13, 77/50 ^4A, v5B♭
68 755.556 ^5A, v4B♭
69 766.667 14/9, 39/25 ^6A, v3B♭
70 777.778 ^7A, vvB♭
71 788.889 52/33 ^8A, vB♭
72 800 35/22 A♯, B♭
73 811.111 8/5 ^A♯, v8B
74 822.222 45/28, 77/48 ^^A♯, v7B
75 833.333 ^3A♯, v6B
76 844.444 13/8 ^4A♯, v5B
77 855.556 64/39 ^5A♯, v4B
78 866.667 33/20, 81/49 ^6A♯, v3B
79 877.778 ^7A♯, vvB
80 888.889 ^8A♯, vB
81 900 B
82 911.111 22/13, 56/33 ^B, v8C
83 922.222 75/44 ^^B, v7C
84 933.333 12/7, 77/45 ^3B, v6C
85 944.444 ^4B, v5C
86 955.556 26/15 ^5B, v4C
87 966.667 7/4 ^6B, v3C
88 977.778 44/25 ^7B, vvC
89 988.889 39/22 ^8B, vC
90 1000 16/9 C
91 1011.111 70/39 ^C, v8D♭
92 1022.222 ^^C, v7D♭
93 1033.333 20/11, 49/27 ^3C, v6D♭
94 1044.444 64/35 ^4C, v5D♭
95 1055.556 ^5C, v4D♭
96 1066.667 ^6C, v3D♭
97 1077.778 28/15 ^7C, vvD♭
98 1088.889 15/8 ^8C, vD♭
99 1100 66/35 C♯, D♭
100 1111.111 ^C♯, v8D
101 1122.222 ^^C♯, v7D
102 1133.333 25/13, 27/14, 77/40 ^3C♯, v6D
103 1144.444 64/33 ^4C♯, v5D
104 1155.556 35/18, 39/20 ^5C♯, v4D
105 1166.667 ^6C♯, v3D
106 1177.778 65/33, 77/39 ^7C♯, vvD
107 1188.889 ^8C♯, vD
108 1200 2/1 D