# 90edo

 ← 89edo 90edo 91edo →
Prime factorization 2 × 32 × 5
Step size 13.3333¢
Fifth 53\90 (706.667¢)
Semitones (A1:m2) 11:5 (146.7¢ : 66.67¢)
Dual sharp fifth 53\90 (706.667¢)
Dual flat fifth 52\90 (693.333¢) (→26\45)
Dual major 2nd 15\90 (200¢) (→1\6)
Consistency limit 7
Distinct consistency limit 7

90 equal divisions of the octave (abbreviated 90edo or 90ed2), also called 90-tone equal temperament (90tet) or 90 equal temperament (90et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 90 equal parts of about 13.3 ¢ each. Each step represents a frequency ratio of 21/90, or the 90th root of 2.

## Theory

As an equal temperament, it tempers out 2048/2025 (diaschisma) in the 5-limit, 245/243 and 3125/3087 in the 7-limit, 121/120 and 176/175 in the 11-limit, and 169/168 and 275/273 in the 13-limit. It provides the optimal patent val for the 31 & 90 temperament in the 7-, 11- and 13-limit. Notably, it is the second lowest in a series of four consecutive edos to temper out the quartisma.

### Odd harmonics

Approximation of odd harmonics in 90edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +4.71 +0.35 +4.51 -3.91 -4.65 -0.53 +5.06 +1.71 -4.18 -4.11 -1.61
Relative (%) +35.3 +2.6 +33.8 -29.3 -34.9 -4.0 +38.0 +12.8 -31.3 -30.9 -12.1
Steps
(reduced)
143
(53)
209
(29)
253
(73)
285
(15)
311
(41)
333
(63)
352
(82)
368
(8)
382
(22)
395
(35)
407
(47)

## Interval table

Steps Cents Approximate Ratios Ups and Downs Notation
(Dual Flat Fifth 52\90)
Ups and Downs Notation
(Dual Sharp Fifth 53\90)
0 0 1/1 D D
1 13.333 ^D, vE♭♭♭ ^D, v4E♭
2 26.667 65/64, 66/65, 78/77 ^^D, E♭♭♭ ^^D, v3E♭
3 40 40/39, 49/48 ^3D, v3E♭♭ ^3D, vvE♭
4 53.333 33/32, 36/35 D♯, vvE♭♭ ^4D, vE♭
5 66.667 25/24, 26/25, 80/77 ^D♯, vE♭♭ ^5D, E♭
6 80 ^^D♯, E♭♭ ^6D, v10E
7 93.333 55/52 ^3D♯, v3E♭ ^7D, v9E
8 106.667 16/15, 35/33 D𝄪, vvE♭ ^8D, v8E
9 120 15/14 ^D𝄪, vE♭ ^9D, v7E
10 133.333 13/12, 14/13 ^^D𝄪, E♭ ^10D, v6E
11 146.667 49/45 ^3D𝄪, v3E D♯, v5E
12 160 11/10, 35/32 D♯𝄪, vvE ^D♯, v4E
13 173.333 54/49 ^D♯𝄪, vE ^^D♯, v3E
14 186.667 39/35 E ^3D♯, vvE
15 200 28/25 ^E, vF♭♭ ^4D♯, vE
16 213.333 ^^E, F♭♭ E
17 226.667 8/7 ^3E, v3F♭ ^E, v4F
18 240 E♯, vvF♭ ^^E, v3F
19 253.333 65/56 ^E♯, vF♭ ^3E, vvF
20 266.667 7/6, 64/55 ^^E♯, F♭ ^4E, vF
21 280 33/28 ^3E♯, v3F F
22 293.333 13/11, 77/65 E𝄪, vvF ^F, v4G♭
23 306.667 ^E𝄪, vF ^^F, v3G♭
24 320 6/5, 77/64 F ^3F, vvG♭
25 333.333 40/33 ^F, vG♭♭♭ ^4F, vG♭
26 346.667 39/32, 60/49 ^^F, G♭♭♭ ^5F, G♭
27 360 16/13 ^3F, v3G♭♭ ^6F, v10G
28 373.333 56/45 F♯, vvG♭♭ ^7F, v9G
29 386.667 5/4 ^F♯, vG♭♭ ^8F, v8G
30 400 49/39 ^^F♯, G♭♭ ^9F, v7G
31 413.333 33/26 ^3F♯, v3G♭ ^10F, v6G
32 426.667 32/25, 50/39, 77/60 F𝄪, vvG♭ F♯, v5G
33 440 9/7 ^F𝄪, vG♭ ^F♯, v4G
34 453.333 13/10 ^^F𝄪, G♭ ^^F♯, v3G
35 466.667 ^3F𝄪, v3G ^3F♯, vvG
36 480 33/25 F♯𝄪, vvG ^4F♯, vG
37 493.333 4/3 ^F♯𝄪, vG G
38 506.667 75/56 G ^G, v4A♭
39 520 65/48 ^G, vA♭♭♭ ^^G, v3A♭
40 533.333 49/36 ^^G, A♭♭♭ ^3G, vvA♭
41 546.667 11/8, 48/35 ^3G, v3A♭♭ ^4G, vA♭
42 560 G♯, vvA♭♭ ^5G, A♭
43 573.333 39/28 ^G♯, vA♭♭ ^6G, v10A
44 586.667 7/5 ^^G♯, A♭♭ ^7G, v9A
45 600 ^3G♯, v3A♭ ^8G, v8A
46 613.333 10/7 G𝄪, vvA♭ ^9G, v7A
47 626.667 56/39 ^G𝄪, vA♭ ^10G, v6A
48 640 ^^G𝄪, A♭ G♯, v5A
49 653.333 16/11, 35/24 ^3G𝄪, v3A ^G♯, v4A
50 666.667 72/49 G♯𝄪, vvA ^^G♯, v3A
51 680 65/44, 77/52 ^G♯𝄪, vA ^3G♯, vvA
52 693.333 A ^4G♯, vA
53 706.667 3/2 ^A, vB♭♭♭ A
54 720 50/33 ^^A, B♭♭♭ ^A, v4B♭
55 733.333 75/49 ^3A, v3B♭♭ ^^A, v3B♭
56 746.667 20/13, 77/50 A♯, vvB♭♭ ^3A, vvB♭
57 760 14/9 ^A♯, vB♭♭ ^4A, vB♭
58 773.333 25/16, 39/25 ^^A♯, B♭♭ ^5A, B♭
59 786.667 52/33 ^3A♯, v3B♭ ^6A, v10B
60 800 78/49 A𝄪, vvB♭ ^7A, v9B
61 813.333 8/5, 77/48 ^A𝄪, vB♭ ^8A, v8B
62 826.667 45/28 ^^A𝄪, B♭ ^9A, v7B
63 840 13/8 ^3A𝄪, v3B ^10A, v6B
64 853.333 49/30, 64/39 A♯𝄪, vvB A♯, v5B
65 866.667 33/20 ^A♯𝄪, vB ^A♯, v4B
66 880 5/3 B ^^A♯, v3B
67 893.333 ^B, vC♭♭ ^3A♯, vvB
68 906.667 22/13 ^^B, C♭♭ ^4A♯, vB
69 920 56/33 ^3B, v3C♭ B
70 933.333 12/7, 55/32 B♯, vvC♭ ^B, v4C
71 946.667 ^B♯, vC♭ ^^B, v3C
72 960 ^^B♯, C♭ ^3B, vvC
73 973.333 7/4 ^3B♯, v3C ^4B, vC
74 986.667 B𝄪, vvC C
75 1000 25/14 ^B𝄪, vC ^C, v4D♭
76 1013.333 70/39 C ^^C, v3D♭
77 1026.667 49/27 ^C, vD♭♭♭ ^3C, vvD♭
78 1040 20/11, 64/35 ^^C, D♭♭♭ ^4C, vD♭
79 1053.333 ^3C, v3D♭♭ ^5C, D♭
80 1066.667 13/7, 24/13 C♯, vvD♭♭ ^6C, v10D
81 1080 28/15 ^C♯, vD♭♭ ^7C, v9D
82 1093.333 15/8, 66/35 ^^C♯, D♭♭ ^8C, v8D
83 1106.667 ^3C♯, v3D♭ ^9C, v7D
84 1120 C𝄪, vvD♭ ^10C, v6D
85 1133.333 25/13, 48/25, 77/40 ^C𝄪, vD♭ C♯, v5D
86 1146.667 35/18, 64/33 ^^C𝄪, D♭ ^C♯, v4D
87 1160 39/20 ^3C𝄪, v3D ^^C♯, v3D
88 1173.333 65/33, 77/39 C♯𝄪, vvD ^3C♯, vvD
89 1186.667 ^C♯𝄪, vD ^4C♯, vD
90 1200 2/1 D D