90edo

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← 89edo90edo91edo →
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. It tempers out 2048/2025 in the 5-limit, 3125/3087 and 245/243 in the 7-limit, 121/120 and 176/175 in the 11-limit, and 275/273 and 169/168 in the 13-limit. It provides the optimal patent val for the 31&90 temperament in the 7-, 11- and 13-limits. Notably, it is the second lowest in a series of four consecutive EDOs to temper out 117440512/117406179.

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 Ups and Downs Notation
(Dual Flat Fifth 52\90)
Ups and Downs Notation
(Dual Sharp Fifth 53\90)
Approximate Ratios
0 0 D D 1/1
1 13.3333 ^D, vE♭♭♭ ^D, v4E♭
2 26.6667 ^^D, E♭♭♭ ^^D, v3E♭ 65/64, 66/65, 78/77
3 40 ^3D, v3E♭♭ ^3D, vvE♭ 40/39, 49/48
4 53.3333 D♯, vvE♭♭ ^4D, vE♭ 33/32, 36/35
5 66.6667 ^D♯, vE♭♭ ^5D, E♭ 25/24, 26/25, 80/77
6 80 ^^D♯, E♭♭ ^6D, v10E
7 93.3333 ^3D♯, v3E♭ ^7D, v9E 55/52
8 106.667 D𝄪, vvE♭ ^8D, v8E 16/15, 35/33
9 120 ^D𝄪, vE♭ ^9D, v7E 15/14
10 133.333 ^^D𝄪, E♭ ^10D, v6E 13/12, 14/13
11 146.667 ^3D𝄪, v3E D♯, v5E 49/45
12 160 D♯𝄪, vvE ^D♯, v4E 11/10, 35/32
13 173.333 ^D♯𝄪, vE ^^D♯, v3E 54/49
14 186.667 E ^3D♯, vvE 39/35
15 200 ^E, vF♭♭ ^4D♯, vE 28/25
16 213.333 ^^E, F♭♭ E
17 226.667 ^3E, v3F♭ ^E, v4F 8/7
18 240 E♯, vvF♭ ^^E, v3F
19 253.333 ^E♯, vF♭ ^3E, vvF 65/56
20 266.667 ^^E♯, F♭ ^4E, vF 7/6, 64/55
21 280 ^3E♯, v3F F 33/28
22 293.333 E𝄪, vvF ^F, v4G♭ 13/11, 77/65
23 306.667 ^E𝄪, vF ^^F, v3G♭
24 320 F ^3F, vvG♭ 6/5, 77/64
25 333.333 ^F, vG♭♭♭ ^4F, vG♭ 40/33
26 346.667 ^^F, G♭♭♭ ^5F, G♭ 39/32, 60/49
27 360 ^3F, v3G♭♭ ^6F, v10G 16/13
28 373.333 F♯, vvG♭♭ ^7F, v9G 56/45
29 386.667 ^F♯, vG♭♭ ^8F, v8G 5/4
30 400 ^^F♯, G♭♭ ^9F, v7G 49/39
31 413.333 ^3F♯, v3G♭ ^10F, v6G 33/26
32 426.667 F𝄪, vvG♭ F♯, v5G 32/25, 50/39, 77/60
33 440 ^F𝄪, vG♭ ^F♯, v4G 9/7
34 453.333 ^^F𝄪, G♭ ^^F♯, v3G 13/10
35 466.667 ^3F𝄪, v3G ^3F♯, vvG
36 480 F♯𝄪, vvG ^4F♯, vG 33/25
37 493.333 ^F♯𝄪, vG G 4/3
38 506.667 G ^G, v4A♭ 75/56
39 520 ^G, vA♭♭♭ ^^G, v3A♭ 65/48
40 533.333 ^^G, A♭♭♭ ^3G, vvA♭ 49/36
41 546.667 ^3G, v3A♭♭ ^4G, vA♭ 11/8, 48/35
42 560 G♯, vvA♭♭ ^5G, A♭
43 573.333 ^G♯, vA♭♭ ^6G, v10A 39/28
44 586.667 ^^G♯, A♭♭ ^7G, v9A 7/5
45 600 ^3G♯, v3A♭ ^8G, v8A
46 613.333 G𝄪, vvA♭ ^9G, v7A 10/7
47 626.667 ^G𝄪, vA♭ ^10G, v6A 56/39
48 640 ^^G𝄪, A♭ G♯, v5A
49 653.333 ^3G𝄪, v3A ^G♯, v4A 16/11, 35/24
50 666.667 G♯𝄪, vvA ^^G♯, v3A 72/49
51 680 ^G♯𝄪, vA ^3G♯, vvA 65/44, 77/52
52 693.333 A ^4G♯, vA
53 706.667 ^A, vB♭♭♭ A 3/2
54 720 ^^A, B♭♭♭ ^A, v4B♭ 50/33
55 733.333 ^3A, v3B♭♭ ^^A, v3B♭ 75/49
56 746.667 A♯, vvB♭♭ ^3A, vvB♭ 20/13, 77/50
57 760 ^A♯, vB♭♭ ^4A, vB♭ 14/9
58 773.333 ^^A♯, B♭♭ ^5A, B♭ 25/16, 39/25
59 786.667 ^3A♯, v3B♭ ^6A, v10B 52/33
60 800 A𝄪, vvB♭ ^7A, v9B 78/49
61 813.333 ^A𝄪, vB♭ ^8A, v8B 8/5, 77/48
62 826.667 ^^A𝄪, B♭ ^9A, v7B 45/28
63 840 ^3A𝄪, v3B ^10A, v6B 13/8
64 853.333 A♯𝄪, vvB A♯, v5B 49/30, 64/39
65 866.667 ^A♯𝄪, vB ^A♯, v4B 33/20
66 880 B ^^A♯, v3B 5/3
67 893.333 ^B, vC♭♭ ^3A♯, vvB
68 906.667 ^^B, C♭♭ ^4A♯, vB 22/13
69 920 ^3B, v3C♭ B 56/33
70 933.333 B♯, vvC♭ ^B, v4C 12/7, 55/32
71 946.667 ^B♯, vC♭ ^^B, v3C
72 960 ^^B♯, C♭ ^3B, vvC
73 973.333 ^3B♯, v3C ^4B, vC 7/4
74 986.667 B𝄪, vvC C
75 1000 ^B𝄪, vC ^C, v4D♭ 25/14
76 1013.33 C ^^C, v3D♭ 70/39
77 1026.67 ^C, vD♭♭♭ ^3C, vvD♭ 49/27
78 1040 ^^C, D♭♭♭ ^4C, vD♭ 20/11, 64/35
79 1053.33 ^3C, v3D♭♭ ^5C, D♭
80 1066.67 C♯, vvD♭♭ ^6C, v10D 13/7, 24/13
81 1080 ^C♯, vD♭♭ ^7C, v9D 28/15
82 1093.33 ^^C♯, D♭♭ ^8C, v8D 15/8, 66/35
83 1106.67 ^3C♯, v3D♭ ^9C, v7D
84 1120 C𝄪, vvD♭ ^10C, v6D
85 1133.33 ^C𝄪, vD♭ C♯, v5D 25/13, 48/25, 77/40
86 1146.67 ^^C𝄪, D♭ ^C♯, v4D 35/18, 64/33
87 1160 ^3C𝄪, v3D ^^C♯, v3D 39/20
88 1173.33 C♯𝄪, vvD ^3C♯, vvD 65/33, 77/39
89 1186.67 ^C♯𝄪, vD ^4C♯, vD
90 1200 D D 2/1