90edo
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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
← 89edo | 90edo | 91edo → |
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.
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 | +34 | -29 | -35 | -4 | +38 | +13 | -31 | -31 | -12 | |
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, ↓E♭♭♭ | ↑D, ↓4E♭ | |
2 | 26.6667 | ↑↑D, E♭♭♭ | ↑↑D, ↓3E♭ | 65/64, 66/65, 78/77 |
3 | 40 | ↑3D, ↓3E♭♭ | ↑3D, ↓↓E♭ | 40/39, 49/48 |
4 | 53.3333 | D♯, ↓↓E♭♭ | ↑4D, ↓E♭ | 33/32, 36/35 |
5 | 66.6667 | ↑D♯, ↓E♭♭ | ↑5D, E♭ | 25/24, 26/25, 80/77 |
6 | 80 | ↑↑D♯, E♭♭ | ↑6D, ↓10E | |
7 | 93.3333 | ↑3D♯, ↓3E♭ | ↑7D, ↓9E | 55/52 |
8 | 106.667 | D𝄪, ↓↓E♭ | ↑8D, ↓8E | 16/15, 35/33 |
9 | 120 | ↑D𝄪, ↓E♭ | ↑9D, ↓7E | 15/14 |
10 | 133.333 | ↑↑D𝄪, E♭ | ↑10D, ↓6E | 13/12, 14/13 |
11 | 146.667 | ↑3D𝄪, ↓3E | D♯, ↓5E | 49/45 |
12 | 160 | D♯𝄪, ↓↓E | ↑D♯, ↓4E | 11/10, 35/32 |
13 | 173.333 | ↑D♯𝄪, ↓E | ↑↑D♯, ↓3E | 54/49 |
14 | 186.667 | E | ↑3D♯, ↓↓E | 39/35 |
15 | 200 | ↑E, ↓F♭♭ | ↑4D♯, ↓E | 28/25 |
16 | 213.333 | ↑↑E, F♭♭ | E | |
17 | 226.667 | ↑3E, ↓3F♭ | ↑E, ↓4F | 8/7 |
18 | 240 | E♯, ↓↓F♭ | ↑↑E, ↓3F | |
19 | 253.333 | ↑E♯, ↓F♭ | ↑3E, ↓↓F | 65/56 |
20 | 266.667 | ↑↑E♯, F♭ | ↑4E, ↓F | 7/6, 64/55 |
21 | 280 | ↑3E♯, ↓3F | F | 33/28 |
22 | 293.333 | E𝄪, ↓↓F | ↑F, ↓4G♭ | 13/11, 77/65 |
23 | 306.667 | ↑E𝄪, ↓F | ↑↑F, ↓3G♭ | |
24 | 320 | F | ↑3F, ↓↓G♭ | 6/5, 77/64 |
25 | 333.333 | ↑F, ↓G♭♭♭ | ↑4F, ↓G♭ | 40/33 |
26 | 346.667 | ↑↑F, G♭♭♭ | ↑5F, G♭ | 39/32, 60/49 |
27 | 360 | ↑3F, ↓3G♭♭ | ↑6F, ↓10G | 16/13 |
28 | 373.333 | F♯, ↓↓G♭♭ | ↑7F, ↓9G | 56/45 |
29 | 386.667 | ↑F♯, ↓G♭♭ | ↑8F, ↓8G | 5/4 |
30 | 400 | ↑↑F♯, G♭♭ | ↑9F, ↓7G | 49/39 |
31 | 413.333 | ↑3F♯, ↓3G♭ | ↑10F, ↓6G | 33/26 |
32 | 426.667 | F𝄪, ↓↓G♭ | F♯, ↓5G | 32/25, 50/39, 77/60 |
33 | 440 | ↑F𝄪, ↓G♭ | ↑F♯, ↓4G | 9/7 |
34 | 453.333 | ↑↑F𝄪, G♭ | ↑↑F♯, ↓3G | 13/10 |
35 | 466.667 | ↑3F𝄪, ↓3G | ↑3F♯, ↓↓G | |
36 | 480 | F♯𝄪, ↓↓G | ↑4F♯, ↓G | 33/25 |
37 | 493.333 | ↑F♯𝄪, ↓G | G | 4/3 |
38 | 506.667 | G | ↑G, ↓4A♭ | 75/56 |
39 | 520 | ↑G, ↓A♭♭♭ | ↑↑G, ↓3A♭ | 65/48 |
40 | 533.333 | ↑↑G, A♭♭♭ | ↑3G, ↓↓A♭ | 49/36 |
41 | 546.667 | ↑3G, ↓3A♭♭ | ↑4G, ↓A♭ | 11/8, 48/35 |
42 | 560 | G♯, ↓↓A♭♭ | ↑5G, A♭ | |
43 | 573.333 | ↑G♯, ↓A♭♭ | ↑6G, ↓10A | 39/28 |
44 | 586.667 | ↑↑G♯, A♭♭ | ↑7G, ↓9A | 7/5 |
45 | 600 | ↑3G♯, ↓3A♭ | ↑8G, ↓8A | |
46 | 613.333 | G𝄪, ↓↓A♭ | ↑9G, ↓7A | 10/7 |
47 | 626.667 | ↑G𝄪, ↓A♭ | ↑10G, ↓6A | 56/39 |
48 | 640 | ↑↑G𝄪, A♭ | G♯, ↓5A | |
49 | 653.333 | ↑3G𝄪, ↓3A | ↑G♯, ↓4A | 16/11, 35/24 |
50 | 666.667 | G♯𝄪, ↓↓A | ↑↑G♯, ↓3A | 72/49 |
51 | 680 | ↑G♯𝄪, ↓A | ↑3G♯, ↓↓A | 65/44, 77/52 |
52 | 693.333 | A | ↑4G♯, ↓A | |
53 | 706.667 | ↑A, ↓B♭♭♭ | A | 3/2 |
54 | 720 | ↑↑A, B♭♭♭ | ↑A, ↓4B♭ | 50/33 |
55 | 733.333 | ↑3A, ↓3B♭♭ | ↑↑A, ↓3B♭ | 75/49 |
56 | 746.667 | A♯, ↓↓B♭♭ | ↑3A, ↓↓B♭ | 20/13, 77/50 |
57 | 760 | ↑A♯, ↓B♭♭ | ↑4A, ↓B♭ | 14/9 |
58 | 773.333 | ↑↑A♯, B♭♭ | ↑5A, B♭ | 25/16, 39/25 |
59 | 786.667 | ↑3A♯, ↓3B♭ | ↑6A, ↓10B | 52/33 |
60 | 800 | A𝄪, ↓↓B♭ | ↑7A, ↓9B | 78/49 |
61 | 813.333 | ↑A𝄪, ↓B♭ | ↑8A, ↓8B | 8/5, 77/48 |
62 | 826.667 | ↑↑A𝄪, B♭ | ↑9A, ↓7B | 45/28 |
63 | 840 | ↑3A𝄪, ↓3B | ↑10A, ↓6B | 13/8 |
64 | 853.333 | A♯𝄪, ↓↓B | A♯, ↓5B | 49/30, 64/39 |
65 | 866.667 | ↑A♯𝄪, ↓B | ↑A♯, ↓4B | 33/20 |
66 | 880 | B | ↑↑A♯, ↓3B | 5/3 |
67 | 893.333 | ↑B, ↓C♭♭ | ↑3A♯, ↓↓B | |
68 | 906.667 | ↑↑B, C♭♭ | ↑4A♯, ↓B | 22/13 |
69 | 920 | ↑3B, ↓3C♭ | B | 56/33 |
70 | 933.333 | B♯, ↓↓C♭ | ↑B, ↓4C | 12/7, 55/32 |
71 | 946.667 | ↑B♯, ↓C♭ | ↑↑B, ↓3C | |
72 | 960 | ↑↑B♯, C♭ | ↑3B, ↓↓C | |
73 | 973.333 | ↑3B♯, ↓3C | ↑4B, ↓C | 7/4 |
74 | 986.667 | B𝄪, ↓↓C | C | |
75 | 1000 | ↑B𝄪, ↓C | ↑C, ↓4D♭ | 25/14 |
76 | 1013.33 | C | ↑↑C, ↓3D♭ | 70/39 |
77 | 1026.67 | ↑C, ↓D♭♭♭ | ↑3C, ↓↓D♭ | 49/27 |
78 | 1040 | ↑↑C, D♭♭♭ | ↑4C, ↓D♭ | 20/11, 64/35 |
79 | 1053.33 | ↑3C, ↓3D♭♭ | ↑5C, D♭ | |
80 | 1066.67 | C♯, ↓↓D♭♭ | ↑6C, ↓10D | 13/7, 24/13 |
81 | 1080 | ↑C♯, ↓D♭♭ | ↑7C, ↓9D | 28/15 |
82 | 1093.33 | ↑↑C♯, D♭♭ | ↑8C, ↓8D | 15/8, 66/35 |
83 | 1106.67 | ↑3C♯, ↓3D♭ | ↑9C, ↓7D | |
84 | 1120 | C𝄪, ↓↓D♭ | ↑10C, ↓6D | |
85 | 1133.33 | ↑C𝄪, ↓D♭ | C♯, ↓5D | 25/13, 48/25, 77/40 |
86 | 1146.67 | ↑↑C𝄪, D♭ | ↑C♯, ↓4D | 35/18, 64/33 |
87 | 1160 | ↑3C𝄪, ↓3D | ↑↑C♯, ↓3D | 39/20 |
88 | 1173.33 | C♯𝄪, ↓↓D | ↑3C♯, ↓↓D | 65/33, 77/39 |
89 | 1186.67 | ↑C♯𝄪, ↓D | ↑4C♯, ↓D | |
90 | 1200 | D | D | 2/1 |