90edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 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 +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
Retrieved from "https://en.xen.wiki/index.php?title=90edo&oldid=118432"