88edo

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← 87edo88edo89edo →
Prime factorization 23 × 11
Step size 13.6364¢ 
Fifth 51\88 (695.455¢)
Semitones (A1:m2) 5:9 (68.18¢ : 122.7¢)
Dual sharp fifth 52\88 (709.091¢) (→13\22)
Dual flat fifth 51\88 (695.455¢)
Dual major 2nd 15\88 (204.545¢)
Consistency limit 7
Distinct consistency limit 7

88 equal divisions of the octave (abbreviated 88edo or 88ed2), also called 88-tone equal temperament (88tet) or 88 equal temperament (88et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 88 equal parts of about 13.6 ¢ each. Each step represents a frequency ratio of 21/88, or the 88th root of 2.

Theory

Using two different approximations to the perfect fifth (one of 51 steps and one of 52 steps), 88edo is compatible with both meantone and the particular variety of superpyth supported by 22 equal temperament, respectively. The meantone fifth is 0.0384 cents flatter than that of Lucy Tuning and, thus, audibly indistinguishable from it. It also gives the optimal patent val for the 11-limit mothra and euterpe temperaments.

Odd harmonics

Approximation of odd harmonics in 88edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -6.50 -4.50 -0.64 +0.64 -5.86 +4.93 +2.64 +4.14 +2.49 +6.49 -1.00
Relative (%) -47.7 -33.0 -4.7 +4.7 -43.0 +36.1 +19.4 +30.3 +18.2 +47.6 -7.3
Steps
(reduced)
139
(51)
204
(28)
247
(71)
279
(15)
304
(40)
326
(62)
344
(80)
360
(8)
374
(22)
387
(35)
398
(46)

Subsets and supersets

Since 88 factors into 23 × 11, 88edo has subset edos 2, 4, 8, 11, 22, and 44. 176edo, which doubles it, provides correction for the approximation to harmonic 3.

Interval table

Steps Cents Approximate Ratios Ups and Downs Notation
(Dual Flat Fifth 51\88)
Ups and Downs Notation
(Dual Sharp Fifth 52\88)
0 0 1/1 D D
1 13.636 ^D, v3E♭♭ ^D, v3E♭
2 27.273 ^^D, vvE♭♭ ^^D, vvE♭
3 40.909 41/40, 42/41 ^3D, vE♭♭ ^3D, vE♭
4 54.545 32/31 ^4D, E♭♭ ^4D, E♭
5 68.182 25/24 D♯, v4E♭ ^5D, v11E
6 81.818 21/20, 22/21 ^D♯, v3E♭ ^6D, v10E
7 95.455 37/35 ^^D♯, vvE♭ ^7D, v9E
8 109.091 ^3D♯, vE♭ ^8D, v8E
9 122.727 ^4D♯, E♭ ^9D, v7E
10 136.364 40/37 D𝄪, v4E ^10D, v6E
11 150 12/11 ^D𝄪, v3E ^11D, v5E
12 163.636 11/10 ^^D𝄪, vvE D♯, v4E
13 177.273 31/28, 41/37 ^3D𝄪, vE ^D♯, v3E
14 190.909 19/17, 29/26 E ^^D♯, vvE
15 204.545 ^E, v3F♭ ^3D♯, vE
16 218.182 42/37 ^^E, vvF♭ E
17 231.818 8/7 ^3E, vF♭ ^E, v3F
18 245.455 ^4E, F♭ ^^E, vvF
19 259.091 E♯, v4F ^3E, vF
20 272.727 34/29, 41/35 ^E♯, v3F F
21 286.364 ^^E♯, vvF ^F, v3G♭
22 300 19/16, 25/21 ^3E♯, vF ^^F, vvG♭
23 313.636 6/5 F ^3F, vG♭
24 327.273 ^F, v3G♭♭ ^4F, G♭
25 340.909 28/23, 39/32 ^^F, vvG♭♭ ^5F, v11G
26 354.545 38/31 ^3F, vG♭♭ ^6F, v10G
27 368.182 ^4F, G♭♭ ^7F, v9G
28 381.818 F♯, v4G♭ ^8F, v8G
29 395.455 39/31 ^F♯, v3G♭ ^9F, v7G
30 409.091 ^^F♯, vvG♭ ^10F, v6G
31 422.727 ^3F♯, vG♭ ^11F, v5G
32 436.364 ^4F♯, G♭ F♯, v4G
33 450 F𝄪, v4G ^F♯, v3G
34 463.636 17/13 ^F𝄪, v3G ^^F♯, vvG
35 477.273 ^^F𝄪, vvG ^3F♯, vG
36 490.909 ^3F𝄪, vG G
37 504.545 G ^G, v3A♭
38 518.182 31/23 ^G, v3A♭♭ ^^G, vvA♭
39 531.818 ^^G, vvA♭♭ ^3G, vA♭
40 545.455 26/19 ^3G, vA♭♭ ^4G, A♭
41 559.091 ^4G, A♭♭ ^5G, v11A
42 572.727 32/23, 39/28 G♯, v4A♭ ^6G, v10A
43 586.364 ^G♯, v3A♭ ^7G, v9A
44 600 ^^G♯, vvA♭ ^8G, v8A
45 613.636 ^3G♯, vA♭ ^9G, v7A
46 627.273 23/16 ^4G♯, A♭ ^10G, v6A
47 640.909 G𝄪, v4A ^11G, v5A
48 654.545 19/13, 35/24 ^G𝄪, v3A G♯, v4A
49 668.182 ^^G𝄪, vvA ^G♯, v3A
50 681.818 ^3G𝄪, vA ^^G♯, vvA
51 695.455 A ^3G♯, vA
52 709.091 ^A, v3B♭♭ A
53 722.727 ^^A, vvB♭♭ ^A, v3B♭
54 736.364 26/17 ^3A, vB♭♭ ^^A, vvB♭
55 750 37/24 ^4A, B♭♭ ^3A, vB♭
56 763.636 A♯, v4B♭ ^4A, B♭
57 777.273 ^A♯, v3B♭ ^5A, v11B
58 790.909 ^^A♯, vvB♭ ^6A, v10B
59 804.545 35/22 ^3A♯, vB♭ ^7A, v9B
60 818.182 ^4A♯, B♭ ^8A, v8B
61 831.818 A𝄪, v4B ^9A, v7B
62 845.455 31/19 ^A𝄪, v3B ^10A, v6B
63 859.091 23/14, 41/25 ^^A𝄪, vvB ^11A, v5B
64 872.727 ^3A𝄪, vB A♯, v4B
65 886.364 5/3 B ^A♯, v3B
66 900 32/19, 37/22, 42/25 ^B, v3C♭ ^^A♯, vvB
67 913.636 39/23 ^^B, vvC♭ ^3A♯, vB
68 927.273 29/17, 41/24 ^3B, vC♭ B
69 940.909 ^4B, C♭ ^B, v3C
70 954.545 B♯, v4C ^^B, vvC
71 968.182 7/4 ^B♯, v3C ^3B, vC
72 981.818 37/21 ^^B♯, vvC C
73 995.455 ^3B♯, vC ^C, v3D♭
74 1009.091 34/19 C ^^C, vvD♭
75 1022.727 ^C, v3D♭♭ ^3C, vD♭
76 1036.364 20/11 ^^C, vvD♭♭ ^4C, D♭
77 1050 11/6 ^3C, vD♭♭ ^5C, v11D
78 1063.636 37/20 ^4C, D♭♭ ^6C, v10D
79 1077.273 41/22 C♯, v4D♭ ^7C, v9D
80 1090.909 ^C♯, v3D♭ ^8C, v8D
81 1104.545 ^^C♯, vvD♭ ^9C, v7D
82 1118.182 21/11, 40/21 ^3C♯, vD♭ ^10C, v6D
83 1131.818 ^4C♯, D♭ ^11C, v5D
84 1145.455 31/16 C𝄪, v4D C♯, v4D
85 1159.091 41/21 ^C𝄪, v3D ^C♯, v3D
86 1172.727 ^^C𝄪, vvD ^^C♯, vvD
87 1186.364 ^3C𝄪, vD ^3C♯, vD
88 1200 2/1 D D