88edo: Difference between revisions

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→Instruments: Insert Music section after this, starting with Bryan Deister's ''microtonal improvisation in 88edo'' (2025)
 
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In music, '''88 equal temperament''' is the scale derived by dividing the octave into 88 equally large steps, 13.6363 [[cent|cent]]s in size. Using two different approximations to the perfect fifth (one of 51 steps and one of 52 steps), it is compatible with both [[Meantone|meantone temperament]] and the particular variety of [[Superpyth|superpyth]] temperament(s) supported by [[22edo|22 equal temperament]], respectively. The meantone fifth is 0.0384 cents flatter than that of [[Lucy_Tuning|Lucy Tuning]] and, thus, audibly indistinguishable from it. It also gives the [[Optimal_patent_val|optimal patent val]] for the 11-limit [[Meantone_family|mothra]] and [[Didymus_rank_three_family|euterpe]] temperaments.
{{Infobox ET}}
{{ED intro}}
 
== Theory ==
Using two different approximations to the [[3/2|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 [[22edo|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 ===
{{Harmonics in equal|88}}
{{Harmonics in equal|88}}


[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
=== Subsets and supersets ===
[[Category:Euterpe]]
Since 88 factors into {{factorization|88}}, 88edo has subset edos {{EDOs| 2, 4, 8, 11, 22, and 44 }}. [[176edo]], which doubles it, provides correction for the approximation to harmonic 3.
 
== Intervals ==
{{Interval table}}
 
== Instruments ==
=== Lumatone ===
[[Lumatone mapping for 88edo]]
=== Skip fretting ===
'''Skip fretting system 88 6 13''' is a [[skip fretting]] system for [[88edo]]. All examples on this page are for 7-string [[guitar]].
 
; Prime intervals
 
1/1: string 2 open
 
2/1: string 6 fret 6
 
3/2: string 5 fret 2
 
5/4: not easily accessible
 
7/4: string 7 fret 1
 
11/8: not easily accessible
 
13/8: string 4 fret 6
 
17/16: not easily accessible
 
19/16: not easily accessible
 
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=JQly-kX6kcM ''microtonal improvisation in 88edo''] (2025)
 
[[Category:Lucy tuning]]
[[Category:Lucy tuning]]
[[Category:Meantone]]
[[Category:Meantone]]
[[Category:Mothra]]
[[Category:Mothra]]

Latest revision as of 19:59, 30 July 2025

← 87edo 88edo 89edo →
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.

Intervals

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 13.6 ^D, ^^E♭♭♭ ^D, v3E♭
2 27.3 ^^D, vvE♭♭ ^^D, vvE♭
3 40.9 41/40, 42/41 vvD♯, vE♭♭ ^3D, vE♭
4 54.5 32/31 vD♯, E♭♭ ^4D, E♭
5 68.2 25/24 D♯, ^E♭♭ ^5D, ^E♭
6 81.8 21/20, 22/21 ^D♯, ^^E♭♭ ^6D, ^^E♭
7 95.5 37/35 ^^D♯, vvE♭ v5D♯, ^3E♭
8 109.1 vvD𝄪, vE♭ v4D♯, ^4E♭
9 122.7 vD𝄪, E♭ v3D♯, ^5E♭
10 136.4 40/37 D𝄪, ^E♭ vvD♯, v6E
11 150 12/11 ^D𝄪, ^^E♭ vD♯, v5E
12 163.6 11/10 ^^D𝄪, vvE D♯, v4E
13 177.3 31/28, 41/37 vvD♯𝄪, vE ^D♯, v3E
14 190.9 19/17, 29/26 E ^^D♯, vvE
15 204.5 ^E, ^^F♭♭ ^3D♯, vE
16 218.2 42/37 ^^E, vvF♭ E
17 231.8 8/7 vvE♯, vF♭ ^E, v3F
18 245.5 vE♯, F♭ ^^E, vvF
19 259.1 E♯, ^F♭ ^3E, vF
20 272.7 34/29, 41/35 ^E♯, ^^F♭ F
21 286.4 ^^E♯, vvF ^F, v3G♭
22 300 19/16, 25/21 vvE𝄪, vF ^^F, vvG♭
23 313.6 6/5 F ^3F, vG♭
24 327.3 ^F, ^^G♭♭♭ ^4F, G♭
25 340.9 28/23, 39/32 ^^F, vvG♭♭ ^5F, ^G♭
26 354.5 38/31 vvF♯, vG♭♭ ^6F, ^^G♭
27 368.2 vF♯, G♭♭ v5F♯, ^3G♭
28 381.8 F♯, ^G♭♭ v4F♯, ^4G♭
29 395.5 39/31 ^F♯, ^^G♭♭ v3F♯, ^5G♭
30 409.1 ^^F♯, vvG♭ vvF♯, v6G
31 422.7 vvF𝄪, vG♭ vF♯, v5G
32 436.4 vF𝄪, G♭ F♯, v4G
33 450 F𝄪, ^G♭ ^F♯, v3G
34 463.6 17/13 ^F𝄪, ^^G♭ ^^F♯, vvG
35 477.3 ^^F𝄪, vvG ^3F♯, vG
36 490.9 vvF♯𝄪, vG G
37 504.5 G ^G, v3A♭
38 518.2 31/23 ^G, ^^A♭♭♭ ^^G, vvA♭
39 531.8 ^^G, vvA♭♭ ^3G, vA♭
40 545.5 26/19 vvG♯, vA♭♭ ^4G, A♭
41 559.1 vG♯, A♭♭ ^5G, ^A♭
42 572.7 32/23, 39/28 G♯, ^A♭♭ ^6G, ^^A♭
43 586.4 ^G♯, ^^A♭♭ v5G♯, ^3A♭
44 600 ^^G♯, vvA♭ v4G♯, ^4A♭
45 613.6 vvG𝄪, vA♭ v3G♯, ^5A♭
46 627.3 23/16 vG𝄪, A♭ vvG♯, v6A
47 640.9 G𝄪, ^A♭ vG♯, v5A
48 654.5 19/13, 35/24 ^G𝄪, ^^A♭ G♯, v4A
49 668.2 ^^G𝄪, vvA ^G♯, v3A
50 681.8 vvG♯𝄪, vA ^^G♯, vvA
51 695.5 A ^3G♯, vA
52 709.1 ^A, ^^B♭♭♭ A
53 722.7 ^^A, vvB♭♭ ^A, v3B♭
54 736.4 26/17 vvA♯, vB♭♭ ^^A, vvB♭
55 750 37/24 vA♯, B♭♭ ^3A, vB♭
56 763.6 A♯, ^B♭♭ ^4A, B♭
57 777.3 ^A♯, ^^B♭♭ ^5A, ^B♭
58 790.9 ^^A♯, vvB♭ ^6A, ^^B♭
59 804.5 35/22 vvA𝄪, vB♭ v5A♯, ^3B♭
60 818.2 vA𝄪, B♭ v4A♯, ^4B♭
61 831.8 A𝄪, ^B♭ v3A♯, ^5B♭
62 845.5 31/19 ^A𝄪, ^^B♭ vvA♯, v6B
63 859.1 23/14, 41/25 ^^A𝄪, vvB vA♯, v5B
64 872.7 vvA♯𝄪, vB A♯, v4B
65 886.4 5/3 B ^A♯, v3B
66 900 32/19, 37/22, 42/25 ^B, ^^C♭♭ ^^A♯, vvB
67 913.6 39/23 ^^B, vvC♭ ^3A♯, vB
68 927.3 29/17, 41/24 vvB♯, vC♭ B
69 940.9 vB♯, C♭ ^B, v3C
70 954.5 B♯, ^C♭ ^^B, vvC
71 968.2 7/4 ^B♯, ^^C♭ ^3B, vC
72 981.8 37/21 ^^B♯, vvC C
73 995.5 vvB𝄪, vC ^C, v3D♭
74 1009.1 34/19 C ^^C, vvD♭
75 1022.7 ^C, ^^D♭♭♭ ^3C, vD♭
76 1036.4 20/11 ^^C, vvD♭♭ ^4C, D♭
77 1050 11/6 vvC♯, vD♭♭ ^5C, ^D♭
78 1063.6 37/20 vC♯, D♭♭ ^6C, ^^D♭
79 1077.3 41/22 C♯, ^D♭♭ v5C♯, ^3D♭
80 1090.9 ^C♯, ^^D♭♭ v4C♯, ^4D♭
81 1104.5 ^^C♯, vvD♭ v3C♯, ^5D♭
82 1118.2 21/11, 40/21 vvC𝄪, vD♭ vvC♯, v6D
83 1131.8 vC𝄪, D♭ vC♯, v5D
84 1145.5 31/16 C𝄪, ^D♭ C♯, v4D
85 1159.1 41/21 ^C𝄪, ^^D♭ ^C♯, v3D
86 1172.7 ^^C𝄪, vvD ^^C♯, vvD
87 1186.4 vvC♯𝄪, vD ^3C♯, vD
88 1200 2/1 D D

Instruments

Lumatone

Lumatone mapping for 88edo

Skip fretting

Skip fretting system 88 6 13 is a skip fretting system for 88edo. All examples on this page are for 7-string guitar.

Prime intervals

1/1: string 2 open

2/1: string 6 fret 6

3/2: string 5 fret 2

5/4: not easily accessible

7/4: string 7 fret 1

11/8: not easily accessible

13/8: string 4 fret 6

17/16: not easily accessible

19/16: not easily accessible

Music

Bryan Deister
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