81edo

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← 80edo81edo82edo →
Prime factorization 34
Step size 14.8148¢ 
Fifth 47\81 (696.296¢)
Semitones (A1:m2) 5:8 (74.07¢ : 118.5¢)
Dual sharp fifth 48\81 (711.111¢) (→16\27)
Dual flat fifth 47\81 (696.296¢)
Dual major 2nd 14\81 (207.407¢)
Consistency limit 7
Distinct consistency limit 7

81 equal divisions of the octave (abbreviated 81edo or 81ed2), also called 81-tone equal temperament (81tet) or 81 equal temperament (81et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 81 equal parts of about 14.8 ¢ each. Each step represents a frequency ratio of 21/81, or the 81st root of 2.

Theory

By Tom Winspear, utilizing the Accidents shown below. Left: Chain of 4ths/5ths , Right: Chromatic view. Black font represents the '6 accidentals deep' notation that covers the chromatic scale with enharmonics only across EF & BC. White text displays deep enharmonics in the ambiguous infrared & ultraviolet area of the colour notation.

81edo is notable as a tuning for meantone and related temperaments and is the optimal patent val for a number of them. In particular it is the optimal patent val for 5-limit meantone, 7-limit meantone, 11-limit meanpop, 13-limit meanpop, and the rank three temperament erato. The electronic music pioneer Daphne Oram was interested in 81edo. As a step in the Golden meantone series of EDOs, 81 EDO marks the point at which the series ceases to display audible changes to meantone temperament, and is also the EDO with the lowest average and most evenly spread Just-error across the scale (though 31 EDO does have the best harmonic 7th).

Besides meantone, 81edo is a tuning for the cobalt temperament, since it contains 27 as a divisor. It also tunes the unnamed 15 & 51 temperament which divides the octave into 3 equal parts, and is a member of the augmented-cloudy equivalence continuum. 81bd val is a tuning for the septimal porcupine temperament.

In the higher limits, it is a strong tuning for the 2.5.17.19 subgroup, and also can be used to map 19/17 to the meantone major second resulting from stacking of two patent val fifths (13\81).

Odd harmonics

Approximation of odd harmonics in 81edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -5.66 -1.13 -5.86 +3.50 -3.17 +3.92 -6.79 -1.25 -1.22 +3.29 -6.05
Relative (%) -38.2 -7.6 -39.6 +23.6 -21.4 +26.4 -45.8 -8.4 -8.2 +22.2 -40.9
Steps
(reduced)
128
(47)
188
(26)
227
(65)
257
(14)
280
(37)
300
(57)
316
(73)
331
(7)
344
(20)
356
(32)
366
(42)

Subsets and supersets

Since 81 is equal to 34, a perfect power of 3, 81edo contains subset edos 1, 3, 9, 27.

Intervals

Steps Cents Ups and Downs Notation
(Dual Flat Fifth 47\81)
Ups and Downs Notation
(Dual Sharp Fifth 48\81)
Approximate Ratios
0 0 D D 1/1
1 14.8148 ^D, vvE♭♭ ^D, vvE♭
2 29.6296 ^^D, vE♭♭ ^^D, vE♭ 56/55, 65/64
3 44.4444 ^3D, E♭♭ ^3D, E♭ 36/35, 40/39, 77/75
4 59.2593 ^4D, v4E♭ ^4D, v11E
5 74.0741 D♯, v3E♭ ^5D, v10E 25/24
6 88.8889 ^D♯, vvE♭ ^6D, v9E
7 103.704 ^^D♯, vE♭ ^7D, v8E 35/33
8 118.519 ^3D♯, E♭ ^8D, v7E 15/14, 77/72
9 133.333 ^4D♯, v4E ^9D, v6E
10 148.148 D𝄪, v3E ^10D, v5E 12/11, 49/45
11 162.963 ^D𝄪, vvE ^11D, v4E 11/10, 54/49
12 177.778 ^^D𝄪, vE D♯, v3E
13 192.593 E ^D♯, vvE 28/25, 39/35
14 207.407 ^E, vvF♭ ^^D♯, vE 44/39
15 222.222 ^^E, vF♭ E 25/22
16 237.037 ^3E, F♭ ^E, vvF 8/7, 55/48
17 251.852 ^4E, v4F ^^E, vF
18 266.667 E♯, v3F F 7/6, 64/55
19 281.481 ^E♯, vvF ^F, vvG♭ 33/28
20 296.296 ^^E♯, vF ^^F, vG♭
21 311.111 F ^3F, G♭ 6/5
22 325.926 ^F, vvG♭♭ ^4F, v11G
23 340.741 ^^F, vG♭♭ ^5F, v10G 39/32
24 355.556 ^3F, G♭♭ ^6F, v9G 16/13, 60/49
25 370.37 ^4F, v4G♭ ^7F, v8G
26 385.185 F♯, v3G♭ ^8F, v7G 5/4
27 400 ^F♯, vvG♭ ^9F, v6G 44/35
28 414.815 ^^F♯, vG♭ ^10F, v5G 14/11
29 429.63 ^3F♯, G♭ ^11F, v4G 9/7, 32/25, 50/39, 77/60
30 444.444 ^4F♯, v4G F♯, v3G
31 459.259 F𝄪, v3G ^F♯, vvG 13/10
32 474.074 ^F𝄪, vvG ^^F♯, vG
33 488.889 ^^F𝄪, vG G
34 503.704 G ^G, vvA♭ 4/3, 75/56
35 518.519 ^G, vvA♭♭ ^^G, vA♭ 66/49
36 533.333 ^^G, vA♭♭ ^3G, A♭ 15/11, 49/36
37 548.148 ^3G, A♭♭ ^4G, v11A 11/8, 48/35
38 562.963 ^4G, v4A♭ ^5G, v10A
39 577.778 G♯, v3A♭ ^6G, v9A 7/5, 39/28
40 592.593 ^G♯, vvA♭ ^7G, v8A 55/39
41 607.407 ^^G♯, vA♭ ^8G, v7A 78/55
42 622.222 ^3G♯, A♭ ^9G, v6A 10/7, 56/39
43 637.037 ^4G♯, v4A ^10G, v5A
44 651.852 G𝄪, v3A ^11G, v4A 16/11, 35/24
45 666.667 ^G𝄪, vvA G♯, v3A 22/15, 72/49
46 681.481 ^^G𝄪, vA ^G♯, vvA 49/33
47 696.296 A ^^G♯, vA 3/2
48 711.111 ^A, vvB♭♭ A
49 725.926 ^^A, vB♭♭ ^A, vvB♭
50 740.741 ^3A, B♭♭ ^^A, vB♭ 20/13, 75/49
51 755.556 ^4A, v4B♭ ^3A, B♭
52 770.37 A♯, v3B♭ ^4A, v11B 14/9, 25/16, 39/25
53 785.185 ^A♯, vvB♭ ^5A, v10B 11/7
54 800 ^^A♯, vB♭ ^6A, v9B 35/22
55 814.815 ^3A♯, B♭ ^7A, v8B 8/5, 77/48
56 829.63 ^4A♯, v4B ^8A, v7B
57 844.444 A𝄪, v3B ^9A, v6B 13/8, 49/30
58 859.259 ^A𝄪, vvB ^10A, v5B 64/39
59 874.074 ^^A𝄪, vB ^11A, v4B
60 888.889 B A♯, v3B 5/3
61 903.704 ^B, vvC♭ ^A♯, vvB
62 918.519 ^^B, vC♭ ^^A♯, vB 56/33, 75/44
63 933.333 ^3B, C♭ B 12/7, 55/32, 77/45
64 948.148 ^4B, v4C ^B, vvC
65 962.963 B♯, v3C ^^B, vC 7/4
66 977.778 ^B♯, vvC C 44/25
67 992.593 ^^B♯, vC ^C, vvD♭ 39/22
68 1007.41 C ^^C, vD♭ 25/14, 70/39
69 1022.22 ^C, vvD♭♭ ^3C, D♭
70 1037.04 ^^C, vD♭♭ ^4C, v11D 20/11, 49/27
71 1051.85 ^3C, D♭♭ ^5C, v10D 11/6
72 1066.67 ^4C, v4D♭ ^6C, v9D
73 1081.48 C♯, v3D♭ ^7C, v8D 28/15
74 1096.3 ^C♯, vvD♭ ^8C, v7D 66/35
75 1111.11 ^^C♯, vD♭ ^9C, v6D
76 1125.93 ^3C♯, D♭ ^10C, v5D 48/25
77 1140.74 ^4C♯, v4D ^11C, v4D
78 1155.56 C𝄪, v3D C♯, v3D 35/18, 39/20
79 1170.37 ^C𝄪, vvD ^C♯, vvD 55/28
80 1185.19 ^^C𝄪, vD ^^C♯, vD
81 1200 D D 2/1

Notation

Tom Winspear's notation

81 EDO Accidentals created and used by Tom Winspear, based on those provided in Scala though with a logic correction. The innermost accidentals represent one EDOstep, followed by two, then the bracket representing three. Conventional sharp/doublesharp/flat/doubleflat accidentals are reached in steps of five and the pattern repeats itself on them. The chromatic scale can be notated utilizing only six accidentals in either direction - the rest are for enharmonics.

Regular temperament properties

Commas

  • 5-limit commas: 81/80, [-48 1 20
  • 7-limit commas: 81/80, 126/125, [-24 1 0 8
  • 11-limit commas: 81/80, 126/125, 385/384, 12005/11979
  • 13-limit commas: 81/80, 105/104, 144/143, 196/195, 6655/6591

Scales