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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 3<sup>4</sup>
{{ED intro}}
| Step size = 14.815¢
| Fifth = 47\81 = 696.30¢
| Major 2nd = 13\81 = 192.59¢
| Semitones = 5:8 (74.1¢:118.5¢)
| Consistency = 7
}}


== Theory ==
[[File:81 EDO structure-accidentals-notation-colour notation.png|thumb|
[[File:81 EDO structure-accidentals-notation-colour notation.png|thumb|
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.  
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''' divides the octave into 81 steps of 14.815 [[cent]]s each. It is notable as a tuning for [[meantone family|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 [[Didymus rank three family|erato]]. The electronic music pioneer [https://web.archive.org/web/20120211182601/http://daphneoram.org/2012/01/13/letter-from-yehudi-menuhin/ Daphne Oram] was interested in 81edo.
81edo is notable as a tuning for [[meantone family|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-3 temperament [[erato]]. The electronic music pioneer {{w|Daphne Oram}} was interested in 81edo<ref>[https://web.archive.org/web/20120211182601/http://daphneoram.org/2012/01/13/letter-from-yehudi-menuhin/ Letter from Yehudi Menuhin to Daphne Oram]</ref>. As a step in the [[Golden meantone]] series of edos, 81edo 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 31edo does have the best [[7/4|harmonic 7th]]). However, it is no longer [[consistent]] in the [[9-odd-limit]], as the best direct approximations of [[9/8]] and [[10/9]] are one step above and below the patent val mapping.


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 81 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]]. The 81bd val is a tuning for the [[Porcupine family#Septimal porcupine|septimal porcupine]] temperament.


[[File:81 EDO Accidentals.png|thumb|
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).
81 EDO Accidentals created and used by Tom Winspear.
 
Based on those provided in Scala though with a logic correction.
=== Odd harmonics ===
The innermost accidentals represent the 15 cent 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.  
{{Harmonics in equal|81}}
The chromatic scale can be notated utilizing only six accidentals in either direction - the rest are for enharmonics.
 
]]
=== Subsets and supersets ===
Since 81 is equal to 3<sup>4</sup>, a perfect power of 3, 81edo contains subset edos {{EDOs| 3, 9, and 27 }}.
 
== Intervals ==
{{Interval table}}
 
== Notation ==
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
{{Sharpness-sharp5-szg}}
 
=== Kite's ups and downs notation ===
81edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
{{Sharpness-sharp5a}}
 
=== Tom Winspear's notation ===
[[File:81 EDO Accidentals.png|alt=|959x959px]]
 
81edo 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. What appears to be a half-sharp is actually a fifth-sharp. 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.


{{primes in edo|81}}
== Approximation to JI ==
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|81|15}}


== Comma list ==
== Regular temperament properties ==
=== Commas ===
* 5-limit commas: 81/80, {{monzo|-48 1 20}}
* 5-limit commas: 81/80, {{monzo|-48 1 20}}
* 7-limit commas: 81/80, 126/125, {{monzo|-24 1 0 8}}
* 7-limit commas: 81/80, 126/125, {{monzo|-24 1 0 8}}
Line 34: Line 50:
* [[Leantone6]]
* [[Leantone6]]
* [[Leantone7]]
* [[Leantone7]]
* [[Leantone13]]    
* [[Leantone13]]
* [[User:BudjarnLambeth/Antechinus|Antechinus]]
 
== Instruments ==
=== Skip fretting ===
'''Skip fretting system 81 7 20''' is a [[skip fretting]] system for [[81edo]]. All examples on this page are for 7-string [[guitar]].
 
; Prime harmonics
 
1/1: string 2 open
 
2/1: string 5 fret 3
 
3/2: string 4 fret 1 and string 7 fret 4
 
5/4: string 7 fret 1
 
7/4: string 3 fret 18
 
11/8: string 3 fret 14
 
13/8: string 1 fret 11 and string 4 fret 14
 
17/16: string 2 fret 1 and string 5 fret 4
 
19/16: string 3 open
 
23/16: string 2 fret 6 and string 5 fret 9
 
29/16: string 3 fret 7 and string 6 fret 10
 
31/16: string 2 fret 11 and string 5 fret 14
 
; Chords
 
Minor 7th: 100133x
 
Dominant 7th: 10113xx
 
=== Keyboards ===
A [[Lumatone mapping for 81edo]] is available.
 
== Music ==
 
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/5O6Cyawjkd8 ''''microtonal improvisation in 81edo''] (2025)
* [https://www.youtube.com/shorts/yY0TzTiNeB8 ''81edo improv''] (2025)
 
== Notes ==


[[Category:81edo| ]] <!-- main article -->
[[Category:Equal divisions of the octave]]
[[Category:Erato]]
[[Category:Erato]]
[[Category:Golden meantone]]
[[Category:Golden meantone]]
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[[Category:Meantone]]
[[Category:Meantone]]
[[Category:Meanpop]]
[[Category:Meanpop]]
<references />

Latest revision as of 14:09, 12 May 2026

← 80edo 81edo 82edo →
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-3 temperament erato. The electronic music pioneer Daphne Oram was interested in 81edo[1]. As a step in the Golden meantone series of edos, 81edo 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 31edo does have the best harmonic 7th). However, it is no longer consistent in the 9-odd-limit, as the best direct approximations of 9/8 and 10/9 are one step above and below the patent val mapping.

Besides meantone, 81edo is a tuning for the cobalt temperament, since 81 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. The 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 3, 9, and 27.

Intervals

Steps Cents Approximate ratios Ups and downs notation
(Dual flat fifth 47\81) 		Ups and downs notation
(Dual sharp fifth 48\81)
0 0 1/1 D D
1 14.8 ^D, vvE♭♭ ^D, vvE♭
2 29.6 ^^D, vE♭♭ ^^D, vE♭
3 44.4 38/37, 39/38, 40/39 vvD♯, E♭♭ ^3D, E♭
4 59.3 29/28, 30/29, 31/30 vD♯, ^E♭♭ ^4D, ^E♭
5 74.1 23/22, 24/23 D♯, ^^E♭♭ ^5D, ^^E♭
6 88.9 20/19, 39/37 ^D♯, vvE♭ ^6D, ^3E♭
7 103.7 17/16, 35/33 ^^D♯, vE♭ v5D♯, ^4E♭
8 118.5 15/14, 31/29 vvD𝄪, E♭ v4D♯, ^5E♭
9 133.3 40/37 vD𝄪, ^E♭ v3D♯, v6E
10 148.1 12/11, 37/34 D𝄪, ^^E♭ vvD♯, v5E
11 163 11/10, 34/31 ^D𝄪, vvE vD♯, v4E
12 177.8 31/28 ^^D𝄪, vE D♯, v3E
13 192.6 19/17 E ^D♯, vvE
14 207.4 35/31 ^E, vvF♭ ^^D♯, vE
15 222.2 25/22, 33/29 ^^E, vF♭ E
16 237 39/34 vvE♯, F♭ ^E, vvF
17 251.9 22/19, 37/32 vE♯, ^F♭ ^^E, vF
18 266.7 7/6 E♯, ^^F♭ F
19 281.5 20/17, 33/28 ^E♯, vvF ^F, vvG♭
20 296.3 19/16 ^^E♯, vF ^^F, vG♭
21 311.1 F ^3F, G♭
22 325.9 29/24, 35/29 ^F, vvG♭♭ ^4F, ^G♭
23 340.7 28/23, 39/32 ^^F, vG♭♭ ^5F, ^^G♭
24 355.6 38/31 vvF♯, G♭♭ ^6F, ^3G♭
25 370.4 31/25 vF♯, ^G♭♭ v5F♯, ^4G♭
26 385.2 5/4 F♯, ^^G♭♭ v4F♯, ^5G♭
27 400 29/23, 39/31 ^F♯, vvG♭ v3F♯, v6G
28 414.8 14/11 ^^F♯, vG♭ vvF♯, v5G
29 429.6 32/25 vvF𝄪, G♭ vF♯, v4G
30 444.4 22/17, 31/24, 40/31 vF𝄪, ^G♭ F♯, v3G
31 459.3 30/23 F𝄪, ^^G♭ ^F♯, vvG
32 474.1 25/19 ^F𝄪, vvG ^^F♯, vG
33 488.9 ^^F𝄪, vG G
34 503.7 G ^G, vvA♭
35 518.5 31/23 ^G, vvA♭♭ ^^G, vA♭
36 533.3 34/25 ^^G, vA♭♭ ^3G, A♭
37 548.1 11/8 vvG♯, A♭♭ ^4G, ^A♭
38 563 vG♯, ^A♭♭ ^5G, ^^A♭
39 577.8 G♯, ^^A♭♭ ^6G, ^3A♭
40 592.6 31/22 ^G♯, vvA♭ v5G♯, ^4A♭
41 607.4 ^^G♯, vA♭ v4G♯, ^5A♭
42 622.2 33/23 vvG𝄪, A♭ v3G♯, v6A
43 637 vG𝄪, ^A♭ vvG♯, v5A
44 651.9 16/11, 35/24 G𝄪, ^^A♭ vG♯, v4A
45 666.7 25/17 ^G𝄪, vvA G♯, v3A
46 681.5 37/25 ^^G𝄪, vA ^G♯, vvA
47 696.3 A ^^G♯, vA
48 711.1 ^A, vvB♭♭ A
49 725.9 35/23, 38/25 ^^A, vB♭♭ ^A, vvB♭
50 740.7 23/15 vvA♯, B♭♭ ^^A, vB♭
51 755.6 17/11, 31/20 vA♯, ^B♭♭ ^3A, B♭
52 770.4 25/16, 39/25 A♯, ^^B♭♭ ^4A, ^B♭
53 785.2 11/7 ^A♯, vvB♭ ^5A, ^^B♭
54 800 ^^A♯, vB♭ ^6A, ^3B♭
55 814.8 8/5 vvA𝄪, B♭ v5A♯, ^4B♭
56 829.6 vA𝄪, ^B♭ v4A♯, ^5B♭
57 844.4 31/19 A𝄪, ^^B♭ v3A♯, v6B
58 859.3 23/14 ^A𝄪, vvB vvA♯, v5B
59 874.1 ^^A𝄪, vB vA♯, v4B
60 888.9 B A♯, v3B
61 903.7 32/19 ^B, vvC♭ ^A♯, vvB
62 918.5 17/10 ^^B, vC♭ ^^A♯, vB
63 933.3 12/7 vvB♯, C♭ B
64 948.1 19/11 vB♯, ^C♭ ^B, vvC
65 963 B♯, ^^C♭ ^^B, vC
66 977.8 ^B♯, vvC C
67 992.6 39/22 ^^B♯, vC ^C, vvD♭
68 1007.4 34/19 C ^^C, vD♭
69 1022.2 ^C, vvD♭♭ ^3C, D♭
70 1037 20/11, 31/17 ^^C, vD♭♭ ^4C, ^D♭
71 1051.9 11/6 vvC♯, D♭♭ ^5C, ^^D♭
72 1066.7 37/20 vC♯, ^D♭♭ ^6C, ^3D♭
73 1081.5 28/15 C♯, ^^D♭♭ v5C♯, ^4D♭
74 1096.3 32/17 ^C♯, vvD♭ v4C♯, ^5D♭
75 1111.1 19/10 ^^C♯, vD♭ v3C♯, v6D
76 1125.9 23/12 vvC𝄪, D♭ vvC♯, v5D
77 1140.7 29/15 vC𝄪, ^D♭ vC♯, v4D
78 1155.6 37/19, 39/20 C𝄪, ^^D♭ C♯, v3D
79 1170.4 ^C𝄪, vvD ^C♯, vvD
80 1185.2 ^^C𝄪, vD ^^C♯, vD
81 1200 2/1 D D

Notation

Stein–Zimmermann–Gould notation

Stein–Zimmermann–Gould notation uses sharps and flats with arrows:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
Flat symbol

Kite's ups and downs notation

81edo can also be notated with Kite's ups and downs, spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
      
         
    
Flat symbol       
              

Tom Winspear's notation

81edo 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. What appears to be a half-sharp is actually a fifth-sharp. 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.

Approximation to JI

15-odd-limit interval mappings

The following tables show how 15-odd-limit intervals are represented in 81edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 81edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
7/6, 12/7 0.204 1.4
13/9, 18/13 0.419 2.8
15/14, 28/15 0.924 6.2
5/4, 8/5 1.129 7.6
11/10, 20/11 2.041 13.8
11/6, 12/11 2.489 16.8
11/7, 14/11 2.693 18.2
11/8, 16/11 3.170 21.4
9/8, 16/9 3.497 23.6
15/11, 22/15 3.617 24.4
13/8, 16/13 3.917 26.4
15/13, 26/15 4.111 27.7
5/3, 6/5 4.530 30.6
9/5, 10/9 4.626 31.2
7/5, 10/7 4.734 32.0
13/7, 14/13 5.035 34.0
13/10, 20/13 5.045 34.1
13/12, 24/13 5.239 35.4
9/7, 14/9 5.454 36.8
3/2, 4/3 5.659 38.2
7/4, 8/7 5.863 39.6
11/9, 18/11 6.667 45.0
15/8, 16/15 6.787 45.8
13/11, 22/13 7.087 47.8
15-odd-limit intervals in 81edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
7/6, 12/7 0.204 1.4
15/14, 28/15 0.924 6.2
5/4, 8/5 1.129 7.6
11/10, 20/11 2.041 13.8
11/6, 12/11 2.489 16.8
11/7, 14/11 2.693 18.2
11/8, 16/11 3.170 21.4
15/11, 22/15 3.617 24.4
13/8, 16/13 3.917 26.4
5/3, 6/5 4.530 30.6
7/5, 10/7 4.734 32.0
13/10, 20/13 5.045 34.1
9/7, 14/9 5.454 36.8
3/2, 4/3 5.659 38.2
7/4, 8/7 5.863 39.6
15/8, 16/15 6.787 45.8
13/11, 22/13 7.087 47.8
11/9, 18/11 8.148 55.0
13/12, 24/13 9.575 64.6
13/7, 14/13 9.780 66.0
9/5, 10/9 10.189 68.8
15/13, 26/15 10.704 72.3
9/8, 16/9 11.317 76.4
13/9, 18/13 15.234 102.8

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

Instruments

Skip fretting

Skip fretting system 81 7 20 is a skip fretting system for 81edo. All examples on this page are for 7-string guitar.

Prime harmonics

1/1: string 2 open

2/1: string 5 fret 3

3/2: string 4 fret 1 and string 7 fret 4

5/4: string 7 fret 1

7/4: string 3 fret 18

11/8: string 3 fret 14

13/8: string 1 fret 11 and string 4 fret 14

17/16: string 2 fret 1 and string 5 fret 4

19/16: string 3 open

23/16: string 2 fret 6 and string 5 fret 9

29/16: string 3 fret 7 and string 6 fret 10

31/16: string 2 fret 11 and string 5 fret 14

Chords

Minor 7th: 100133x

Dominant 7th: 10113xx

Keyboards

A Lumatone mapping for 81edo is available.

Music

Bryan Deister

Notes

  1. ↑ Letter from Yehudi Menuhin to Daphne Oram
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