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== Theory ==

[[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

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 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.

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.

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 ===

=== 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 ==

== Notation ==

===Ups and downs notation===

81edo can be notated with [[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).

Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:

=== 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.

== Regular temperament properties ==

=== Commas ===

* 5-limit commas: 81/80, {{monzo|-48 1 20}}

* 7-limit commas: 81/80, 126/125, {{monzo|-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 ==

* [[Leantone6]]

* [[Leantone7]]

* [[Leantone13]]

== 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

~~Black font represents the '6 accidentals deep' notation that covers the chromatic scale. White text displays enharmonics in the ambiguous infrared & ultraviolet area of the colour notation.~~

19/16: string 3 open

]]

''81edo'' divides the octave into 81 steps of 14.815 cents each. It is notable as a tuning for [[Meantone_family|meantone]] and related temperaments and is the [[Optimal_patent_val|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 [http:~~//daphneoram.org/2012/01/13/letter-from-yehudi-menuhin/ 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).~~

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

~~[[File~~:~~81 EDO Accidentals.png|thumb|~~

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

~~81 EDO Accidentals created~~ and ~~used by Tom Winspear.~~

~~Based on those provided in Scala though with a logic correction.~~

~~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.~~

~~The chromatic scale can be notated utilizing only six accidentals in either direction - the rest are for enharmonics.~~

]]

~~=Commas=~~

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

~~5-limit commas: 81/80, |-48 1 20&gt~~;

; Chords

~~7-limit commas~~: ~~81/80, 126/125, |-24 1 0 8>~~

Minor 7th: 100133x

~~11-limit commas~~: ~~81/80, 126/125, 385/384, 12005/11979~~

Dominant 7th: 10113xx

~~13-limit commas: 81/80, 105/104, 144/143, 196/195, 6655/6591~~

=== Keyboards ===

A [[Lumatone mapping for 81edo]] is available.

=~~Scales~~=

== Music ==

[[~~leantone6|leantone6~~]]

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/5O6Cyawjkd8 ''''microtonal improvisation in 81edo''] (2025)

~~[[leantone7|leantone7]]~~

== Notes ==

~~[[leantone13|leantone13]]~~

[[Category:Erato]]

[[Category:~~81edo~~]]

[[Category:Golden meantone]]

[[Category:~~edo~~]]

[[Category:Leantone]]

[[Category:~~erato~~]]

[[Category:Meantone]]

[[Category:~~meanpop~~]]

[[Category:Meanpop]]

[[Category:~~meantone~~]]

~~[[Category:theory]]~~

@@ Line 1: / Line 1: @@
+{{Infobox ET}}
+{{ED intro}}
+== Theory ==
 [[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
+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.
+]]
+edo 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.
+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.
+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 ===
+{{Harmonics in equal|81}}
+=== 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 ==
+===Ups and downs notation===
+edo can be notated with [[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}}
+Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
+{{Sharpness-sharp5}}
+=== Tom Winspear's notation ===
+[[File:81 EDO Accidentals.png|alt=|959x959px]]
+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. 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.
+== Regular temperament properties ==
+=== Commas ===
+* 5-limit commas: 81/80, {{monzo|-48 1 20}}
+* 7-limit commas: 81/80, 126/125, {{monzo|-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 ==
+* [[Leantone6]]
+* [[Leantone7]]
+* [[Leantone13]]
+== 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: string 2 open
+/1: string 5 fret 3
+/2: string 4 fret 1 and string 7 fret 4
+/4: string 7 fret 1
+/4: string 3 fret 18
+/8: string 3 fret 14
+/8: string 1 fret 11 and string 4 fret 14
+/16: string 2 fret 1 and string 5 fret 4
-Black font represents the '6 accidentals deep' notation that covers the chromatic scale. White text displays enharmonics in the ambiguous infrared & ultraviolet area of the colour notation.
+/16: string 3 open
-]]
-''81edo'' divides the octave into 81 steps of 14.815 cents each. It is notable as a tuning for [[Meantone_family|meantone]] and related temperaments and is the [[Optimal_patent_val|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 [http://daphneoram.org/2012/01/13/letter-from-yehudi-menuhin/ 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).
+/16: string 2 fret 6 and string 5 fret 9
-[[File:81 EDO Accidentals.png|thumb|
+/16: string 3 fret 7 and string 6 fret 10
-EDO Accidentals created and used by Tom Winspear.
-Based on those provided in Scala though with a logic correction.
-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.
-The chromatic scale can be notated utilizing only six accidentals in either direction - the rest are for enharmonics.
-]]
-=Commas=
+/16: string 2 fret 11 and string 5 fret 14
--limit commas: 81/80, |-48 1 20&gt;
+; Chords
--limit commas: 81/80, 126/125, |-24 1 0 8&gt;
+Minor 7th: 100133x
--limit commas: 81/80, 126/125, 385/384, 12005/11979
+Dominant 7th: 10113xx
--limit commas: 81/80, 105/104, 144/143, 196/195, 6655/6591
+=== Keyboards ===
+A [[Lumatone mapping for 81edo]] is available.
-=Scales=
+== Music ==
-[[leantone6|leantone6]]
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/5O6Cyawjkd8 ''''microtonal improvisation in 81edo''] (2025)
-[[leantone7|leantone7]]
+== Notes ==
-[[leantone13|leantone13]]
+[[Category:Erato]]
-[[Category:81edo]]
+[[Category:Golden meantone]]
-[[Category:edo]]
+[[Category:Leantone]]
-[[Category:erato]]
+[[Category:Meantone]]
-[[Category:meanpop]]
+[[Category:Meanpop]]
-[[Category:meantone]]
+<references />
-[[Category:theory]]