81edo: Difference between revisions

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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. 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 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.

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).

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

=== Odd harmonics ===

81 EDO Accidentals created and used by Tom Winspear.

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

== Notation ==

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.

[[File:81 EDO Accidentals.png]]

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 ===

~~{{primes in edo|81}}~~

== ~~Comma list~~ ==

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

* 7-limit commas: 81/80, 126/125, {{monzo|-24 1 0 8}}

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
+{{EDO intro|81}}
+== 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. 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.
+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 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.
 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).
-[[File:81 EDO Accidentals.png|thumb|
+=== Odd harmonics ===
-EDO Accidentals created and used by Tom Winspear.
+{{harmonics in equal|81}}
-Based on those provided in Scala though with a logic correction.
+== Notation ==
-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.
+[[File:81 EDO Accidentals.png]]
+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 ===
-{{primes in edo|81}}
-== Comma list ==
 * 5-limit commas: 81/80, {{monzo|-48 1 20}}
 * 7-limit commas: 81/80, 126/125, {{monzo|-24 1 0 8}}