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.

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 ~~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).

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]]).

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 [[Porcupine family#Septimal porcupine|septimal porcupine]] temperament.

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]]. 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|1, 3, 9, 27}}.

Since 81 is equal to 3<sup>4</sup>, a perfect power of 3, 81edo contains subset edos {{EDOs| 3, 9, and 27 }}.

== Intervals ==

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== Notation ==

=== Tom Winspear's notation ===

[[File:81 EDO Accidentals.png|alt=|959x959px]]

~~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.

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. 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 ===

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* [[Leantone6]]

* [[Leantone7]]

* [[Leantone13]]

== Notes ==

[[Category:Erato]]

@@ Line 4: / Line 4: @@
 == 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.
+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 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).
+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]]).
-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 [[Porcupine family#Septimal porcupine|septimal porcupine]] temperament.
+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]]. 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}}
+{{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|1, 3, 9, 27}}.
+Since 81 is equal to 3<sup>4</sup>, a perfect power of 3, 81edo contains subset edos {{EDOs| 3, 9, and 27 }}.
 == Intervals ==
@@ Line 23: / Line 23: @@
 == Notation ==
 === 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. 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.
+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 ===
@@ Line 38: / Line 38: @@
 * [[Leantone6]]
 * [[Leantone7]]
 * [[Leantone13]]
+== Notes ==
 [[Category:Erato]]