87edo: Difference between revisions

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~~The '''~~87 equal divisions of the octave''' ('''87edo'''), or the '''87(-tone) equal temperament''' ('''87tet''', '''87et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 87 [[equal]]ly-sized steps, where each step is about 13.8 [[cent]]s.

== Theory ==

87edo is solid as both a [[13-limit]] (or [[15-odd-limit]]) and as a [[5-limit]] system, and of course does well enough in any limit in between. It represents the [[13-odd-limit]] [[tonality diamond]] both uniquely and [[consistent]]ly (see [[87edo/13-limit detempering]]), and is the smallest edo to do so. It is a [[zeta peak integer edo]].

87edo also shows some potential in limits beyond 13. The next four prime harmonics 17, 19, 23 and 29 are all near-critically sharp, but the feature of it is that the overtones and undertones are distinct, and most intervals are usable as long as they ~~don't~~ combine with 7, which is flat. Actually, as a no-sevens system, it is consistent in the 33-odd-limit.

87edo also shows some potential in limits beyond 13. The next four prime harmonics 17, 19, 23 and 29 are all near-critically sharp, but the feature of it is that the overtones and undertones are distinct, and most intervals are usable as long as they do not combine with 7, which is flat. Actually, as a no-sevens system, it is consistent in the 33-odd-limit.

87et [[tempering out|tempers out]] [[196/195]], [[325/324]], [[352/351]], [[364/363]], [[385/384]], [[441/440]], [[625/624]], [[676/675]], and [[1001/1000]] as well as the [[29-comma]], {{val| 46 -29 }}, the [[misty comma]], {{val| 26 -12 -3 }}, the [[kleisma]], 15625/15552, in addition to [[245/243]], [[1029/1024]], [[3136/3125]], and [[5120/5103]].

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=== Prime harmonics ===

== Intervals ==

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== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" | Subgroup

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Comma list|Comma List]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~stretch~~ (¢)

! rowspan="2" | Optimal 8ve Stretch (¢)

! colspan="2" | Tuning ~~error~~

! colspan="2" | Tuning Error

|-

! [[TE error|Absolute]] (¢)

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=== 13-limit detempering ===

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+ Table of rank-2 temperaments by generator

|-

! Periods per ~~octave~~

! Periods per 8ve

! Generator (Reduced)

! Cents (Reduced)

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== Music ==

* [http://www.archive.org/details/Pianodactyl Pianodactyl] [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play~~] by [[Gene Ward Smith]~~]

; [[Gene Ward Smith]]

* [http://www.archive.org/details/Pianodactyl ''Pianodactyl''] [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play]

[[Category:~~87edo| ]] ~~

[[Category:Zeta|##]]

~~[[Category:Equal divisions of the octave~~|##]]

[[Category:Listen]]

[[Category:Clyde]]

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[[Category:Rodan]]

[[Category:Tritikleismic]]

~~[[Category:Zeta]]~~

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-The '''87 equal divisions of the octave''' ('''87edo'''), or the '''87(-tone) equal temperament''' ('''87tet''', '''87et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 87 [[equal]]ly-sized steps, where each step is about 13.8 [[cent]]s.
+{{EDO intro|87}}
 == Theory ==
 edo is solid as both a [[13-limit]] (or [[15-odd-limit]]) and as a [[5-limit]] system, and of course does well enough in any limit in between. It represents the [[13-odd-limit]] [[tonality diamond]] both uniquely and [[consistent]]ly (see [[87edo/13-limit detempering]]), and is the smallest edo to do so. It is a [[zeta peak integer edo]].
-edo also shows some potential in limits beyond 13. The next four prime harmonics 17, 19, 23 and 29 are all near-critically sharp, but the feature of it is that the overtones and undertones are distinct, and most intervals are usable as long as they don't combine with 7, which is flat. Actually, as a no-sevens system, it is consistent in the 33-odd-limit.
+edo also shows some potential in limits beyond 13. The next four prime harmonics 17, 19, 23 and 29 are all near-critically sharp, but the feature of it is that the overtones and undertones are distinct, and most intervals are usable as long as they do not combine with 7, which is flat. Actually, as a no-sevens system, it is consistent in the 33-odd-limit.
 et [[tempering out|tempers out]] [[196/195]], [[325/324]], [[352/351]], [[364/363]], [[385/384]], [[441/440]], [[625/624]], [[676/675]], and [[1001/1000]] as well as the [[29-comma]], {{val| 46 -29 }}, the [[misty comma]], {{val| 26 -12 -3 }}, the [[kleisma]], 15625/15552, in addition to [[245/243]], [[1029/1024]], [[3136/3125]], and [[5120/5103]].
@@ Line 12: / Line 12: @@
 === Prime harmonics ===
-{{Primes in edo|87|columns=13}}
+{{Harmonics in equal|87|columns=13}}
 == Intervals ==
@@ Line 342: / Line 342: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" | Subgroup
+! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Comma list|Comma List]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
-! colspan="2" | Tuning error
+! colspan="2" | Tuning Error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 395: / Line 395: @@
 === 13-limit detempering ===
+{{Main|87edo/13-limit detempering}}
-{{main|87edo/13-limit detempering}}
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
 |+ Table of rank-2 temperaments by generator
 |-
-! Periods<br> per octave
+! Periods<br>per 8ve
 ! Generator<br>(Reduced)
 ! Cents<br>(Reduced)
@@ Line 628: / Line 626: @@
 == Music ==
-* [http://www.archive.org/details/Pianodactyl Pianodactyl] [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] by [[Gene Ward Smith]]
+; [[Gene Ward Smith]]
+* [http://www.archive.org/details/Pianodactyl ''Pianodactyl''] [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play]
-[[Category:87edo| ]] <!-- main article -->
+[[Category:Zeta|##]] <!-- 2-digit number -->
-[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
 [[Category:Listen]]
 [[Category:Clyde]]
@@ Line 639: / Line 637: @@
 [[Category:Rodan]]
 [[Category:Tritikleismic]]
-[[Category:Zeta]]