87edo: Difference between revisions

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| Augmented 1sn = 9\87 = 124¢

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The '''87 equal ~~temperament~~''', ~~often abbreviated~~ '''87-~~tET~~''', '''~~87-EDO~~''', or '''~~87-ET~~''', is the ~~scale~~ derived by dividing the octave into 87 ~~equally~~-sized steps, where each step is 13.79 [[cent]]s.

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 13.79 [[cent]]s.

== Theory ==

~~87et~~ 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-limit]] [[tonality diamond]] both uniquely and [[consistent~~|consistently~~]] (see [[87edo/13-limit detempering]]), and is the smallest ~~equal temperament~~ to do so. It is a [[zeta peak integer edo]].

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]].

~~87et~~ 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 don't 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, 245/243, 1029/1024, 3136/3125, and 5120/5103.

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]].

~~87et~~ is a particularly good tuning for [[Gamelismic clan #Rodan|rodan temperament]]. The 8/7 generator of 17\87 is a remarkable 0.00062 cents sharper than the 13-limit [[POTE ~~tuning|POTE~~]] ~~generator~~ and is close to the [[11-limit]] POTE generator also. Also, the 32\87 generator for [[Kleismic family #Clyde|clyde temperament]] is 0.04455 cents sharp of the 7-limit POTE generator.

87edo is a particularly good tuning for [[Gamelismic clan #Rodan|rodan temperament]]. The 8/7 generator of 17\87 is a remarkable 0.00062 cents sharper than the 13-limit [[POTE generator]] and is close to the [[11-limit]] POTE generator also. Also, the 32\87 generator for [[Kleismic family #Clyde|clyde temperament]] is 0.04455 cents sharp of the 7-limit POTE generator.

=== Prime harmonics ===

@@ Line 7: / Line 7: @@
 | Augmented 1sn = 9\87 = 124¢
 }}
-The '''87 equal temperament''', often abbreviated '''87-tET''', '''87-EDO''', or '''87-ET''', is the scale derived by dividing the octave into 87 equally-sized steps, where each step is 13.79 [[cent]]s.
+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 13.79 [[cent]]s.
 == Theory ==
-et 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-limit]] [[tonality diamond]] both uniquely and [[consistent|consistently]] (see [[87edo/13-limit detempering]]), and is the smallest equal temperament to do so. It is a [[zeta peak integer edo]].
+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]].
-et 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 don't 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, 245/243, 1029/1024, 3136/3125, and 5120/5103.
+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]].
-et is a particularly good tuning for [[Gamelismic clan #Rodan|rodan temperament]]. The 8/7 generator of 17\87 is a remarkable 0.00062 cents sharper than the 13-limit [[POTE tuning|POTE]] generator and is close to the [[11-limit]] POTE generator also. Also, the 32\87 generator for [[Kleismic family #Clyde|clyde temperament]] is 0.04455 cents sharp of the 7-limit POTE generator.
+edo is a particularly good tuning for [[Gamelismic clan #Rodan|rodan temperament]]. The 8/7 generator of 17\87 is a remarkable 0.00062 cents sharper than the 13-limit [[POTE generator]] and is close to the [[11-limit]] POTE generator also. Also, the 32\87 generator for [[Kleismic family #Clyde|clyde temperament]] is 0.04455 cents sharp of the 7-limit POTE generator.
 === Prime harmonics ===