87edo: Difference between revisions
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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. | 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. | ||
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, | 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 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. | 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. | ||
=== Prime harmonics === | |||
{{Primes in edo|87|prec=2}} | |||
== Intervals == | == Intervals == | ||
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|} | |} | ||
== | == Regular temperament properties == | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
! colspan="2" | | ! colspan="2" | | ||
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|} | |} | ||
== 13-limit detempering | === 13-limit detempering === | ||
{{main|87edo/13-limit detempering}} | {{main|87edo/13-limit detempering}} | ||
== Rank-2 temperaments == | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
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== Scales == | == Scales == | ||
=== Harmonic scale === | |||
=== Harmonic | |||
87edo accurately approximates the mode 8 of [[harmonic series]], and the only intervals not distinct are 14/13 and 15/14. It does mode 16 fairly decent, with the only anomaly at 28/27 (4 steps) and 29/28 (5 steps). | 87edo accurately approximates the mode 8 of [[harmonic series]], and the only intervals not distinct are 14/13 and 15/14. It does mode 16 fairly decent, with the only anomaly at 28/27 (4 steps) and 29/28 (5 steps). | ||
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== Music == | == Music == | ||
* [http://www.archive.org/details/Pianodactyl Pianodactyl] [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] by [[Gene Ward Smith]] | * [http://www.archive.org/details/Pianodactyl Pianodactyl] [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] by [[Gene Ward Smith]] | ||
[[Category: | [[Category:Theory]] | ||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||
[[Category:87edo]] | [[Category:87edo]] | ||
[[Category: | [[Category:Listen]] | ||
[[Category: | [[Category:Clyde]] | ||
[[Category: | [[Category:Countercata]] | ||
[[Category: | [[Category:Hemithirds]] | ||
[[Category: | [[Category:Mystery]] | ||
[[Category: | [[Category:Rodan]] | ||
[[Category: | [[Category:Tritikleismic]] | ||
[[Category:Zeta]] | [[Category:Zeta]] | ||