87edo: Difference between revisions
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=== Prime harmonics === | === Prime harmonics === | ||
In higher limits it excels as a [[subgroup]] temperament, especially as an incomplete 71-limit temperament with [[128/127]] and [[129/128]] (the subharmonic and harmonic hemicomma-sized intervals, respectively) mapped accurately to a single step. Generalizing a single step of 87edo harmonically yields harmonics 115 through 138, which when detempered is the beginning of the construction of [[Ringer scale|Ringer]] 87, thus tempering [[ | In higher limits it excels as a [[subgroup]] temperament, especially as an incomplete 71-limit temperament with [[128/127]] and [[129/128]] (the subharmonic and harmonic hemicomma-sized intervals, respectively) mapped accurately to a single step. Generalizing a single step of 87edo harmonically yields harmonics 115 through 138, which when detempered is the beginning of the construction of [[Ringer scale|Ringer]] 87, thus tempering [[S-expression|S116 through S137]] by patent val and corresponding to the gravity of the fact that 87edo is a circle of [[126/125]]'s, meaning ([[126/125]])<sup>87</sup> only very slightly exceeds the octave. | ||
{{Harmonics in equal|87|columns=12}} | {{Harmonics in equal|87|columns=12}} | ||
{{Harmonics in equal|87|columns=12|start=13|collapsed=1|title=Approximation of prime harmonics in 87edo (continued)}} | {{Harmonics in equal|87|columns=12|start=13|collapsed=1|title=Approximation of prime harmonics in 87edo (continued)}} | ||
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=== Interval mappings === | === Interval mappings === | ||
{{Q-odd-limit intervals|87}} | {{Q-odd-limit intervals|87}} | ||
== Regular temperament properties == | == Regular temperament properties == | ||
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87 can serve as a mos in these: | 87 can serve as a mos in these: | ||
* [[Avicenna (temperament)|Avicenna]] ([[Breed|87 & 270]]) | * [[Avicenna (temperament)|Avicenna]] ([[Breed|87 & 270]]) | ||
* [[Breed|87 & 494]] | * [[Breed|87 & 494]] | ||
== Scales == | == Scales == | ||
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{{main|List of MOS scales in 87edo}} | {{main|List of MOS scales in 87edo}} | ||
=== Harmonic | === Harmonic scales === | ||
87edo accurately approximates the mode 8 of [[harmonic series]], and the only interval pair not distinct is 14/13 and 15/14. It can also do mode 12 decently. | 87edo accurately approximates the mode 8 of [[harmonic series]], and the only interval pair not distinct is 14/13 and 15/14. It can also do mode 12 decently. | ||
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|} | |} | ||
The scale in adjacent steps is 15, 13, 12, 11, 10, 9, 9, 8. | |||
==== (Mode 12) ==== | ==== (Mode 12) ==== | ||
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|} | |} | ||
The scale in adjacent steps is 10, 9, 9, 8, 7, 7, 6, 6, 6, 6, 5. | |||
13, 15, 16, 18, 20, and 22 are close matches. | |||
14 and 21 are flat; 17, 19, and 23 are sharp. Still decent all things considered. | |||
=== Other scales === | === Other scales === | ||
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== Instruments == | == Instruments == | ||
* [[Lumatone mapping for 87edo]] | |||
* [[Skip fretting system 87 2 17]] | * [[Skip fretting system 87 2 17]] | ||
== Music == | == Music == | ||
; [[Bryan Deister]] | |||
* [https://www.youtube.com/shorts/ecxELXmkYAs ''microtonal improvisation in 87edo''] (2025) | |||
; [[Gene Ward Smith]] | ; [[Gene Ward Smith]] | ||
* ''Pianodactyl'' (archived 2010) – [https://soundcloud.com/genewardsmith/pianodactyl SoundCloud] | [http://www.archive.org/details/Pianodactyl detail] | [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] – rodan[26] in 87edo tuning | * ''Pianodactyl'' (archived 2010) – [https://soundcloud.com/genewardsmith/pianodactyl SoundCloud] | [http://www.archive.org/details/Pianodactyl detail] | [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] – rodan[26] in 87edo tuning | ||