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 [[Square superparticular|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. | 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 [[Square superparticular|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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== Intervals == | == Intervals == | ||
{| class="wikitable center-all right-2 left-3 left-4" | {| class="wikitable center-all right-2 left-3 left-4" | ||
! rowspan="2" | # | ! rowspan="2" | # | ||
! rowspan="2" | Cents | ! rowspan="2" | Cents | ||
! colspan="2" | Approximated Ratios | ! colspan="2" | Approximated Ratios | ||
! colspan="2" rowspan="2" |[[Ups and Downs Notation]] | ! colspan="2" rowspan="2" | [[Ups and Downs Notation]] | ||
|- | |- | ||
! 13-Limit | ! 13-Limit | ||
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=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+ Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
|- | |- | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
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| [[Mystery]] | | [[Mystery]] | ||
|} | |} | ||
{{asterisk}} [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
87 can serve as a MOS in these: | 87 can serve as a MOS in these: | ||
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== Music == | == Music == | ||
; [[Gene Ward Smith]] | ; [[Gene Ward Smith]] | ||
* ''Pianodactyl'' (archived 2010) | * ''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 | ||
[[Category:Zeta|##]] <!-- 2-digit number --> | [[Category:Zeta|##]] <!-- 2-digit number --> | ||