87edo: Difference between revisions
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== Theory == | == Theory == | ||
87edo is solid as both a [[13-limit]] (or [[15-odd-limit]]) and as a [[5-limit]] system, and does well enough in any limit in between. It is the smallest edo that is [[distinctly consistent]] in the [[13-odd-limit]] [[tonality diamond]], the smallest edo that is [[purely consistent]]{{idiosyncratic}} in the [[15-odd-limit]] (maintains [[relative interval error]]s of no greater than 25% on all of the first 16 [[harmonic]]s of the [[harmonic series]]). It is also a [[zeta peak integer edo]]. Since {{nowrap|87 {{=}} 3 × 29}}, 87edo shares the same perfect fifth with [[29edo]]. | 87edo is solid as both a [[13-limit]] (or [[15-odd-limit]]) and as a [[5-limit]] system, and does well enough in any limit in between. It is the smallest edo that is [[distinctly consistent]] in the [[13-odd-limit]] [[tonality diamond]], and the smallest edo that is [[purely consistent]]{{idiosyncratic}} in the [[15-odd-limit]] (maintains [[relative interval error]]s of no greater than 25% on all of the first 16 [[harmonic]]s of the [[harmonic series]]). It is also a [[zeta peak integer edo]]. Since {{nowrap|87 {{=}} 3 × 29}}, 87edo shares the same perfect fifth with [[29edo]]. | ||
87edo also shows some potential in limits beyond 13. The next four prime harmonics [[17/1|17]], [[19/1|19]], [[23/1|23]], and [[29/1|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 [[7/1|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 [[17/1|17]], [[19/1|19]], [[23/1|23]], and [[29/1|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 [[7/1|7]], which is flat. Actually, as a no-sevens system, it is consistent in the 33-odd-limit. | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
87edo contains [[3edo]] and [[29edo]] as subset edos. | 87edo contains [[3edo]] and [[29edo]] as subset edos. | ||
[[348edo]], which slices the edostep in four, provides a good correction of the 7th harmonic. | |||
== Intervals == | == Intervals == | ||
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87edo can be written using [[Kite's ups and downs notation]]. Note that quudsharp (quadruple-down sharp) is equivalent to quip (quintuple-up) and that quupflat (quadruple-up flat) is equivalent to quid (quintuple-down): | 87edo can be written using [[Kite's ups and downs notation]]. Note that quudsharp (quadruple-down sharp) is equivalent to quip (quintuple-up) and that quupflat (quadruple-up flat) is equivalent to quid (quintuple-down): | ||
{{Ups and downs sharpness}} | {{Ups and downs sharpness}} | ||
Mapping an arrow to 2\87 rather than 1\87 is an alternative approach which takes advantage of 87edo being a tuning of akea temperament. This way, one arrow is equivalent to 81/80~64/63, and two arrows are equivalent to 33/32~1053/1024. | |||
== Approximation to JI == | == Approximation to JI == | ||
| Line 369: | Line 372: | ||
| 2.3.5 | | 2.3.5 | ||
| 15625/15552, 67108864/66430125 | | 15625/15552, 67108864/66430125 | ||
| {{ | | {{Mapping| 87 138 202 }} | ||
| −0.299 | | −0.299 | ||
| 0.455 | | 0.455 | ||
| Line 376: | Line 379: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 245/243, 1029/1024, 3136/3125 | | 245/243, 1029/1024, 3136/3125 | ||
| {{ | | {{Mapping| 87 138 202 244 }} | ||
| +0.070 | | +0.070 | ||
| 0.752 | | 0.752 | ||
| Line 383: | Line 386: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 245/243, 385/384, 441/440, 3136/3125 | | 245/243, 385/384, 441/440, 3136/3125 | ||
| {{ | | {{Mapping| 87 138 202 244 301 }} | ||
| +0.033 | | +0.033 | ||
| 0.676 | | 0.676 | ||
| Line 390: | Line 393: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 196/195, 245/243, 352/351, 364/363, 625/624 | | 196/195, 245/243, 352/351, 364/363, 625/624 | ||
| {{ | | {{Mapping| 87 138 202 244 301 322 }} | ||
| −0.011 | | −0.011 | ||
| 0.625 | | 0.625 | ||
| Line 397: | Line 400: | ||
| 2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
| 154/153, 196/195, 245/243, 273/272, 364/363, 375/374 | | 154/153, 196/195, 245/243, 273/272, 364/363, 375/374 | ||
| {{ | | {{Mapping| 87 138 202 244 301 322 356 }} | ||
| −0.198 | | −0.198 | ||
| 0.738 | | 0.738 | ||
| Line 404: | Line 407: | ||
| 2.3.5.7.11.13.17.19 | | 2.3.5.7.11.13.17.19 | ||
| 154/153, 196/195, 210/209, 245/243, 273/272, 286/285, 364/363 | | 154/153, 196/195, 210/209, 245/243, 273/272, 286/285, 364/363 | ||
| {{ | | {{Mapping| 87 138 202 244 301 322 356 370 }} | ||
| −0.348 | | −0.348 | ||
| 0.796 | | 0.796 | ||
| Line 513: | Line 516: | ||
| [[Mystery]] | | [[Mystery]] | ||
|} | |} | ||
<nowiki/>* [[Normal | <nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | ||
87 can serve as a mos in these: | 87 can serve as a mos in these: | ||
| Line 681: | Line 684: | ||
== Music == | == Music == | ||
; [[ALLY195]] | |||
* [https://www.bilibili.com/video/BV16h411g7QM/ ''Root note and subharmonic series cadence - 103EDO, 87EDO, 94EDO''] (2023) | |||
* [https://www.bilibili.com/video/BV1N84y1T792/ ''A comparison between 87edo and 12edo''] (2023) | |||
; [[Bryan Deister]] | ; [[Bryan Deister]] | ||
* [https://www.youtube.com/shorts/ecxELXmkYAs ''microtonal improvisation in 87edo''] (2025) | * [https://www.youtube.com/shorts/ecxELXmkYAs ''microtonal improvisation in 87edo''] (2025) | ||
* [https://www.youtube.com/shorts/5OH9OOGeuX4 ''87edo waltz''] (2025) | * [https://www.youtube.com/shorts/5OH9OOGeuX4 ''87edo waltz''] (2025) | ||
* [https://www.youtube.com/shorts/rINJKiMQE78 ''Circuit Bent - Stomach Book (microtonal cover 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 | ||
[[Category:Listen]] | [[Category:Listen]] | ||
[[Category:Clyde]] | [[Category:Clyde]] | ||