87edo - Revision history

Tristanbay: /* Ups and downs notation */typo fix

2026-05-10T18:47:33Z

Ups and downs notation: typo fix

← Older revision		Revision as of 18:47, 10 May 2026
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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 ~~eqivalent~~ to 33/32~1053/1024.		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 ==

Tristanbay: /* Ups and downs notation */Akea notation with ups and downs

2026-05-10T18:47:06Z

Ups and downs notation: Akea notation with ups and downs

← Older revision		Revision as of 18:47, 10 May 2026
Line 352:		Line 352:
	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 eqivalent to 33/32~1053/1024.

	== Approximation to JI ==		== Approximation to JI ==

FloraC: /* Music */ cleanup, not sure if these count as music tho

2026-03-24T13:30:20Z

Music: cleanup, not sure if these count as music tho

← Older revision		Revision as of 13:30, 24 March 2026
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	== 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)
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	; [[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


	~~'''ALLY195'''~~

	* [https://www.bilibili.com/video/BV16h411g7QM/?spm_id_from=333.1387.homepage.video_card.click&vd_source=3e5765d9963bdf3370712d765d3ff3e1 Root note and subharmonic series cadence - 103EDO,87EDO,94EDO] (2023)
	* [https://www.bilibili.com/video/BV1N84y1T792/?spm_id_from=333.337.search-card.all.click&vd_source=3e5765d9963bdf3370712d765d3ff3e1 A comparison between 87edo and 12edo] (2023)

	[[Category:Listen]]		[[Category:Listen]]

Haidilao at 08:03, 24 March 2026

2026-03-24T08:03:27Z

← Older revision		Revision as of 08:03, 24 March 2026
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	; [[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


			'''ALLY195'''

			* [https://www.bilibili.com/video/BV16h411g7QM/?spm_id_from=333.1387.homepage.video_card.click&vd_source=3e5765d9963bdf3370712d765d3ff3e1 Root note and subharmonic series cadence - 103EDO,87EDO,94EDO] (2023)
			* [https://www.bilibili.com/video/BV1N84y1T792/?spm_id_from=333.337.search-card.all.click&vd_source=3e5765d9963bdf3370712d765d3ff3e1 A comparison between 87edo and 12edo] (2023)

	[[Category:Listen]]		[[Category:Listen]]

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:32:44Z

Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

← Older revision		Revision as of 13:32, 13 March 2026
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	\| [[Mystery]]		\| [[Mystery]]
	\|}		\|}
	<nowiki/>* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~\|minimal form]] in parentheses if distinct		<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:

Contribution: /* Subsets and supersets */

2025-12-15T17:46:33Z

Subsets and supersets

← Older revision		Revision as of 17:46, 15 December 2025
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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 ==

Overthink: /* Theory */ fix grammar

2025-12-13T01:12:43Z

Theory: fix grammar

← Older revision		Revision as of 01:12, 13 December 2025
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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.

Lucius Chiaraviglio: /* Music */ Add Bryan Deister's ''Circuit Bent - Stomach Book (microtonal cover in 87edo)'' (2025)

2025-11-15T08:41:34Z

Music: Add Bryan Deister's ''Circuit Bent - Stomach Book (microtonal cover in 87edo)'' (2025)

← Older revision		Revision as of 08:41, 15 November 2025
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	* [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]]

FloraC: - redundant category

2025-10-02T16:53:59Z

- redundant category

← Older revision		Revision as of 16:53, 2 October 2025
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	\| 2.3.5		\| 2.3.5
	\| 15625/15552, 67108864/66430125		\| 15625/15552, 67108864/66430125
	\| {{~~mapping~~\| 87 138 202 }}		\| {{Mapping\| 87 138 202 }}
	\| −0.299		\| −0.299
	\| 0.455		\| 0.455
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	\| 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 }}		\| {{Mapping\| 87 138 202 244 }}
	\| +0.070		\| +0.070
	\| 0.752		\| 0.752
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	\| 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 }}		\| {{Mapping\| 87 138 202 244 301 }}
	\| +0.033		\| +0.033
	\| 0.676		\| 0.676
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	\| 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 }}		\| {{Mapping\| 87 138 202 244 301 322 }}
	\| −0.011		\| −0.011
	\| 0.625		\| 0.625
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	\| 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 }}		\| {{Mapping\| 87 138 202 244 301 322 356 }}
	\| −0.198		\| −0.198
	\| 0.738		\| 0.738
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	\| 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 }}		\| {{Mapping\| 87 138 202 244 301 322 356 370 }}
	\| −0.348		\| −0.348
	\| 0.796		\| 0.796
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	* ''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:Zeta\|##]] <!-- 2-digit number -->~~
	[[Category:Listen]]		[[Category:Listen]]
	[[Category:Clyde]]		[[Category:Clyde]]

ArrowHead294: /* Notation */ Functionality added to template

2025-09-25T16:17:56Z

Notation: Functionality added to template

← Older revision		Revision as of 16:17, 25 September 2025
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	== Notation ==		== Notation ==
			=== Ups and downs notation ===
	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~~\|87~~}}		{{Ups and downs sharpness}}

	== Approximation to JI ==		== Approximation to JI ==