Gjaeck - Revision history

Lériendil: Text replacement - "Category:Tempered_scales↵Category:Pages with mostly numerical content" to "Category:Tempered scales Category:Pages with mostly numerical content"

2025-06-24T03:04:36Z

Text replacement - "Category:Tempered_scales↵Category:Pages with mostly numerical content" to "Category:Tempered scales Category:Pages with mostly numerical content"

← Older revision		Revision as of 03:04, 24 June 2025
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	[[Yer]], another tuning in the [[2.11.13.17.19 subgroup\|2.11.13.17.19 domain]], JI, also designed by [[Douglas Blumeyer]]		[[Yer]], another tuning in the [[2.11.13.17.19 subgroup\|2.11.13.17.19 domain]], JI, also designed by [[Douglas Blumeyer]]
	[[Category:13-tone scales]]		[[Category:13-tone scales]]
	[[Category:~~Tempered_scales~~]]		[[Category:Tempered scales]]
	[[Category:Pages with mostly numerical content]]		[[Category:Pages with mostly numerical content]]
	[[Category:MOS scales]]		[[Category:MOS scales]]

Lériendil: Text replacement - "Category:Tempered scales↵Category:Pages with mostly numerical content" to "Category:Tempered_scales Category:Pages with mostly numerical content"

2025-06-24T02:29:31Z

Text replacement - "Category:Tempered scales↵Category:Pages with mostly numerical content" to "Category:Tempered_scales Category:Pages with mostly numerical content"

← Older revision		Revision as of 02:29, 24 June 2025
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	[[Yer]], another tuning in the [[2.11.13.17.19 subgroup\|2.11.13.17.19 domain]], JI, also designed by [[Douglas Blumeyer]]		[[Yer]], another tuning in the [[2.11.13.17.19 subgroup\|2.11.13.17.19 domain]], JI, also designed by [[Douglas Blumeyer]]
	[[Category:13-tone scales]]		[[Category:13-tone scales]]
	[[Category:~~Tempered scales~~]]		[[Category:Tempered_scales]]
	[[Category:Pages with mostly numerical content]]		[[Category:Pages with mostly numerical content]]
	[[Category:MOS scales]]		[[Category:MOS scales]]

Lériendil: Text replacement - "Category:Tempered scales" to "Category:Tempered scales Category:Pages with mostly numerical content"

2025-06-24T02:25:07Z

Text replacement - "Category:Tempered scales" to "Category:Tempered scales Category:Pages with mostly numerical content"

← Older revision		Revision as of 02:25, 24 June 2025
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	[[Category:13-tone scales]]		[[Category:13-tone scales]]
	[[Category:Tempered scales]]		[[Category:Tempered scales]]
			[[Category:Pages with mostly numerical content]]
	[[Category:MOS scales]]		[[Category:MOS scales]]
	[[Category:19-limit]]		[[Category:19-limit]]
	[[Category:57edo]]		[[Category:57edo]]
	[[Category:Pages with Scala files]]		[[Category:Pages with Scala files]]

Cmloegcmluin: clean up

2024-08-13T14:54:23Z

clean up

← Older revision		Revision as of 14:54, 13 August 2024
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	Gjaeck is a [[Tridecatonic MOS\|tridecatonic]] ([[5L 8s]]) [[MOS scale]] of [[57edo\|~~57ed2~~]] optimized for the prime harmonics 11, 13, 17, and 19, while not optimizing for the simpler prime harmonics 3, 5, and 7.		Gjaeck is a [[Tridecatonic MOS\|tridecatonic]] ([[5L 8s]]) [[MOS scale]] of [[57edo\|57-EDO]] optimized for the prime harmonics 11, 13, 17, and 19, while not optimizing for the simpler prime harmonics 3, 5, and 7.
	[[File:Gjaeck - cycle view.jpg\|thumb\|		[[File:Gjaeck - cycle view.jpg\|thumb\|
	Gjaeck		Gjaeck
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	== Temperings ==		== Temperings ==

	The [[~~patent val~~]] for 2.11.13.17.19 ~~prime limit 57ed2~~ is {{~~Val~~\| 57 197 211 233 242 }}, resulting in the following mappings:		The [[simple map]] for 57-ET (57p) in the 2.11.13.17.19 domain is {{map\| 57 197 211 233 242 }}, resulting in the following mappings:
	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+
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	\|294.737 - 297.513 = -2.776		\|294.737 - 297.513 = -2.776
	\|}		\|}
	The Blumeyer comma is [[Tempering out\|tempered out]] by ~~57ed2~~.
			The [[Blumeyer comma]] is [[Tempering out\|tempered out]] by 57p.

	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+
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	</math>		</math>

	~~Gjaeck~~ tempers out some other high-limit commas too:		57p tempers out some other high-limit commas too:
	* The Blume comma, 2057/2048; moving up by two 11’s and then an 17 gets you nowhere: 26 + 26 + 5 = 57.		* The [[Blume comma]], 2057/2048; moving up by two 11’s and then an 17 gets you nowhere: 26 + 26 + 5 = 57.
	* The yama comma, 209/208, since 26 + 14 = 40; moving by an 11 and 19 is the same as moving by a 13		* The [[yama comma]], 209/208, since 26 + 14 = 40; moving by an 11 and 19 is the same as moving by a 13

	== The MOS scale ==		== The MOS scale ==

	Gjaeck has a small step equal to 4 steps of ~~57ed2~~ and a large step equal to 5 steps. It follows the small and large step sequence:		Gjaeck has a small step equal to 4 steps of 57-EDO and a large step equal to 5 steps. It follows the small and large step sequence:

	ssLssLsLssLsL		ssLssLsLssLsL
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	While the 11th harmonic is the most prevalent sound, the 19th is the rarest. But since the 19th is the key to the Blumeyer comma, being the one of the four higher harmonics that sits by its lonesome on one side of the ratio while the other three party together, we should pay special attention to the moments in the scale with LsL.		While the 11th harmonic is the most prevalent sound, the 19th is the rarest. But since the 19th is the key to the Blumeyer comma, being the one of the four higher harmonics that sits by its lonesome on one side of the ratio while the other three party together, we should pay special attention to the moments in the scale with LsL.

	The generator for this MOS is 35th step of 57, which is 736.8421¢, an interval associated with the 21st harmonic.		The generator for this MOS is 35th step of 57-EDO, which is 736.8421¢, an interval associated with the 21st harmonic.

	While 57 = 19⋅3 and thus ~~57ed2~~ extends [[19edo\|~~19ed2~~]], Gjaeck bears little resemblance to ~~19ed2~~; it feels more like a whacked out [[13edo\|~~13ed2~~]], pinched and pulled in a particular way to more perfectly capture the four higher harmonic primes.		While 57 = 19⋅3 and thus 57-EDO extends [[19edo\|19-EDO]], Gjaeck bears little resemblance to 19-EDO; it feels more like a whacked out [[13edo\|13-EDO]], pinched and pulled in a particular way to more perfectly capture the four higher harmonic primes.

	== Scala file ==		== Scala file ==
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	! blumeyer_tempered.scl		! blumeyer_tempered.scl
	!		!
	Blumeyer comma scale, 5L8s MOS of ~~57ed2~~		Blumeyer comma scale, 5L8s MOS of 57-EDO
	13		13
	!		!
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	== See also ==		== See also ==

	[[Hilim13]], another tridecatonic tuning in the [[2.11.13.17.19 subgroup]], [[Just intonation\|JI]], designed by [[Gene Ward Smith]]		[[Hilim13]], another tridecatonic tuning in the [[2.11.13.17.19 subgroup\|2.11.13.17.19 domain]], [[Just intonation\|JI]], designed by [[Gene Ward Smith]]

	[[Yer]], another tuning in the [[2.11.13.17.19 subgroup]], JI, also designed by [[Douglas Blumeyer]]		[[Yer]], another tuning in the [[2.11.13.17.19 subgroup\|2.11.13.17.19 domain]], JI, also designed by [[Douglas Blumeyer]]
	[[Category:13-tone scales]]		[[Category:13-tone scales]]
	[[Category:Tempered scales]]		[[Category:Tempered scales]]

BudjarnLambeth: Added categories

2023-04-18T08:40:56Z

Added categories

← Older revision		Revision as of 08:40, 18 April 2023
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	[[Yer]], another tuning in the [[2.11.13.17.19 subgroup]], JI, also designed by [[Douglas Blumeyer]]		[[Yer]], another tuning in the [[2.11.13.17.19 subgroup]], JI, also designed by [[Douglas Blumeyer]]
			[[Category:13-tone scales]]
			[[Category:Tempered scales]]
			[[Category:MOS scales]]
			[[Category:19-limit]]
			[[Category:57edo]]
	[[Category:Pages with Scala files]]		[[Category:Pages with Scala files]]
	~~[[Category:13-tone scales]]~~

Cmloegcmluin: /* See Also */ fix section capitalization

2021-11-11T19:11:14Z

Cmloegcmluin: /* The MOS Scale */ fix section capitalization

2021-11-11T19:10:58Z

The MOS Scale: fix section capitalization

← Older revision		Revision as of 19:10, 11 November 2021
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	* The yama comma, 209/208, since 26 + 14 = 40; moving by an 11 and 19 is the same as moving by a 13		* The yama comma, 209/208, since 26 + 14 = 40; moving by an 11 and 19 is the same as moving by a 13

	== The MOS ~~Scale~~ ==		== The MOS scale ==

	Gjaeck has a small step equal to 4 steps of 57ed2 and a large step equal to 5 steps. It follows the small and large step sequence:		Gjaeck has a small step equal to 4 steps of 57ed2 and a large step equal to 5 steps. It follows the small and large step sequence:

Flirora: cleanup +cat

2021-03-18T22:35:43Z

cleanup +cat

← Older revision		Revision as of 22:35, 18 March 2021
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	Gjaeck is a [[Tridecatonic MOS\|tridecatonic]] ([[5L 8s~~\|5L8s~~]]) [[MOS scale]] of [[57edo\|~~57ed~~]]2 optimized for the prime harmonics 11, 13, 17, and 19, while not optimizing for the simpler prime harmonics 3, 5, and 7.		Gjaeck is a [[Tridecatonic MOS\|tridecatonic]] ([[5L 8s]]) [[MOS scale]] of [[57edo\|57ed2]] optimized for the prime harmonics 11, 13, 17, and 19, while not optimizing for the simpler prime harmonics 3, 5, and 7.
	[[File:Gjaeck - cycle view.jpg\|thumb\|		[[File:Gjaeck - cycle view.jpg\|thumb\|
	Gjaeck		Gjaeck
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	== Temperings ==		== Temperings ==

	The [[patent val]] for 2.11.13.17.19 prime limit 57ed2 is ~~<code><~~ 57 197 211 233 242 ~~\|</code>~~, resulting in the following mappings:		The [[patent val]] for 2.11.13.17.19 prime limit 57ed2 is {{Val\| 57 197 211 233 242 }}, resulting in the following mappings:
	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+
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	\|2432/2431		\|2432/2431
	\|0.71200249782		\|0.71200249782
	\|~~<nowiki>~~\| 7 0 0 0 -1 -1 -1 1 ~~></nowiki>~~		\|{{Monzo\| 7 0 0 0 -1 -1 -1 1 }}
	\|}		\|}

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	[[Yer]], another tuning in the [[2.11.13.17.19 subgroup]], JI, also designed by [[Douglas Blumeyer]]		[[Yer]], another tuning in the [[2.11.13.17.19 subgroup]], JI, also designed by [[Douglas Blumeyer]]

			[[Category:Pages with Scala files]]
			[[Category:13-tone scales]]

Cmloegcmluin at 22:31, 18 March 2021

2021-03-18T22:31:46Z

← Older revision		Revision as of 22:31, 18 March 2021
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	!value		!value
	!cents		!cents
	!~~monzo~~		!prime exponent vector
	\|-		\|-
	\|Blumeyer comma		\|Blumeyer comma

Cmloegcmluin: /* Temperings */

2020-02-21T02:54:26Z

Temperings

← Older revision		Revision as of 02:54, 21 February 2020
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	\|294.737 - 297.513 = -2.776		\|294.737 - 297.513 = -2.776
	\|}		\|}
	The Blumeyer comma is tempered out by 57ed2.		The Blumeyer comma is [[Tempering out\|tempered out]] by 57ed2.
	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+

← Older revision		Revision as of 19:11, 11 November 2021
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	2/1		2/1

	== See ~~Also~~ ==		== See also ==

	[[Hilim13]], another tridecatonic tuning in the [[2.11.13.17.19 subgroup]], [[Just intonation\|JI]], designed by [[Gene Ward Smith]]		[[Hilim13]], another tridecatonic tuning in the [[2.11.13.17.19 subgroup]], [[Just intonation\|JI]], designed by [[Gene Ward Smith]]