Yer - Revision history

Overthink: reformat some stuff

2026-01-18T22:42:22Z

reformat some stuff

← Older revision		Revision as of 22:42, 18 January 2026
Line 1:		Line 1:
	~~== Introduction ==~~
	Yer is an octave-reduced EFG ([[wikipedia:Euler–Fokker_genus\|Euler-Fokker genus]]) of 11, 13, 17, and 19. As such it is a 2.11.13.17.19 limit just intonation tuning (ignoring the 3rd, 5th, and 7th harmonics) which is octave-repeating.		Yer is an octave-reduced EFG ([[wikipedia:Euler–Fokker_genus\|Euler-Fokker genus]]) of 11, 13, 17, and 19. As such it is a 2.11.13.17.19 limit just intonation tuning (ignoring the 3rd, 5th, and 7th harmonics) which is octave-repeating.

Line 6:		Line 5:
	Yer was developed by [[Douglas Blumeyer\|Douglas Blumeyer]] as a system to explore the [[unnoticeable comma]] 2432/2431, the Blumeyer comma, which appears twice as a step in the scale.		Yer was developed by [[Douglas Blumeyer\|Douglas Blumeyer]] as a system to explore the [[unnoticeable comma]] 2432/2431, the Blumeyer comma, which appears twice as a step in the scale.

			== The scale ==
	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	\|+		\|+ style="font-size: 105%" \| Steps of Yer
	!~~name~~		!Name
	!11		!11
	!13		!13
Line 14:		Line 14:
	!19		!19
	!EFG composition		!EFG composition
	!~~frequency~~ multiplier (definition)		!Frequency multiplier (definition)
	!~~frequency~~ multiplier (decimal)		!Frequency multiplier (decimal)
	!~~pitch~~ (¢)		!Pitch (¢)
	!EHEJIPN (on C)		!EHEJIPN (on C)
	\|-		\|-
Line 243:		Line 243:
	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+
	!~~name~~		!Name
	!~~value~~		!Value
	!~~cents~~		!Cents
	!~~prime~~ exponent vector		!Prime exponent vector
	\|-		\|-
	\|Blumeyer comma		\|Blumeyer comma
Line 279:		Line 279:
	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+
	!~~name~~		!Name
	!~~value~~		!Value
	!~~cents~~		!Cents
	!~~prime~~ exponent vector		!Prime exponent vector
	\|-		\|-
	\|~~yama~~ comma		\|Yama comma
	\|[[209/208]]		\|[[209/208]]
	\|8.303296728		\|8.303296728
Line 293:		Line 293:
	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+
	!~~name~~		!Name
	!~~value~~		!Value
	!~~cents~~		!Cents
	!~~prime~~ exponent vector		!Prime exponent vector
	\|-		\|-
	\|Blume comma		\|Blume comma
Line 709:		Line 709:
	{\| class="wikitable"		{\| class="wikitable"
	\|+Yer interval classes		\|+Yer interval classes
	!~~ratio~~		!Ratio
	!~~type~~		!Type
	!~~count~~		!Count
	!~~prime~~ exponent vector		!Prime exponent vector
	!~~cents~~		!Cents
	!~~gap~~ between available interval classes		!Gap between available interval classes
	!~~lower~~ limit interval to nearest in Yer		!Lower limit interval to nearest in Yer
	!~~cents~~ difference up		!Cents difference up
	!~~cents different~~ down (to prove how they're right in between		!Cents difference down (to prove how they're right in between)
	!~~inverse~~ ratio		!Inverse ratio
	!~~inverse~~ cents		!Inverse cents
	\|-		\|-
	\|2431:2432		\|2431:2432

Xenllium at 12:58, 23 February 2025

2025-02-23T12:58:55Z

← Older revision		Revision as of 12:58, 23 February 2025
Line 1:		Line 1:
	== Introduction ==		== Introduction ==

	Yer is an octave-reduced EFG ([[wikipedia:Euler–Fokker_genus\|Euler-Fokker genus]]) of 11, 13, 17, and 19. As such it is a 2.11.13.17.19 limit just intonation tuning (ignoring the 3rd, 5th, and 7th harmonics) which is octave-repeating.		Yer is an octave-reduced EFG ([[wikipedia:Euler–Fokker_genus\|Euler-Fokker genus]]) of 11, 13, 17, and 19. As such it is a 2.11.13.17.19 limit just intonation tuning (ignoring the 3rd, 5th, and 7th harmonics) which is octave-repeating.

Line 240:		Line 239:

	== Blumeyer comma ==		== Blumeyer comma ==

	Those pitches right on top of each other are a feature, not a bug. The pitch 11 * 13 * 17 is 2431, only one off from the pitch 19, which when octave adjusted is 2432. So we end up with this superparticular ratio, 2432:2431, called the Blumeyer comma.		Those pitches right on top of each other are a feature, not a bug. The pitch 11 * 13 * 17 is 2431, only one off from the pitch 19, which when octave adjusted is 2432. So we end up with this superparticular ratio, 2432:2431, called the Blumeyer comma.

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	\|-		\|-
	\|Blumeyer comma		\|Blumeyer comma
	\|2432/2431		\|[[2432/2431]]
	\|0.71200249782		\|0.71200249782
	\|<nowiki>\| 7 0 0 0 -1 -1 -1 1 ></nowiki>		\|<nowiki>[ 7 0 0 0 -1 -1 -1 1 ></nowiki>
	\|}		\|}
	[[File:Blumeyer comma.png\|none\|thumb\|		[[File:Blumeyer comma.png\|none\|thumb\|
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	\|-		\|-
	\|yama comma		\|yama comma
	\|209/208		\|[[209/208]]
	\|8.303296728		\|8.303296728
	\|<nowiki>\| -4 0 0 0 1 -1 0 1 ></nowiki>		\|<nowiki>[ -4 0 0 0 1 -1 0 1 ></nowiki>
	\|}		\|}
	Another consequence of the yama comma is that a couple “extra” lattice connections appear. These are drawn in dotted green on the lattices in the introduction.		Another consequence of the yama comma is that a couple “extra” lattice connections appear. These are drawn in dotted green on the lattices in the introduction.
Line 301:		Line 299:
	\|-		\|-
	\|Blume comma		\|Blume comma
	\|2057/2048		\|[[2057/2048]]
	\|7.59129422992		\|7.59129422992
	\|<nowiki>\| -11 0 0 0 2 0 1 ></nowiki>		\|<nowiki>[ -11 0 0 0 2 0 1 ></nowiki>
	\|}		\|}
	[[File:Blume comma.png\|none\|thumb\|		[[File:Blume comma.png\|none\|thumb\|
Line 322:		Line 320:

	== Comma pumps ==		== Comma pumps ==

	Yer is pure JI, but due to the five places where it boasts two pitches very close together but with very different harmonic compositions, it can achieve [[Zero comma pump\|zero comma pumps]] by “comma shifting” at those key points, returning to exactly their original pitch.		Yer is pure JI, but due to the five places where it boasts two pitches very close together but with very different harmonic compositions, it can achieve [[Zero comma pump\|zero comma pumps]] by “comma shifting” at those key points, returning to exactly their original pitch.

Line 727:		Line 724:
	\|3v1		\|3v1
	\|1		\|1
	\|<nowiki>\| 7 0 0 0 -1 -1 -1 1 ></nowiki>		\|<nowiki>[ 7 0 0 0 -1 -1 -1 1 ></nowiki>
	\|0.7120024978		\|0.7120024978
	\|0.7120024978		\|0.7120024978
Line 1,195:		Line 1,192:
	\|4v0		\|4v0
	\|1		\|1
	\|<nowiki>\| 15 0 0 0 1 1 1 1 ></nowiki>		\|<nowiki>[ 15 0 0 0 1 1 1 1 ></nowiki>
	\|594.3140298		\|594.3140298
	\|7.5912943		\|7.5912943
Line 1,206:		Line 1,203:

	== Scala file ==		== Scala file ==
			<pre>
	! yer.scl		! yer.scl
	!		!
Line 1,228:		Line 1,225:
	247/128		247/128
	2/1		2/1
			</pre>
	== Video explanation ==		== Video explanation ==

	More details are presented in this video: [https://vimeo.com/184148765 Yer (pitch system)]		More details are presented in this video: [https://vimeo.com/184148765 Yer (pitch system)]

	== Listening ==		== Listening ==
			* [https://soundcloud.com/cmloegcmluin/tsraxcfaubdj Douglas Blumeyer - Tsraxcfaubdj]
	[https://soundcloud.com/cmloegcmluin/tsraxcfaubdj Douglas Blumeyer - Tsraxcfaubdj]		* [https://soundcloud.com/cmloegcmluin/blumeyer-comma-ji-unpump Douglas Blumeyer - Blumeyer Comma JI Unpump]
			* [https://chrisvaisvil.com/the-figment-for-woodwinds-muted-strings-and-choir-in-just-intonation-tuning Chris Vaisvil - The Figment]
	[https://soundcloud.com/cmloegcmluin/blumeyer-comma-ji-unpump Douglas Blumeyer - Blumeyer Comma JI Unpump]		* [http://chrisvaisvil.com/now-yer-talkin-ji-piano Chris Vaisvil - Now Yer Talkin']

	[https://chrisvaisvil.com/the-figment-for-woodwinds-muted-strings-and-choir-in-just-intonation-tuning Chris Vaisvil - The Figment]

	[http://chrisvaisvil.com/now-yer-talkin-ji-piano Chris Vaisvil - Now Yer Talkin']

	== Tritave-Based Yer ==		== Tritave-Based Yer ==
Line 1,247:		Line 1,239:

	== Yer as a temperament ==		== Yer as a temperament ==

	If you [[temper out]] only the Blumeyer comma, you get this 2.11.13.17.19 [[subgroup]] [[mapping]], which naturally should be called "Blumeyer [[regular temperament\|temperament]]":		If you [[temper out]] only the Blumeyer comma, you get this 2.11.13.17.19 [[subgroup]] [[mapping]], which naturally should be called "Blumeyer [[regular temperament\|temperament]]":

Line 1,255:		Line 1,246:
	⟨ 0 0 0 1 1 ] ⟩		⟨ 0 0 0 1 1 ] ⟩

	Expressed as a [[map-merging\|map-merge]] of [[ET]]s, that's 13&113&137&194. So one ~19 is up one each of the ~11, ~13, and ~17 here.		Expressed as a [[map-merging\|map-merge]] of [[ET]]s, that's 13&113&137&194. So one ~19 is up one each of the ~11, ~13, and ~17 here.

	Now if you temper out the Blumeyer comma and the yama comma (and therefore also the blume comma; in fact, the [[canonical form]] of the [[comma basis]] appears to be blume and yama, with the Blumeyer comma being a linear combination of them), then you get a [[rank-3 temperament]], with mapping:		Now if you temper out the Blumeyer comma and the yama comma (and therefore also the blume comma; in fact, the [[canonical form]] of the [[comma basis]] appears to be blume and yama, with the Blumeyer comma being a linear combination of them), then you get a [[rank-3 temperament]], with mapping:
Line 1,267:		Line 1,258:
	A 13-note scale with only three scale step sizes is possible here. It's 7L 4M 2s, with the pattern LMLLMLsLMLMLs, where L ≈ 103.4¢, M ≈ 85.9¢, and s ≈ 66.2¢:		A 13-note scale with only three scale step sizes is possible here. It's 7L 4M 2s, with the pattern LMLLMLsLMLMLs, where L ≈ 103.4¢, M ≈ 85.9¢, and s ≈ 66.2¢:

	<~~nowiki~~>		<pre>
	! yerMV3.scl		! yerMV3.scl
	!		!
Line 1,286:		Line 1,277:
	1133.7		1133.7
	(2/1)		(2/1)
	</~~nowiki~~>		</pre>

	However this is not quite an [[MV3]] scale. It is a rotation of the "LLMLsLMLMLsLM" pattern which is identified [[User:Xenoindex/Tridecatonic_MV3\|here]] as "Conditional (MM=Ls)", and while it's quite close, L + s = 103.4 + 66.2 = 169.6 but 169.6/2 = 84.8, not 85.9 which is the M here.		However this is not quite an [[MV3]] scale. It is a rotation of the "LLMLsLMLMLsLM" pattern which is identified [[User:Xenoindex/Tridecatonic_MV3\|here]] as "Conditional (MM=Ls)", and while it's quite close, L + s = 103.4 + 66.2 = 169.6 but 169.6/2 = 84.8, not 85.9 which is the M here.
Line 1,379:		Line 1,370:
	\|}		\|}

	==See also==		== See also ==
			* [https://musical-patterns.douglasblumeyer.com Musical Patterns] - select Tsraxcfaubdj (GitHub page is here to explore the programmatic reasoning behind how this piece uses Yer: https://github.com/MusicalPatterns/pattern-tsraxcfaubdj)
	[https://musical-patterns.douglasblumeyer.com Musical Patterns] - select Tsraxcfaubdj (GitHub page is here to explore the programmatic reasoning behind how this piece uses Yer: https://github.com/MusicalPatterns/pattern-tsraxcfaubdj)		* [[Hilim13]], another tridecatonic tuning in the [[2.11.13.17.19 subgroup]], [[Just intonation\|JI]], designed by [[Gene Ward Smith]]
			* [[Gjaeck]], another tuning in the [[2.11.13.17.19 subgroup]], a tempered [[MOS scale\|MOS]] of [[57edo\|57ed2]], also designed by [[Douglas Blumeyer]]
	[[Hilim13]], another tridecatonic tuning in the [[2.11.13.17.19 subgroup]], [[Just intonation\|JI]], designed by [[Gene Ward Smith]]

	~~[[Gjaeck]], another tuning in the [[2.11.13.17.19 subgroup]], a tempered [[MOS scale\|MOS]] of [[57edo\|57ed2]], also designed by [[Douglas Blumeyer]]~~
	[[Category:Euler-Fokker genera]]		[[Category:Euler-Fokker genera]]
	[[Category:Tables]]		[[Category:Tables]]
	[[Category:19-limit]]		[[Category:19-limit]]
	[[Category:Just intonation scales]]		[[Category:Just intonation scales]]
	[[Category:~~2432/2431~~]]		[[Category:Blumeyer]]
	[[Category:Combination product sets]]		[[Category:Combination product sets]]
	[[Category:Comma pump]]		[[Category:Comma pump]]

BudjarnLambeth at 11:37, 25 April 2023

2023-04-25T11:37:21Z

← Older revision		Revision as of 11:37, 25 April 2023
Line 1,391:		Line 1,391:
	[[Category:Just intonation scales]]		[[Category:Just intonation scales]]
	[[Category:2432/2431]]		[[Category:2432/2431]]
			[[Category:Combination product sets]]
			[[Category:Comma pump]]

BudjarnLambeth: Categorised this uncategorised page

2023-04-25T11:36:57Z

Categorised this uncategorised page

← Older revision		Revision as of 11:36, 25 April 2023
Line 1,386:		Line 1,386:

	[[Gjaeck]], another tuning in the [[2.11.13.17.19 subgroup]], a tempered [[MOS scale\|MOS]] of [[57edo\|57ed2]], also designed by [[Douglas Blumeyer]]		[[Gjaeck]], another tuning in the [[2.11.13.17.19 subgroup]], a tempered [[MOS scale\|MOS]] of [[57edo\|57ed2]], also designed by [[Douglas Blumeyer]]
			[[Category:Euler-Fokker genera]]
			[[Category:Tables]]
			[[Category:19-limit]]
			[[Category:Just intonation scales]]
			[[Category:2432/2431]]

Cmloegcmluin: /* Yer as a temperament */ update link

2023-02-16T19:38:33Z

Yer as a temperament: update link

← Older revision		Revision as of 19:38, 16 February 2023
Line 1,263:		Line 1,263:
	⟨ 0 0 1 0 1 ] ⟩		⟨ 0 0 1 0 1 ] ⟩

	Again that's still in the 2.11.13.17.19 subgroup. And this one is the map-merge of 13&24&33, with [[generators]] 1200.4457, 4149.8305, 4442.6991 (that 1st generator modulo the [[period]] is 548.4934 and the 2nd generator is 841.362). So one ~17 is down two ~11, and one ~19 is down one ~11 and up a ~13. Cool! This you'd call "Yer temperament" then. It has been registered here: [[~~Subgroup_temperaments~~#~~Yer_.28rank_3.29~~]]		Again that's still in the 2.11.13.17.19 subgroup. And this one is the map-merge of 13&24&33, with [[generators]] 1200.4457, 4149.8305, 4442.6991 (that 1st generator modulo the [[period]] is 548.4934 and the 2nd generator is 841.362). So one ~17 is down two ~11, and one ~19 is down one ~11 and up a ~13. Cool! This you'd call "Yer temperament" then. It has been registered here: [[No-threes subgroup temperaments#Yer (rank 3)]]

	A 13-note scale with only three scale step sizes is possible here. It's 7L 4M 2s, with the pattern LMLLMLsLMLMLs, where L ≈ 103.4¢, M ≈ 85.9¢, and s ≈ 66.2¢:		A 13-note scale with only three scale step sizes is possible here. It's 7L 4M 2s, with the pattern LMLLMLsLMLMLs, where L ≈ 103.4¢, M ≈ 85.9¢, and s ≈ 66.2¢:

Cmloegcmluin: fix link to map-merging

2022-01-16T00:33:39Z

fix link to map-merging

← Older revision		Revision as of 00:33, 16 January 2022
Line 1,255:		Line 1,255:
	⟨ 0 0 0 1 1 ] ⟩		⟨ 0 0 0 1 1 ] ⟩

	Expressed as a [[map-merge]] of [[ET]]s, that's 13&113&137&194. So one ~19 is up one each of the ~11, ~13, and ~17 here.		Expressed as a [[map-merging\|map-merge]] of [[ET]]s, that's 13&113&137&194. So one ~19 is up one each of the ~11, ~13, and ~17 here.

	Now if you temper out the Blumeyer comma and the yama comma (and therefore also the blume comma; in fact, the [[canonical form]] of the [[comma basis]] appears to be blume and yama, with the Blumeyer comma being a linear combination of them), then you get a [[rank-3 temperament]], with mapping:		Now if you temper out the Blumeyer comma and the yama comma (and therefore also the blume comma; in fact, the [[canonical form]] of the [[comma basis]] appears to be blume and yama, with the Blumeyer comma being a linear combination of them), then you get a [[rank-3 temperament]], with mapping:

Cmloegcmluin: /* Yer as a temperament */ join, meet → merge

2022-01-03T00:13:37Z

Yer as a temperament: join, meet → merge

← Older revision		Revision as of 00:13, 3 January 2022
Line 1,255:		Line 1,255:
	⟨ 0 0 0 1 1 ] ⟩		⟨ 0 0 0 1 1 ] ⟩

	Expressed as a [[~~join~~]] of [[ET]]s, that's 13&113&137&194. So one ~19 is up one each of the ~11, ~13, and ~17 here.		Expressed as a [[map-merge]] of [[ET]]s, that's 13&113&137&194. So one ~19 is up one each of the ~11, ~13, and ~17 here.

	Now if you temper out the Blumeyer comma and the yama comma (and therefore also the blume comma; in fact, the [[canonical form]] of the [[comma basis]] appears to be blume and yama, with the Blumeyer comma being a linear combination of them), then you get a [[rank-3 temperament]], with mapping:		Now if you temper out the Blumeyer comma and the yama comma (and therefore also the blume comma; in fact, the [[canonical form]] of the [[comma basis]] appears to be blume and yama, with the Blumeyer comma being a linear combination of them), then you get a [[rank-3 temperament]], with mapping:
Line 1,263:		Line 1,263:
	⟨ 0 0 1 0 1 ] ⟩		⟨ 0 0 1 0 1 ] ⟩

	Again that's still in the 2.11.13.17.19 subgroup. And this one is the ~~join~~ of 13&24&33, with [[generators]] 1200.4457, 4149.8305, 4442.6991 (that 1st generator modulo the [[period]] is 548.4934 and the 2nd generator is 841.362). So one ~17 is down two ~11, and one ~19 is down one ~11 and up a ~13. Cool! This you'd call "Yer temperament" then. It has been registered here: [[Subgroup_temperaments#Yer_.28rank_3.29]]		Again that's still in the 2.11.13.17.19 subgroup. And this one is the map-merge of 13&24&33, with [[generators]] 1200.4457, 4149.8305, 4442.6991 (that 1st generator modulo the [[period]] is 548.4934 and the 2nd generator is 841.362). So one ~17 is down two ~11, and one ~19 is down one ~11 and up a ~13. Cool! This you'd call "Yer temperament" then. It has been registered here: [[Subgroup_temperaments#Yer_.28rank_3.29]]

	A 13-note scale with only three scale step sizes is possible here. It's 7L 4M 2s, with the pattern LMLLMLsLMLMLs, where L ≈ 103.4¢, M ≈ 85.9¢, and s ≈ 66.2¢:		A 13-note scale with only three scale step sizes is possible here. It's 7L 4M 2s, with the pattern LMLLMLsLMLMLs, where L ≈ 103.4¢, M ≈ 85.9¢, and s ≈ 66.2¢:

Cmloegcmluin: unhyphenate "comma basis"

2021-11-12T20:24:45Z

unhyphenate "comma basis"

← Older revision		Revision as of 20:24, 12 November 2021
Line 1,257:		Line 1,257:
	Expressed as a [[join]] of [[ET]]s, that's 13&113&137&194. So one ~19 is up one each of the ~11, ~13, and ~17 here.		Expressed as a [[join]] of [[ET]]s, that's 13&113&137&194. So one ~19 is up one each of the ~11, ~13, and ~17 here.

	Now if you temper out the Blumeyer comma and the yama comma (and therefore also the blume comma; in fact, the [[canonical form]] of the [[comma-basis]] appears to be blume and yama, with the Blumeyer comma being a linear combination of them), then you get a [[rank-3 temperament]], with mapping:		Now if you temper out the Blumeyer comma and the yama comma (and therefore also the blume comma; in fact, the [[canonical form]] of the [[comma basis]] appears to be blume and yama, with the Blumeyer comma being a linear combination of them), then you get a [[rank-3 temperament]], with mapping:

	[ ⟨ 1 0 0 11 4 ]		[ ⟨ 1 0 0 11 4 ]

Cmloegcmluin: /* Yer as a temperament */

2021-11-08T21:10:06Z

Yer as a temperament

@@ Line 7: / Line 7: @@
 Yer was developed by [[Douglas Blumeyer|Douglas Blumeyer]] as a system to explore the [[unnoticeable comma]] 2432/2431, the Blumeyer comma, which appears twice as a step in the scale.
-{| class="wikitable"
+{| class="wikitable center-all"
 |+
 !name
@@ Line 1,265: / Line 1,265: @@
 Again that's still in the 2.11.13.17.19 subgroup. And this one is the join of 13&24&33, with [[generators]] 1200.4457, 4149.8305, 4442.6991 (that 1st generator modulo the [[period]] is 548.4934 and the 2nd generator is 841.362). So one ~17 is down two ~11, and one ~19 is down one ~11 and up a ~13. Cool! This you'd call "Yer temperament" then. It has been registered here: [[Subgroup_temperaments#Yer_.28rank_3.29]]
-== See also ==
+A 13-note scale with only three scale step sizes is possible here. It's 7L 4M 2s, with the pattern LMLLMLsLMLMLs, where L ≈ 103.4¢, M ≈ 85.9¢, and s ≈ 66.2¢:
 [https://musical-patterns.douglasblumeyer.com Musical Patterns] - select Tsraxcfaubdj (GitHub page is here to explore the programmatic reasoning behind how this piece uses Yer: https://github.com/MusicalPatterns/pattern-tsraxcfaubdj)

Cmloegcmluin: Yer as a temperament

2021-11-08T21:02:28Z

Yer as a temperament

@@ Line 1,245: / Line 1,245: @@
 == Tritave-Based Yer ==
 One could imagine a tritave-repeating variation of Yer, where 2's are out but 3's are in (the EFG of 11, 13, 17, 19 remains but is tritave-reduced instead).
 == See also ==