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== 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.  


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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
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!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 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.


{| class="wikitable"
{| class="wikitable"
|+
|+
!name
!Name
!value
!Value
!cents
!Cents
!prime exponent vector
!Prime exponent vector
|-
|-
|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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{| 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
|<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.
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{| class="wikitable"
{| class="wikitable"
|+
|+
!name
!Name
!value
!Value
!cents
!Cents
!prime exponent vector
!Prime exponent vector
|-
|-
|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|
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== 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.


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{| 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
|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&amp;113&amp;137&amp;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:
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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:Comma pump]]
Retrieved from "https://en.xen.wiki/w/Yer"