Yer: Difference between revisions

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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.

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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.

== The scale ==

{| class="wikitable center-all"

|+

|+ style="font-size: 105%" | Steps of Yer

!~~name~~

!Name

!11

!13

Line 15:

Line 14:

!19

!EFG composition

!~~frequency~~ multiplier (definition)

!Frequency multiplier (definition)

!~~frequency~~ multiplier (decimal)

!Frequency multiplier (decimal)

!~~pitch~~ (¢)

!Pitch (¢)

!EHEJIPN (on C)

|-

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== 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"

|+

!~~name~~

!Name

!~~value~~

!Value

!~~cents~~

!Cents

!~~prime~~ exponent vector

!Prime exponent vector

|-

|Blumeyer comma

|2432/2431

|[[2432/2431]]

|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|

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{| class="wikitable"

|+

!~~name~~

!Name

!~~value~~

!Value

!~~cents~~

!Cents

!~~prime~~ exponent vector

!Prime exponent vector

|-

|~~yama~~ comma

|Yama comma

|209/208

|[[209/208]]

|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.

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{| class="wikitable"

|+

!~~name~~

!Name

!~~value~~

!Value

!~~cents~~

!Cents

!~~prime~~ exponent vector

!Prime exponent vector

|-

|Blume comma

|2057/2048

|[[2057/2048]]

|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|

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== 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.

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{| class="wikitable"

|+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

|3v1

|1

|<nowiki>| 7 0 0 0 -1 -1 -1 1 ></nowiki>

|<nowiki>[ 7 0 0 0 -1 -1 -1 1 ></nowiki>

|0.7120024978

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Line 1,192:

|4v0

|1

|<nowiki>| 15 0 0 0 1 1 1 1 ></nowiki>

|<nowiki>[ 15 0 0 0 1 1 1 1 ></nowiki>

|594.3140298

|7.5912943

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== Scala file ==

<pre>

! yer.scl

!

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247/128

2/1

</pre>

== Video explanation ==

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

== 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 ==

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== 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]]":

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⟨ 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.

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¢:

<~~nowiki~~>

<pre>

! yerMV3.scl

!

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1133.7

(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.

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|}

==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:Tables]]

[[Category:19-limit]]

[[Category:Just intonation scales]]

[[Category:~~2432/2431~~]]

[[Category:Blumeyer]]

[[Category:Combination product sets]]

[[Category:Comma pump]]

@@ 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.
@@ Line 7: / 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.
+== The scale ==
 {| class="wikitable center-all"
-|+
+|+ style="font-size: 105%" | Steps of Yer
-!name
+!Name
 !11
 !13
@@ Line 15: / Line 14: @@
 !19
 !EFG composition
-!frequency multiplier (definition)
+!Frequency multiplier (definition)
-!frequency multiplier (decimal)
+!Frequency multiplier (decimal)
-!pitch (¢)
+!Pitch (¢)
 !EHEJIPN (on C)
 |-
@@ Line 240: / Line 239: @@
 == 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"
 |+
-!name
+!Name
-!value
+!Value
-!cents
+!Cents
-!prime exponent vector
+!Prime exponent vector
 |-
 |Blumeyer comma
-|2432/2431
+|[[2432/2431]]
 |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|
@@ Line 281: / Line 279: @@
 {| class="wikitable"
 |+
-!name
+!Name
-!value
+!Value
-!cents
+!Cents
-!prime exponent vector
+!Prime exponent vector
 |-
-|yama comma
+|Yama comma
-|209/208
+|[[209/208]]
 |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.
@@ Line 295: / Line 293: @@
 {| class="wikitable"
 |+
-!name
+!Name
-!value
+!Value
-!cents
+!Cents
-!prime exponent vector
+!Prime exponent vector
 |-
 |Blume comma
-|2057/2048
+|[[2057/2048]]
 |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|
@@ Line 322: / Line 320: @@
 == 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.
@@ Line 712: / Line 709: @@
 {| class="wikitable"
 |+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
 |3v1
 |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
@@ Line 1,195: / Line 1,192: @@
 |4v0
 |1
-|<nowiki>| 15 0 0 0 1 1 1 1 ></nowiki>
+|<nowiki>[ 15 0 0 0 1 1 1 1 ></nowiki>
 |594.3140298
 |7.5912943
@@ Line 1,206: / Line 1,203: @@
 == Scala file ==
+<pre>
    ! yer.scl
    !
@@ Line 1,228: / Line 1,225: @@
 /128
 /1
+</pre>
 == Video explanation ==
 More details are presented in this video: [https://vimeo.com/184148765 Yer (pitch system)]
 == 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 ==
@@ Line 1,247: / Line 1,239: @@
 == 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]]":
@@ Line 1,255: / Line 1,246: @@
 ⟨	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:
@@ 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¢:
- <nowiki>
+<pre>
    ! yerMV3.scl
    !
@@ Line 1,286: / Line 1,277: @@
 .7
   (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.
@@ 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:Tables]]
 [[Category:19-limit]]
 [[Category:Just intonation scales]]
-[[Category:2432/2431]]
+[[Category:Blumeyer]]
 [[Category:Combination product sets]]
 [[Category:Comma pump]]