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

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!19

!EFG composition

!~~frequency~~ multiplier (definition)

!Frequency multiplier (definition)

!~~frequency~~ multiplier (decimal)

!Frequency multiplier (decimal)

!~~pitch~~ (¢)

!Pitch (¢)

!EHEJIPN (on C)

|-

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== EFG as combination of CPS's ==

Every pitch in the system is a combination of either zero, one, two, three, or four of 11, 13, 17, or 19, for a total of 1 + 4 + 6 + 4 + 1 = 16 pitches. In other words it is the ~~[[wikipedia:Power_set~~|~~powerset]]~~ of {11, 13, 17, 19}. If familiar with [[~~Erv Wilson|~~Erv Wilson]]’s [[~~Combination_product_sets|Combination Product Sets~~]], or CPS's, it may help to think of an Euler-Fokker Genus as the intersection of the choose-1 tetrany, the choose-2 hexany, and the choose-3 tetrany, plus the unison, plus 11 * 13 * 17 * 19 all together which may be called the aota, an acronym for “all of the above”.

Every pitch in the system is a combination of either zero, one, two, three, or four of 11, 13, 17, or 19, for a total of 1 + 4 + 6 + 4 + 1 = 16 pitches. In other words it is the {{w|power set}} of {11, 13, 17, 19}. If familiar with [[Erv Wilson]]’s [[combination product set]]s, or CPS's, it may help to think of an Euler-Fokker Genus as the intersection of the choose-1 tetrany, the choose-2 hexany, and the choose-3 tetrany, plus the unison, plus 11 * 13 * 17 * 19 all together which may be called the aota, an acronym for “all of the above”.

[[File:EFG as CPSs.png|none|thumb|

EFG as combination of CPS's

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{| class="wikitable"

|+

!~~name~~

!Name

!~~value~~

!Value

!~~cents~~

!Cents

!~~prime~~ exponent vector

!Prime exponent vector

|-

|Blumeyer comma

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{| class="wikitable"

|+

!~~name~~

!Name

!~~value~~

!Value

!~~cents~~

!Cents

!~~prime~~ exponent vector

!Prime exponent vector

|-

|~~yama~~ comma

|Yama comma

|[[209/208]]

|8.303296728

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{| class="wikitable"

|+

!~~name~~

!Name

!~~value~~

!Value

!~~cents~~

!Cents

!~~prime~~ exponent vector

!Prime exponent vector

|-

|Blume comma

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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

@@ 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 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.
+== The scale ==
 {| class="wikitable center-all"
-|+
+|+ style="font-size: 105%" | Steps of Yer
-!name
+!Name
 !11
 !13
@@ Line 14: / Line 14: @@
 !19
 !EFG composition
-!frequency multiplier (definition)
+!Frequency multiplier (definition)
-!frequency multiplier (decimal)
+!Frequency multiplier (decimal)
-!pitch (¢)
+!Pitch (¢)
 !EHEJIPN (on C)
 |-
@@ Line 231: / Line 231: @@
 == EFG as combination of CPS's ==
-Every pitch in the system is a combination of either zero, one, two, three, or four of 11, 13, 17, or 19, for a total of 1 + 4 + 6 + 4 + 1 = 16 pitches. In other words it is the [[wikipedia:Power_set|powerset]] of {11, 13, 17, 19}. If familiar with [[Erv Wilson|Erv Wilson]]’s [[Combination_product_sets|Combination Product Sets]], or CPS's, it may help to think of an Euler-Fokker Genus as the intersection of the choose-1 tetrany, the choose-2 hexany, and the choose-3 tetrany, plus the unison, plus 11 * 13 * 17 * 19 all together which may be called the aota, an acronym for “all of the above”.
+Every pitch in the system is a combination of either zero, one, two, three, or four of 11, 13, 17, or 19, for a total of 1 + 4 + 6 + 4 + 1 = 16 pitches. In other words it is the {{w|power set}} of {11, 13, 17, 19}. If familiar with [[Erv Wilson]]’s [[combination product set]]s, or CPS's, it may help to think of an Euler-Fokker Genus as the intersection of the choose-1 tetrany, the choose-2 hexany, and the choose-3 tetrany, plus the unison, plus 11 * 13 * 17 * 19 all together which may be called the aota, an acronym for “all of the above”.
 [[File:EFG as CPSs.png|none|thumb|
 EFG as combination of CPS's
@@ Line 243: / Line 243: @@
 {| class="wikitable"
 |+
-!name
+!Name
-!value
+!Value
-!cents
+!Cents
-!prime exponent vector
+!Prime exponent vector
 |-
 |Blumeyer comma
@@ Line 279: / Line 279: @@
 {| class="wikitable"
 |+
-!name
+!Name
-!value
+!Value
-!cents
+!Cents
-!prime exponent vector
+!Prime exponent vector
 |-
-|yama comma
+|Yama comma
 |[[209/208]]
 |8.303296728
@@ Line 293: / Line 293: @@
 {| class="wikitable"
 |+
-!name
+!Name
-!value
+!Value
-!cents
+!Cents
-!prime exponent vector
+!Prime exponent vector
 |-
 |Blume comma
@@ Line 709: / 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