Gjaeck: Difference between revisions

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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|

Gjaeck

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== 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"

|+

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|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"

|+

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</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 ==

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

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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.

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 ==

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! blumeyer_tempered.scl

!

Blumeyer comma scale, 5L8s MOS of ~~57ed2~~

Blumeyer comma scale, 5L8s MOS of 57-EDO

13

!

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== 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:Tempered scales]]

[[Category:Pages with mostly numerical content]]

[[Category:MOS scales]]

[[Category:19-limit]]

[[Category:57edo]]

[[Category:Pages with Scala files]]

@@ Line 1: / Line 1: @@
-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|
 Gjaeck
@@ Line 6: / Line 6: @@
 == 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"
 |+
@@ Line 54: / Line 54: @@
 |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"
 |+
@@ Line 82: / Line 84: @@
 </math>
-Gjaeck tempers out some other high-limit commas too:
+p 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 ==
-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
@@ Line 99: / Line 101: @@
 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 ==
@@ Line 107: / Line 109: @@
   ! blumeyer_tempered.scl
   !
-  Blumeyer comma scale, 5L8s MOS of 57ed2
+  Blumeyer comma scale, 5L8s MOS of 57-EDO
   !
@@ Line 127: / Line 129: @@
 == 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:Tempered scales]]
+[[Category:Pages with mostly numerical content]]
 [[Category:MOS scales]]
 [[Category:19-limit]]
 [[Category:57edo]]
 [[Category:Pages with Scala files]]