Gjaeck: Difference between revisions
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Gjaeck is a [[Tridecatonic MOS|tridecatonic]] ([[5L 8s|5L8s]]) [[MOS scale]] of [[57edo|57ed]]2 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|5L8s]]) [[MOS scale]] of [[57edo|57ed]]2 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| | [[File:Gjaeck - cycle view.jpg|thumb| | ||
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== Temperings == | == Temperings == | ||
The [[patent val]] for 2.11.13.17.19 prime limit 57ed2 is <code>< 57 197 211 233 242 |</code>, resulting in the following mappings: | The [[patent val]] for 2.11.13.17.19 prime limit 57ed2 is <code>< 57 197 211 233 242 |</code>, resulting in the following mappings: | ||
{| class="wikitable" | {| class="wikitable" | ||
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|<nowiki>| 7 0 0 0 -1 -1 -1 1 ></nowiki> | |<nowiki>| 7 0 0 0 -1 -1 -1 1 ></nowiki> | ||
|} | |} | ||
We can see that both through multiplying the mapped values: | We can see that both through multiplying the mapped values: | ||
10.9749339402 | <math> \qquad 10.9749339402 ⋅ 13.011851757 ⋅ 17.0030222301 / 18.9695563769 ≈ 2^7 | ||
</math> | |||
Or through summing the scale steps: | Or through summing the scale steps: | ||
197 + 211 + 233 - 242 = 399 = 57 | <math> | ||
\begin {align} | |||
26 + 40 + 5 - 14 = 57 | \qquad &197 &+ &211 &+ &233 &- &242 &= &399 = 57 ⋅ 7 \\ | ||
\qquad &26 &+ &40 &+ &5 &- &14 &= &57 | |||
\end {align} | |||
</math> | |||
Gjaeck tempers out some other commas too: | Gjaeck 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 nothulo comma, 209/208, since 26 + 14 = 40; moving by an 11 and 19 is the same as moving by a 13 | * The nothulo comma, 209/208, since 26 + 14 = 40; moving by an 11 and 19 is the same as moving by a 13 | ||
== The MOS Scale == | == 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 57ed2 and a large step equal to 5 steps. It follows the small and large step sequence: | ||
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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, which is 736.8421¢, an interval associated with the 21st harmonic. | ||
While 57 = 19⋅3 and thus 57ed2 extends 19ed2, Gjaeck bears little resemblance to 19ed2; it feels more like a whacked out 13ed2, pinched and pulled in a particular way to more perfectly capture the four higher harmonic primes. | |||
== Scala file == | == Scala file == | ||
2/1 | ! blumeyer_tempered.scl | ||
! | |||
Blumeyer comma scale, 5L8s MOS of 57ed2 | |||
13 | |||
! | |||
84.21053 | |||
168.42105 | |||
273.68421 | |||
357.89474 | |||
442.10526 | |||
547.36842 | |||
631.57895 | |||
736.84211 | |||
821.05263 | |||
905.26316 | |||
1010.52632 | |||
1094.73684 | |||
2/1 | |||