# Gjaeck

Gjaeck is a tridecatonic (5L 8s) MOS scale of 57ed2 optimized for the prime harmonics 11, 13, 17, and 19, while not optimizing for the simpler prime harmonics 3, 5, and 7.

## Temperings

The patent val for 2.11.13.17.19 prime limit 57ed2 is 57 197 211 233 242], resulting in the following mappings:

prime mapped (decimal) mapped (definition) scale steps octave reduced scale steps octave reduced interval pitch (¢) error (¢)
11 10.9749339402 2^(197/57) 197 197 mod 57 = 26 11/8 547.368 547.368 - 551.318 = -3.950
13 13.011851757 2^(211/57) 211 211 mod 57 = 40 13/8 842.105 842.105 - 840.527 = 1.578
17 17.0030222301 2^(233/57) 233 233 mod 57 = 5 17/16 105.263 105.263 - 104.955 = 0.308
19 18.9695563769 2^(242/57) 242 242 mod 57 = 14 19/16 294.737 294.737 - 297.513 = -2.776

The Blumeyer comma is tempered out by 57ed2.

name value cents prime exponent vector
Blumeyer comma 2432/2431 0.71200249782 [7 0 0 0 -1 -1 -1 1

We can see that both through multiplying the mapped values:

$\qquad 10.9749339402 ⋅ 13.011851757 ⋅ 17.0030222301 / 18.9695563769 ≈ 2^7$

Or through summing the scale steps:

\begin {align} \qquad &197 &+ &211 &+ &233 &- &242 &= &399 = 57 ⋅ 7 \\ \qquad &26 &+ &40 &+ &5 &- &14 &= &57 \end {align}

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 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:

ssLssLsLssLsL

In this scale we find varying counts of the tempered higher prime harmonics:

• 17/16 exists five times once for each L.
• 19/16 exists twice, once for each LsL.
• 11/8 exists ten times! For each L, you can take the six numbers to the left or the six numbers to the right (looping back around if you reach an edge; this is a cyclical set), and you’ll get it, so this is an 11th-harmonic-heavy mode.
• 13/8 exists six times! Like the 11/8, you can easily find each instance by looking to the L’s: for each one, you can take the nine numbers to the left or the nine numbers to the right and you’ll get it. We only get six 13/8’s because unlike with the 11/8’s, some of them overlap (they connect two L’s together).

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.

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

! 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
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