A few weeks ago I started reading about the Björklund algorithm, which is a method of distributing a certain number of items over a certain number of positions. I'll not describe that entire procedure here, as it is easily found elsewhere. The concept can be used to generate rhythms. For example, 5 accented notes over a length of 16 total semiquavers (1 = accented, 0 = unaccented) looks like this: 1001001001001000. As it turns out, this is a fairly common rhythm. Performing these accents with a kick drum (in conjunction with a standard rock beat with snare on 2 and 4) it would look like this:
Alternatively, you could spread the accent pattern over two bars of 4/4 by using a "quaver" grid instead of a "semiquaver" grid:
It could also be written like this:
It could also be described as "3 + 3 + 3 + 3 + 4." Shifting can also be applied without any compromise of integrity, and equates to the spinning of a necklace:
The above are all expressions of the Björklund rhythm E(5,16). E stands for Euclidian.
As an anecdotal aside, I remember once upon a party I had my djembe with me and I was trying to figure out how to evenly distribute 5 accents over a bar of 4/4 1/16th notes. I had no knowledge of the Björklund algorithm at that time, but it was pretty easy to figure out anyway. I chose to shift the necklace in such a way that none of the accents fell on beats 2 or 4 to avoid redundancy:
Lately I've been tending away from 4/4 time in favour of more obscure rhythmic adventures. So, when I started learning of the Björklund algorithm almost three weeks ago, it seemed like it would be extremely useful. I went ahead and crunched some numbers. For example, a rhythm in 13/8 time with 5 accents: E(5,13) = 1001001001010. Standard notation looks like this:
Here are a few more interesting examples:
E(6,15) = 100101001010010 (which could be written as 3 bars of 5/8 time),
E(5,13) = 1001010010100,
E(5,9) = 101010100, and
E(7,17) = 10010100101001010.
Next: Goldbach Conjectures. A few days later, I found myself reading about three Goldbach Conjectures:
1) The Strong Goldbach's Conjecture says that "every even integer greater than 2 can be expressed as the sum of two primes." I knew there must be some way to write rhythms with this idea. Furthermore, I wanted to combine Strong Goldbach with Björklund. It suddenly seemed obvious exactly how this should happen: I would use the The Strong Goldbach's Conjecture to "envelope" two different prime-number Björklund rhythms consecutively within an outer "even" rhythm. To help with this, I made a spreadsheet:
For a very "even" rhythm, I wanted to choose a sum that equates to good old 4/4 time, which would contain two of the more obscure prime-number time signatures. In the image above, I chose the number 32 (highlighted in yellow) because four bars of 4/4 time contain a total of 32 quavers - simple, common and even. From the table you can see that 32 is made up by adding the primes 13 and 19. Therefore, the associated Strong Goldbach-Björklund combination would consist of one bar of 13/8 time followed by one bar 19/8 time. The specific accent pattern, not yet established, requires deciding how many accents in each bar. Since I've been using the number 5 a lot, specifically for the next composition for my heavy metal band Volcano Calculator, I decided upon E(5,13) and E(5,19). The result is one bar of 1001010010100 followed by one bar of 1000100010001000100. Volcano Calculator decided our next composition would place snare drums in position 2 and 5, so here is what the standard notation looks like:
This combination of rhythms will sound very odd. However, this is where the idea of "enveloping" the prime rhythms within the "outer" even rhythm comes into play. In a multi-instrumental/band situation, some instruments can play the inner primes while other instruments play the outer even. So, the drums could play a straight 4/4 drumbeat while the guitar and bass play the 13+19 melody ...and/or vice-versa.
There is also the Weak Goldbach Conjecture, which says that "all odd numbers greater than 7 are the sum of three odd primes." This approach lacks the "even-ness" of the procedure above, but rather accentuates more "odd-ness" and supports the assembly of three consecutive sub-rhythms instead of only two. I plan on using this strategy as well... like so:
Booyah!
Q.E.D.
Sean A. Luciw
