Euclidean rhythms

January 15th, 2009  |  Tags: ,

“The Euclidean Algorithm Generates Traditional Musical Rhythms” (pdf link) is a fascinating paper by Godfried Toussaint at McGill. Friends of this site will likely guess that I was powerless to resist reading the whole thing after glancing at the first paragraph:

What do African bell rhythms, spallation neutron source (SNS) accelerators in nuclear physics, Sturmian words and string theory (stringology) in computer science, Markoff numbers and two-distance sequences in number theory, drawing digital straight lines in computer graphics, calculating leap years in calendar design, and an ancient algorithm […] for computing the greatest common divisor of two numbers, originally described by Euclid, have in common? The short answer is: patterns distributed as evenly as possible. For the long answer please read on.

Touissant shows that the Euclidean greatest-common-divisor algorithm has the same structure as an algorithm, due to Bjorklund, for evenly scheduling n pulses in k units of time. (Bjorklund’s application was scheduling high-voltage power over intervals.) He then demonstrates that this algorithm can be used to generate musical rhythms that have appeared in music throughout recorded history; for example, scheduling three pulses over eight units of time results in the tresillo, or 3+3+2/8 rhythm. (This rhythm is familiar in African and Caribbean music; Touissant notes that it is also the rhythm of the bass part in Elvis Presley’s “Hound Dog.”)

This discussion alone would make for an excellent read, but it represents only the first fifth of the paper! Touissant also identifies a wide range of actual rhythms from world music that can be generated using this technique; compares rhythms generated using this technique to rhythms categorized in the aksak system of rhythms (which I first encountered, but not by that name, in the music of Bartók); and finally, makes convincing analogies with finding leap years, drawing straight lines on grids of pixels, and several other problem domains.

I haven’t had so much fun reading a research paper in ages. If you are reading this site, you will probably enjoy Touissant’s paper.