Eucl*gic

A dual-channel Euclidean sequencer plus logic.

Two papers serve as the backbone of our investigation here. Godfried Toussaint¹⁾ was the first to notice that numerous traditional rhythms from around the world can be derived from the Euclidean algorithm. These patterns share the form $E(k,n)$, where $k$ notes are divided as evenly as possible among $n$ steps. When $k\mid n$, distributing those notes is trivial — simply place one every $n/k$ steps. For $k\nmid n$, E. Bjorklund²⁾ provides a generalized algorithm. Actually, he provides a number of algorithms capable of satisfying various degrees of constraints; fortunately, we need only examine the unconstrained case.

While Bjorklund's algorithm is perfectly serviceable, it relies on a number of arrays (count, remainders, and sequence) to get the job done, and the whole process must be re-run any time the parameters change. Is it possible to generate the sequence more efficiently? Better yet, could we calculate each step on the fly using only $n$ and $k$?

As an alternative to Boolean strings, we can represent Euclidean sequences with interval vectors. Using this notation, we would write $E(5,8)=10110110=(2,1,2,1,2)$.