rnd:projects:euclogic [UWiki XXX]

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====== Eucl*gic ======
A dual-channel Euclidean sequencer plus logic. 

===== Background =====
==== Euclidean sequences ====
Two papers serve as the backbone of our investigation here. Godfried Toussaint(("The Euclidean Algorithm Generates Traditional Music Rhythms")) 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. Bjorklund(("The Theory of Rep-Rate Pattern Generation in the SNS Timing System")) 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. 

<WRAP center round todo 60%>
include and discuss Bjorklund's algorithm
</WRAP>


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 $E(5,8)=10110110=(2,1,2,1,2)$.