Euclidean rhythm

The Euclidean Rhythm in music was discovered by Godfried Toussaint in 2004 and is described in a 2005 paper "The Euclidean Algorithm Generates Traditional Musical Rhythms" The greatest common divisor of two numbers is used rhythmically giving the number of beats and silences, generating almost all of the most important World Music rhythms, (except Indian).

Open Source Hardware Projects
Open-source music hardware projects that can generate Euclidean rythms, include Mutable instruments' MIDIPal, RebelTech's Stoicheia and Ruin & Wesen's Minicommand

Other uses of Euclid's algorithm in music
In the 17th century Conrad Henfling writing to Leibniz about music theory and the tuning of musical instruments makes use of Euclid's algorithm in his reasoning.