Like fractals, melodies can be self-similar, i.e., contain a stretched out copy of themselves. The factor by which one stretches need not be an integer, and in such cases interesting polyrhythms can arise. Further, self-similar melodies can act like strange attractors, where pseudo-random melodies evolve into self-similar ones. We explore many examples of self-similar melodies and emphasize how one can use these in compositions, ultimately culminating in musical realizations of both polyrhythmic, self-similar melodies and strange attraction.