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One of the more complex projects. We have many diverse views of rhythm and this project will investigate only a few. In past incarnations, I've got into trouble with this due to my own faulty thinking about the subject. Rhythm is such an integral part of the texture of music that we often forget how important and, for that matter, how inevitable it is. I'm almost tempted to ask you to write a piece that has no rhythm...
(Yes. Think about it.)
So, rhythm is neither attached to harmony or melody, nor is it distinguishable from it. It (in general) forms a part of the basic structure of music - after all, music is a temporal art-form and therefore probably includes some notion of duration. Indeed, much music, especially that of the non-western world, depends on rhythm almost exclusively. Popular musics also rely very heavily on rhythm often to the exclusion of many other musical elements - listen to a lot of rap and hip-hop music...
This leads to one of the first misunderstandings - that rhythm is essentially reiterative. When most people think of a 'rhythmic' piece it will generally include at least an element of reiteration - the rhythm will repeat and repeat, even if not precisely. There is evidence that this is a very basic human function. Many composers, especially in the twentieth century, have taken advantage of this; notable cases include Stravinsky in Le Sacre du Printemps and Bartok in many pieces including his fourth string quartet.
Good luck!