hmm, well, first of all I must say I'm a great fan of melodiousness (because I like that spelling), just to start off my post with a statement of ultimate ambiguity :P
What do I define as melodic? well, the Oxford dictionary defines melody as, among other things:
"sweet music, tunefulness"
and I believe that this might not be as ambiguous as it first seems.
While there is still always going t be argument over whether something sounds sweet or not, it's usually easy to say whether something jars or whether it flows. If it flows, it's melodic, if it doesn't it isn't.
That isn't to say that melody is the be-all and end-all of music though, discord and jars can form some incredible music, just not melodic music. non-melodic music it however a relatively new idea, in terms of music: most classical music will be melodic in nature, it's only when you get to the 20th century that "unmelodic" music starts being experimented with.
Of course, that's just one meaning of melody. Another meaning of melody is the main tune of a song. as in if there's a basic riff or theme that dominates the song at any one time, that is the melody.
Thus, I suppose you could also define unmelodic music as music that lacks (a) central theme(s) (as poliphony is still melodic, in my opinion anyway), and so melodic music is something that isn't a complete and utter mess of sound :P
In that case, though, I would be inclined to say that anything that isn't melodic is purely noise, rather than music...
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