Melody and rhythm work together. For the purpose of a rigid definition, one could consider "melody" as the ordered pitches and "rhythm" as the duration of each pitch. However, the two must be taken together: a given "melody" with different rhythms will sound very different; so, too, a given rhythm with a different melody.
To borrow the curved-line analogy for melody: the x coordinates would be the time (rhythm) elements, and the y coordinates would be the pitch elements. Thus "melody" in this strict sense would be series of ordered pairs.
In fact, there is some music defined along these lines. "Serial music", and specifically "total serialism", uses mod12 arithmetic and corresponding set/group/category theory to determine pitch sequences, durations, harmonies, and any other musical element the composer chooses to define numerically.
But in most music one might commonly encounter, pitch, rhythm, and harmony all work together, and the boundaries between each can easily become blurred. They do not adhere to the kind of mathematically precise definitions that might be desired.
