See also post 285 and 282
Most musical instruments players, memorize their musical pieces, by just playing them sufficient many times, and letting the subconscious memorize them.
There are neverthless short notes about a melody that helps to remember it (especially when the target is an improvisation over the melody) which are not more complicated than the chord progression which underlies the melody.
Here are hints for it
1) Determine and take note of the root of at least one main diatonic scale from which the melody may deviate and the melody has maximal number of notes in it among other diatonic scales (there may be more than one!) .
2) Determine and take a note of the parts A B C etc and repetion (loop) pattern of them
3) Determine initial-final note of each melodic theme in the parts as it varies , as well as lowest and highest note of it and take a note of it attached to the part
4) Determine and take a note of the longest central notes that define the simplicial submelody that should not me more than the chords of the chord progression
After all the above 4) aspects of partial information about the melody , listen to it sometimes to refresh the memory about it, and then improvise on it without changing the information that you took notes. Chose the best version that you feel better , and play it as your own version of the melody.
On the other hand one may have heard about national or world memory completions, where memory athletes memorize in seconds the order of whole decks of cards, or hundreds of random numbers etc. And most people thing that such people have extraordinary memory! WRONG! Al these people have average memory abilities, BUT have learned the standard method of memorizing complex sets of information. Boldly speaking the method (an ancient Greek method, Cicero's method) is to link the elements of the information you want to memorize in to larger pleasant story! This obviously applies to long and complex melodies, and chord progressions, where the basic parts of the melody are linked between them with a story ...with much more information.
Below are two video that explain how Cicero's method may make an average man a memory champion!
http://www.ted.com/talks/joshua_foer_feats_of_memory_anyone_can_do
https://www.youtube.com/watch?v=9ebJlcZMx3c
