I’ve been seriously working on jazz for about 1 year at this point. One of the things that mystified and literally stopped me in my tracks over and over was the notion of what it means to learn the tune.

I’ve been a classical, theater, and pop trumpet player all my life – playing off of sheet music. One skill you need to have for that kind of playing is sight reading. You look at the sacred inscriptions on the paper and create sound in a faithful reproduction of what the composer intended. Usually you need to do this simultaneously with many others and you all have to more or less agree on what is supposed to happen!

You can memorize music so that you are essentially playing it from sheet music in your mind.

This is not what it learning the tune seems to mean at all. The tune is often written out in a fake book along with chords to play. Not much to go on for the classical trumpet player. However, this is where learning the tune really is. You need to be able to play the melody by ear. You can probably sing dozens of melodies easily. You may miss a note or two but you generally sing it correctly. You need to be able to do that on your trumpet too.

Still, that is not learning the tune.

You need to understand how the melody weaves its way through the rhythm and the chords. For every note in the melody there are alternate notes that could have been there which follow a different path through the harmonic space created by the chords. To learn this deeply get the Improvise for Real book and do the exercises.

You need to hear how each chord moves the tune along like the rises and falls, twists and turns of a road through a landscape. Once you start to understand the landscape you can take different paths which criss cross the original or go off on their own.

Learning the tune is understanding the harmonic landscape of it.