Hot answers tagged improvisation
What I think you should be able to do is: Play the melody fluently; if you can learn it by heart, even better, but if not don't worry. Play the melody slightly varied. If you listen to the same jazz song by many artists, you'll see that none of them play it the same. Everyone changes it a bit here and there. That's something you'll have to do yourself. ...
This is a very broad question, and each individual subquestion is very broad by itself, but I'll try to point you in the right direction. You should grab a copy of either Mark Levine's "The Jazz Theory Book" or Robert Rawlins' "Jazzology". In my experience, those are the most recommended general jazz theory books. Check reviews in Amazon and similar sites ...
Following on from Shev's really good answer, other facets are to be able to play each song at different tempos, and in different keys. Often jazz players 'mess around' with standards, and use different tempos, and sometimes time sigs change, just for fun - or a challenge. Keys will change for songs as they become dependent on the vocalist. "I know xyz is in ...
Improvisations is at its core completion of a melody. That questions you do in your theory exams is there for a reason. If you can get some training in the writing of melodies and take that training into your playing then you will have the basics down to such an extent where you can improvise to any given music.
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