improvise on (something)
To use an existing creative work, usually a song or other such piece of music, as the basis for creating something new and different. Hey, tell the piano player to improvise on the theme I play in the intro, OK?
Farlex Dictionary of Idioms. © 2015 Farlex, Inc, all rights reserved.
improvise on something
[for a musician] to create a new piece of music on an existing musical theme. For an encore, the organist improvised on "Mary Had a Little Lamb." She chose to improvise on an old folk theme.
McGraw-Hill Dictionary of American Idioms and Phrasal Verbs. © 2002 by The McGraw-Hill Companies, Inc.