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?
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.