To expand your analogy, I've heard from Shakespearean actors that they improvise while doing a play. That is to say, that they more or less memorize the script, but get into character and speak as the character would, not following the exact script. Players in a professional production can deal with this, and perhaps add their own improvisation, only within strict limits though. On the other hand, I was in an amateur production with one professional actor in the cast; he threw the rest of us by not following the script exactly. For instance, he'd compact two short speeches into one while another actor would be nervously waiting for a cue which never came. I think that's quite similar to the situation of accomplished musicians who understand improvisation, and struggling musicians who fear any movement from the basic structure of a piece.
By the way, my brother who teaches jazz and theory at a prominent conservatory, says that his formally trained students tell him, "I find improvisation hard -- I just don't get it." He tells them, "Well, of course not; it goes against everything you've been taught. You have to learn to approach music differently." He says, most music training teaches people to obey, while to improvise, they have to learn to be creative, cut loose and go their own way. I've learned a few things about improvisation by jamming with Eric Clapton and Wynton Marsalis, who are very nice, love playing, never criticize, and are available 24 hours a day on YouTube.
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