That's exactly it. I have to say that the highest-level classical performers reach that level, but through different means. Not the improvisational part, but ha

That's exactly it. I have to say that the highest-level classical performers reach that level, but through different means. Not the improvisational part, but harmonically, understanding what the voices are doing in a counterpoint Bach passage for example. In classical you're an interpreter, and that's very challenging. In Jazz, you're an interpreter of an improvisation, which is like composing in real-time. I also have to point out that the great classical composers did improvise a lot, and that trend was lost with time but is documented if you research Mozart, Beethoven and Chopin for instance.