Improvisation -- for me that can be expressed several ways.
The classical pianist may read the notes on the score for a Beethoven sonata and play every one of them without error. Still, their rendition will not likely sound like another pianist. They may vary the tempo, either overall or by intervals. They may insert pauses or hold sustain. They may play some passages (or notes) louder or softer. They are following the score (at least the notes) but they are performing with their personal emotion and thus interpretation. Are they not improvising for self expression?
Similarly a jazz pianist will play notes familiar enough that the song is recognizable, at least for the opening and ending bars. But they may also take the same personal interpretation steps with those segments, while adding more spontaneous intervals of some length through the middle. And unless it is "free jazz", those spontaneous intervals will follow certain "rules", the strict and predetermined framework of form and harmony frogman referenced. Otherwise the other musicians in the group could not follow them, nor could the audience. Still, there is great freedom within that framework, and the best musicians create something new and exciting with each playing. That is their improvisation.
And this is why I find some connection with bluegrass. The lead musicians are following a somewhat similar format as the jazz artists. The main difference may be their frameworks are more limited or restrictive. The still improvise, just not to the wide extent or with as many variations as jazz musicians.
More than speaking English, Previn and Peterson conversed easily because they spoke a similar musical language. So it flowed, had cohesion and energy, and was entertaining. Jazz artists communicate the same way musically. One does not know exactly what the other will play (say) next, but if it follows the predetermined framework then they can respond in a way that makes sense and hopefully builds upon that. I think it was simply Miles' talent that allowed him to carry on following Hancock who'd broken the rules of that framework, so that was a testimony to him. Not every good musician could have done that.
At least as a non-musician that's how I see it.
