Gordon said: Did Beethoven improvise?
Of course. Bach, Liszt and others were well known, highly regarded improvisers.
Many Renaissance pieces were merely the starting point, a ground for the musicians. They were expected to demonstrate their sensitivity, creativity and technique with their interpretations.
This tradition remains alive. Here is Jordi Savall riffing on "Folias de EspaÃa," the equivalent of a 12-bar blues progression of the 1600's. Everyone knew it. The chord progression is simple, but listen to the tremendous variation of moods and techniques employed by the ensemble.
http://www.youtube.com/watch?v=5Frq7rjEGzs&feature=player_detailpage
Thanks, I really enjoyed that.
I suspect that, in my case, my minimal knowledge of the “the starting point, a ground for the musicians” makes it difficult [other than obvious morfations [new word] ] for me to recognize subtle character improvisations and therefore my enjoyment of the obvious ones done by crossover musicians.
With the former [ like wine] I appreciate it or I don’t.
I knoweth not whyeth.
As with the blues, it is the harmonic progression that defines the form. You then can do anything on top of it - smooth, aggressive, fast or slow. In the video you can hear that this progression is quite short and the various approaches.
Like everything else, it just takes a bit of experience to get a good sense of what is happening.
Then, just like wine, all that really matters is if you like it. 
I really enjoy understanding our current musical forms have their beginning in historical musical traditions, often hundreds of years old. I find this enriches my appreciation and listening.
Here’s another great album and Japanese contemporary of Yamamoto.
https://www.youtube.com/watch?v=FFlrVuBy5ok