I heard a couple of times about 15 years ago a classical guitarist playing a transcription of the Memories of Tomorrow encore (the last track of Köln Concert) on the Australian classical-music-only radio station ABC Classic FM. They'd never play the original – that would be "jazz", I guess, and inappropriate. But a classical guitarist playing the same notes – classical.
The second argument can be tricky though, if I played you Technical Death Metal in the piano you may think it's classical music too [0]. There is something more to being a genre than the notes that are played, I think...
Köln Concert does seem a very different kind of thing to Facing You though, although the latter is mostly totally improvised also, I believe. I guess due to F You's much shorter tune lengths and absence of extended improvisations over 1 or 2 chords etc. Facing You will play for 10 seconds with what Köln spends 5 minutes with..
> it's an important characteristic of the genre is not strongly shared by other genres.
What is not strongly shared, improvisation, or being entirely improvised?[0]
I also listen to a lot of North Indian classical..which seems about the same percentage entirely improvised as the Köln Concert.
I'm no an expert, but I don't think totally improvised western music is at all common outside "free jazz". i.e. it usually sounds like atonal noise, not pleasingly harmonious. That is very difficult. Classical musicians used to do it, until the 19th C, apparently, but gave up.
> There is something more to being a genre than the notes that are played
Absolutely, I've made this point many times myself, and it's something that was driven home to me when I started writing music with computers. Use jazz instrumentation and the music sounds like jazz. Rock instrumentation and it sounds like rock. Orchestra and it sounds like classical etc.
[0] Just to be clear, the last track isn't "entirely improvised", it's an improvisation over the Jarrett tune Memories of Tomorrow, which he'd been playing since at least 1966. And
> Subtle laughter may be heard from the audience at the very beginning of "Part I", in response to Jarrett's quoting of the melody of the signal bell which announces the beginning of an opera or concert to patrons at the Köln Opera House, the notes of which are G D C G A.
Both available here: http://tela.sugarmegs.org/
Hmm I doubt it. I'd expect that to exert force more in the other direction. Anyway, it's all speculation. There is a book called Keith Jarrett’s The Köln Concert which seems well-written.
p.s. I just keep mentioning Facing You because I think it's his best record! I've talked to quite a few musicians who feel the same.