I am new to this forum, so please don't bite my head off for doing something wrong
I am a 17 year old pianist from Brisbane, Australia.
For me improvisation is one of the most important aspects of Western Music. Bach, Mozart, Beethoven etc used improvisation extensively, both as it's own form or helping to write their own music .The piano introduction of Beethoven's Choral Fantasy op. 80 sounds like a written down improvisation, and many have suggested that the second last variation in the third movement of his op. 109 sonata is also based on his 4-handed improvisational technique. (ie playing a trill with the thumb and index finger and the melody with the pinkie). I have to say that I find the Beethoven Sonatas very very moving and profound.
You can read an essay by Robert Levin discussing Mozart and Improvising at the URL below
I myself spend a lot of the time improvising on the piano, either using a framework that I developed or being in a total free state of mind. In my very very humble, 17 year old opinion, every performer in Western Art Music on any instrument should also be a good improviser. I agree very strong with what Mr. Levin said in his essay.
I have uploaded some of my works/improvisations onto YouTube, I hope links from YouTube are allowed on this site
Below is a sonata in G that is sort of composed and improvised at the same time
Mvt 1 - http://www.youtube.com/watch?v=YJquG8kagLY
Mvt 2 - http://www.youtube.com/watch?v=mqsJlK4ps10
Mvt 3 - http://www.youtube.com/watch?v=lVawSaubAP4
Sorry for the dreadful dreadful audio quality (well TBH it isn't too bad, I heard worse). After the concert many people came up to me and shook my hand and said many thanks for the performance, I was a bit taken aback by that I must admit
The sonata has a rough frame/structure, but is substantially different every time I play it. I use works from the Great masters to guide me. For instance the trills in the 2nd movement were based on the Last Three Sonatas of Beethoven and how he liberated the trill (I was drunk so to speak on the last Beethoven sonatas [op. 109 to 111] when I gave this performance), and the modulation into A major in the third movement was inspired by Clementi's Sonatina op. 36 no. 2.
And here is an example of a free improvisation (where I sat down at the piano and just played anything that was in my mind)
The audio quality is better here. This was recorded 2 years ago, and as one would expect my musical approach has changed (I don't use shallow virtuosity as much as I did then).
I am currently framing (like the sonata above) a new sonata based on Beethoven's op. 109 and 110, but with my own voice so to speak.
Sorry for these ramblings
A lot of talent is on this forum! I actually feel a little intimidated by it, but music should be shard and loved with all of us, whatever your level of skill is.
Best wishes to all