We musicians strive to be creative whatever we are playing. When playing a written piece, it should sound fresh, as if improvised; when improvising, it should sound thoughtful, as if composed.
To improvise is make music up on the spot, within certain constraints. On the other hand, improvisation is not always totally from scratch. Often, an improvisation is building on previous work using the same material or structure. When you improvise, you don’t just play anything. You have to relate to harmony, rhythm, affect, and other players. More important, you have to create music that actually says something (not just a bunch of notes).
Excellent improvisation supports, and demands, mastery of all the elements of music.
Improvisation is not new to the classical tradition. All of the great composers were great improvisers: Bach, Beethoven, Mozart, Chopin, etc. It is only during last 100 years or so that classical musicians stopped improvising.
Furthermore, other traditions from folk, to eastern, to jazz have always influenced classical music.
Think of Mozart’s “Turkish” music, Beethoven contra dances, the Blues movement of Ravel’s violin sonata, eastern influence on the minimalists like Reich and Glass, Yo Yo Ma and the Silk Road Project. It goes on and on.
For a fascinating and beautiful example of violin improvisation in a thoroughly classical vein, at a high artistic level, listen to the album by Hilary Hahn And Hauschka, Silfra.
We play such great music that it is daunting to make something up. Anything that we make up on the spot will sound banal compared to the music that we practice and perform. This situation is exacerbated by trying to improvise in other styles, particularly if those styles have elements that are foreign to most classical playing (less refined tone, very different use of vibrato, focus on rhythmic drive from the bow, less refined approach to intonation, “feel”).
For this reason, I believe it is best not to start improvising by going to another tradition, but rather to develop improvisation as an integral part of your classical playing.
Playing with musicians from other traditions can contribute to this, but don’t start by trying become a fiddler, or a jazz player, etc. Develop by digging into your own roots, including exploration of the possibilities of basic musical elements: rhythm, pitch, harmony, and technique specific to the violin.
What is the difference between jazz and classical music, after all? They are two aspects of the same rich tradition, which have enriched each other for over a hundred years.
Playing with musicians from other traditions—particularly if their approach is totally intuitive and ear-based—can be a real challenge.
For a highly sophisticated rhythmic workout, check out konokol, the South Indian rhythmic system.
I believe it is highly beneficial to play music from other traditions, even if you can’t completely get the authentic feel. I can improvise on a blues progression, but I will never, ever, nail it like B.B. King—so what? I can draw on my own unique sources, and create music that no one else can.