Who says it's in a key at all? =P

Keys as we know them are only relevant in a 12-tone scale. In order to get that infinite number of notes off the violin string, you have to totally ignore the 12-tone chromatic scale. Take a look at eastern music, where sharps and flats and keys are just plain moot. Intervals of 1/3 of a second are common (even 1/4 and 1/8 tones aren't unheard of). Thats why the frets on a sitar are moveable.

Or perhaps it's 12 tone, but serial. Or 19 tone. Or maybe it's fractal. Or perhaps the chords are non-tertiary, to say, based off fourths and fiths instead of thirds and sixths. =P

Check out these sites for some non keyed weirdness.
http://eceserv0.ece.wisc.edu/~sethares/xentone.html
http://eceserv0.ece.wisc.edu/~sethares/otherperson/sierpinski.html

William Sethares also wrote a little book on alternate tunings that you can grab for free somewhere on that site.

Have fun. =P