There is an interesting reference in HoMe I to the instruments played by the 'Teleri' (i.e. Vanyar), 'Noldoli' (Noldor), and 'Solosimpi' (Teleri), respectively:
Vanyar - 'congregated harps'
Noldoli - 'viols and instruments'
Teleri - pipes
I cannot recall the exact quote, nor do I have the book at hand, but if you want to take a look, it occurs during the procession of the Elves to the festival in Valimar.
For what it's worth, I wrote something on music in Tolkien's Legendarium some time ago, which can be found
here.