Matthew, you can hear ol' Ted for free at each Oxonmoot. You just have to be prepared to run when a certain someone gets up to sing with him...
Now I've no idea if Tolkien could actually read music or play anything - there's no record of that as far as I know (but I'll have a rummage in the encyclopaedic Companion & Guide, as it will be in there if it's anywhere). But it's likely he did have an 'ear' for music. He was born in a time when radios and televisions were not around, and gramophones would have been forbiddingly expensive as his family was quite poor, so making your own music would have been the 'norm' - even if he could not read music he could probably sing - my own grandfather could not read music but it didn't stop him from being able to play piano and accordion pretty well. And of course he used to attend a Catholic church and music is always important in services; I imagine they will have had a sung mass.
I actually quite liked that little melody they created for "The Road Goes Ever On" in the films - I found it quite stately.
