I've asked around those gompies out here in cyberland who are Rabid Tolkien Fans and we've come up fruitless for an original tune... the only example I have of Old JRR sining is him doing Namarie... and it's horrible.. he was not a singer...I also have an album of music by Donald Swan of songs that JRR put his stamp of approval on.. and they're mostly really rotten to, so I'm enclined to believe that Prof Tolk' wasn't much in the music department...
Can't be great at everything eh?
{~`