Due to the number of songs in The Lord of the Rings, I have recently been wondering if Tolkien had any musical training and if any "official" music was written (by him or anyone else on his behalf) to accompany the verses of these songs?
Donald Swann's settings of a number of Tolkien's songs, "The Road Goes Ever On", were written with Tolkien's approval and input. I think they're probably about as close to "official" as you can hope for.
He has been recorded singing some of the songs, see for example here. Two albums were released on vinyl, I was lucky enough to have a friend whose mother owned them.
See also this YouTube video of the Tolkien singing 'Troll Sat Alone On His Seat Of Stone'.
Edit: Reading Tolkien's Beowulf and the accompanying commentary by Christopher Tolkien, it turns out that Tolkien wrote a lay in (modern) English based on the Beowulf epic, and CT recalls it being sung to him at a young age. The words are included in that volume--in fact two versions of the lay--but I doubt that music has been preserved.
No, Tolkien himself conceded he had "little musical knowledge" in Letter 260:
I have little musical knowledge. Though I come of a musical family, owing to defects of education and opportunity as an orphan, such music as was in me was submerged (until I married a musician), or transformed into linguistic terms. Music gives me great pleasure and sometimes inspiration, but I remain in the position in reverse of one who likes to read or hear poetry but knows little of its technique or tradition, or of linguistic structure.
Tolkien, with whom I corresponded in the nineteen sixties wrote music for one song, published in the Flanders and Swann volume, but was not musically trained. However, as an author and academic he had a keen sense of rhythm, melody and meaning, so his song words and poems lend themselves very well to music. When producing a twelve episode radio play of The Hobbit in 1967 for KRRC-FM (radio Reed College, Portland, Oregon) I created music for each of the poems that were presented as songs. These were recorded and included in the show, and later re-recorded in a studio in LA. I still have the tapes and am planning to rerecord and release them some time later this year.
I would like readers of J. R. R. Tolkien to consider the possibility that he knew many traditional songs, and had at least some of those melodies in mind when he wrote his poems/songs. Two examples follow:
Samwise Gamgee’s song about the troll eating a shinbone in the first book of the trilogy exactly matches the form and feel of “fox went out on a chilly night.”
Bilbo’s “The road goes ever on“ can be sung to “over the hills and far away,“ which also matches the spirit and feel of this song.
Tolkien’s immersion in ancient Norse which presumably includes knowledge of the ballad traditions and songs of that culture as well.