Tolkien's songs

When Tolkien wrote the songs in the Hobbit and Lord of the Rings, it seems highly likely that he had particular tunes in mind.

eg. If we look at

The Road Goes Ever On

The Road goes ever on and on
Down from the door where it began.
Now far ahead the Road has gone,
And I must follow, if I can,
Pursuing it with eager feet,
Until it joins some larger way
Where many paths and errands meet.
And whither then? I cannot say.

-- J R R Tolkien

What possible tunes might he have had in mind? (I'm ignoring Donald Swann's work as that was original tunes, and they weren't always as 'folky' as what I think Tolkien might have originally had in mind)

The song above works quite well to the tune of Lilibulero.

What suggestions do people have for other songs in the Tolkien canon?