The novels The Lord of the Rings and The Hobbit by J.R.R Tolkien both contain a number of songs, mostly sung by Hobbits, with fully fleshed out lyrics. A few of these songs even made it into the Peter Jackson movie trilogies, The Lord of the Rings and The Hobbit.

However, as far as I can tell, the actual notes and music to each song never appeared in any of the original books.

Did the production crew for The Lord of the Rings and The Hobbit trilogies make up their own music for these songs? Or was there some reference in a book, letter or transcript to how the songs should be sung?

I'm looking for an answer specific to the movies. This answer to a similar question shows that there was someone writing scores for the songs other than Tolkein, but not whether or not they were used in the movies.