Classical Chipmunk Melody?

[Alvin]

Anyone know the name of the piece or composer that has the original melody used in the Chipmunks Christmas Song? I thought the announcers said the name was "The Chipmunk" but when I search, all the results are David Seville's Chipmunk Song with no credit given to the classical composer.

It was played on KUSC classical radio station a few weeks ago but I didn't get the composer's name. Searched under every keyword I can think online and no luck.

Thanks.