And here is sheet music and a midi of what seems to be the original tune, from the late Bruce Olson's terrific collection:


http://abcnotation.com/tunePage?a=www.fresnostate.edu/folklore/Olson/S1.ABC/0041

It is recognizably the current "Larry" tune with 18th century characteristics. But s-l-o-o-o-w.

Francis O'Neill included the "folk version" in O'Neill's Music of Ireland" (1903), as played by fiddler James O'Neill (no relative). Where O'Neill got it from is unknown: maybe tradition, maybe an old publication.