I checked my Playford book, the Peter Barnes edition. The version of Red House that we did is dated 1721. Of course, that's only the date when it was printed. It could have been around a long time before that, and I suspect it was. It's in the belly-dance minor (Am with G#'s), and it's great fun to play.

Our harpist pointed out that the chords are the same in every section, so why not play them all at the same time? So we did. (The jury is still out on that one.)

You can find it at JC's Tune Finder, number 37124. Look for the tune with an octave jump very close to the beginning.