Thanks to Helen's question, I will be playing Kemp's Jig with my friends this Sunday. I have a duet from the Classical Music Archives, and they say that is was written by John Dowland. This would be the version in D, 4/4 time. It is the third piece on the Tunearch page which Helen linked above.

Here's a page with the original Playford dance, (Kemp's Jegg) which is the 6/8 tune in F.

http://imslp.org/wiki/The_Dancing_Master_(Playford,_John)

The second version on the Tunearch page, the version from Walsh, is easier to play if you change the key to F. That makes sense of the E's. But even after I did the work of figuring out how to play it, I didn't think it was worth the effort.