The Irish dance is up? Is that really the origin?
It's really the diminutive of the dance known as a "gigue". A lively number in 3/4 or 6/8 time.
A gigue by Bach played by some Kraut chick: https://www.youtube.com/watch?v=_vJsPC5u3lw
A jig by some Irish guy played by some Irish lass: https://www.youtube.com/watch?v=2ZxAVUsuE4Y