did you really type that?to mend the joist
+1 for the "don't do it" camp.
Remember, you're looking at cross-sectional area. So instead of a 6" pipe running under the joists, what about a shorter but wider box of some type that would not hang so low? a 6" pipe has about 28 sq in of area. a 10"x3" box (inside dimensions) would be a little larger cross-section, but only drop half as much.
I say forget the "straightforward" route and plan a new one - maybe run a trunk along the wall at one end that runs perpendicular to the joists, and then run sections off that up in between the joists to get to where you want to drop down.
Another option would be to cut the concrete floor (assuming it's concrete) and bury duct under that.