Jim, clear something up for us. You were turning a brass rod sticking out 1.5" from the chuck so ... no live center was used, right? You had a 0.002" taper over that 1.5" length, right?
Okay, so the first thing that comes to mind is deflection. What was the OD of the rod? What kind of turning tool were you using and how was that tool oriented to the work? The fact that the OD was larger on the unsupported end makes this a real possibility because it suggests the tool is pushing the work away on the end but is able to cut as it approaches the chuck end that is more rigidly supported.
Next possibility is that the headstock may not be aligned with the ways. When a lathe cuts a taper in a work piece, the knee jerk response is that the lathe is not properly leveled. However, when the work piece is really short and tailstock support is not used, the headstock is the more likely culprit.
It is also possible that something is loose - a gib in the cross slide or compound - or there may be excessive wear somewhere in the structures supporting the tool post that allows the tool to move.
I would have a look at all these things.