I had to use a half ground dead center today that I haven't brought used that much. The part I was machining was simple shaft ~4" long, held in pretty accurate 3J chuck & center drilled for tailstock dead center. I spotted a taper on first measurements about 0.003" larger diameter on headstock side vs. tailstock across the 4". This taper remained about the same as I turned down the diameter. Not happy about this, but I finished off the part just correcting diameter with a file on final finishing. I figured maybe this center was wonky because normally I use live center & did not recall anywhere near that much taper before. So lathe cleaned off I chucked a DTI in 3J & measured ID of my MT3 quill by rotating chuck. I get: TOP=0.000", NEAR = -0.0055" BOT = 0.000 FAR =+0.0050". I did this in a few positions down the inside of the MT taper, always the same. I got thinking maybe my MT bore is elliptic but that is way too much & lathe never abused. I inserted dead center into MT socket & measured around the point section & also back on the full circular section just outside the quill lap. Same readings. Now it may well be that the chuck jaws are gripping the DTI stem off a thou or 2 so zeroing DTI with the dial facing up initially (what I call TOP) may influence clock relative readings, but what explains this measurement mapping & the taper cutting I'm seeing? I know I can screw the tailstock in/out laterally to adjust taper, but shouldn't I be seeing some hint of this on the readings corresponding to the cut taper? I can repeat this with a 5C collet & hold the circular DTI stem in that for better concentricity than the chuck. But I figure once its rotating, it should all be relative about headstock axis, no?