I do tapered barrels an alternate way, I know, I know, I'm not a machinist. But, I've never had to deal with chatter on my light Atlas lathes I used. I don't set over the tailstock. I chuck the barrel on centers, and turn it concentric. If the breech end is concentric with the bore, hold it in a 3 jaw chuck to drive. If it isn't, I indicate the barrel in using a range rod and/or a gauge pin in the bore, and turn it concentric. That way, I can drive the barrel with the breech held in a 3 jaw chuck. Then, I turn the barrel to the largest diameter, using a sharp V tipped HSS cutter. Leaves a rough turned outside, like very fine threading. Next, I take a yardstick, and a sharpie and make a mark every inch along the barrel. Calculate the taper of the barrel in thousandths per inch of barrel. Say, it comes out .016" per inch. Start at the muzzle end, and start with a .008" cut. As the carriage moves, turn out the cross feed at a rate that will take out .008" per inch of travel. You'll cut a taper for 1 inch, and then it won't cut. Restart, and feed in .016". Again, turn out the cross feed at .008" per inch. This time you'll cut 2 inches along the barrel. Now, remark the 1 inch marks you've cut off, so you can estimate the feed. Repeat this, increasing the feed by .008" each time, until you have the entire taper cut. This is faster than the time it takes to set over the tailstock, check it, and reset it and check it when you're done. You can do a taper amazingly accurate by this method. At first, you're not very precise, but by the time you're repeated it 25 times along the barrel taper, you're pretty good and you'll come out with a uniform, smooth taper, ready to clean up with a file and start polishing. For some reason, the changing feed on the cross screw seems to suppress the chatter and make the cut come out better. Probably because non-linearity in the hand feed adjustment cancels out the harmonic chatter. I then file it while turning at slow speed, holding the file at a 20 degree or so angle, and feeding towards the headstock. Then, I take a couple 3 or 4 inch wide pieces of 1" boards, clamp them together, and drill a hole slightly larger than the largest part of the barrel, centered on the seam. You now have a half round hole in the board to hold sandpaper to sand the barrel, again at slow speed and feeding toward the headstock while holding moderate pressure.
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