That's a good idea. But I don't think it actually works, at least with the obvious way to do it. Suppose you have a lathe whose screw thread has errors in pitc

That's a good idea. But I don't think it actually works, at least with the obvious way to do it. Suppose you have a lathe whose screw thread has errors in pitch such that the position of the middle thread is incorrect. With the right gears, you can use that lathe to cut a second thread with a different pitch. But regardless of what pitch you choose, the middle of that second thread will have the exact same absolute error in that position as the original thread. So you haven't actually improved anything. The best I can think of to improve the accuracy of a thread is to measure the thread against multiple length standards, and grind/lap away material by hand to bring the actual position along the thread length as close as possible to your length standards. However, as the number of length standards you can practically measure against is limited, you'll need to already be at a point where relative pitch error is small. Averaging out errors with a long follower nut is probably one way to do that.