Further to that: in die cutting threads, especially small threads, you don't want to start out with the rod at (or, worse, above) the tabulated major dia. for the thread, because the cutting force will go up enormously. Dies aren't really designed to cut on the tops of thread forms. If you were turning the threads on a thread cutting lathe (probably the way they were made at the factory) then you just keep feeding the single point tool in until the thread meets size (in factory work you'd check every 50th part or so with a go-no go thread gage; in repair work making one you're not going to do that). You can tell if the thread on the end of a rod was originally lathe cut; it'll have a short section of reduced diameter at or smaller than the minor diameter of the thread; that's the relief groove where the single point tool runs when you disengage the thread cutting lead screw before you back the carriage out to make the next cut.
If you have an adjustable die, then you can adjust thread size using that feature.
For making a single rod to go in a single place in a single musical instrument, I'm thinking the best plan would be: turn down the OD of the rod to halfway between max and min major diameter, as per a tabulated chart; use an adjustable die (if available in the size you're working); and use the adjustability to bring the thread down till it just fits in the post of interest, or if that's not applicable, into a standard hex nut of the thread's size. You, unlike the factory, don't have to worry about parts being truly interchangeable.