Turning tenons is the area of pipe making that allows for the least amount of error. One must be very precise and accurate with tenon turning in order end up with a well fit stem. Tenon diameter variances of hundredths of an inch have a HUGE impact on the tightness of the fit between tenon and mortise. This is why it is typical to turn tenons with a metal lathe. A metal lathe can turn to very precise diameters. For example, my Taig is capable of accuracies of 5/10,000 of an inch! Not bad for a $400 lathe!
Actually turning the tenon is rather obvious if you are familiar with the use of a metal lathe. Rather than going through a step-by-step explaination of the process like I have in other areas of the site, I will list some pointers that are helpful in the process.
Always keep your cutting tools sharp! I don't think I can emphasize this enough. In order to make clean, smooth cuts, that are almost polished as they are cut, you must have a sharp cutting tool. I use a HSS cutting tool, and I need to resharpen it with every stem I make.
For the nicest cuts, set the lathe to turn at its fastest speed. Unlike drilling, turning does better when the RPM is higher.
Use calipers to assess how close you are getting to the desired diameter. You will want to stop about 3 or 4 thousanths before your final diameter so that you can polish the tenon with some sand paper. (If you have a sharp enough cutting tool, the need for this is minimized.) I usually hit the tenon with some 400, 600, and 800 paper with the speed of the lathe set very slowly. I then polish the tenon with adab of beeswax by gently bumping the spinning tenon with the wax. I then friction polish the tenon by squeezing the tenon with a clean rag. The wax polishes the stem nicely, and I can see if my surface is sufficiently smooth.
Ultimately, each block of briar is a bit different and will not drill identically to the ones before it. Though you know the approximate diameter you will need your tenon, alway use the actual mortise it is to be used in as a final guide. Do this check WITHOUT removing the stem/rod from the lathe. Once the stem/rod is removed from the lathe you will never get it back in precisely as it was before. With the stem/rod in slightly differently, you will need to turn off a relatively large amount of material to get the tenon round and true along its new axis. If you were almost to your desired diameter, there is a chance you will not have the material available to do this, and you will end up with an out-of-round tenon. When you check the fit in the mortise, you want it to be a tight fit. As you continue to work with the pipe, for reasons that are currently a mystery to me, the stem will loosen slightly. BE CAREFUL with thin shanks, I have cracked more than one shank trying to force and overly tight stem into a mortise.
Here is a video of me turning a tenon. It is a Widows Media file, and includes me describing the process as I go along.
VIDEOS Tenon and Inlay