If you take a good look at your headjoint, you will see the top is a bit smaller. There is a taper over nearly the full length of the headjoint. That is what Theobald Boehm calles the “parabola” . Of all the headjoints I have measured I have not really seen one with a parabolic headjoint taper, but allright , the taper does get progressively steeper. Each headjoint maker has its own recipe for this. The taper is neccesary to get the tuning of the higher notes in order, and also affects the sound quality.
In order to impart the taper design on to tube, I have a mandrel, a precision turned bar of tool steel, in the exact shape I want the bore of the headjoint to be. And with that mandrel inside the headjoint I push the tube through an undersize hole in a block of pure polyamide.
How does a player know if the headjoint taper is right? Just play. If the sound is right, and the tuning is good between octaves, then the taper is good.
headjoint and mandrel in press