A private message to me read, "I recently acquired a sax and it has new cork on it. I tried several mouthpieces but they don't go on easily and they all play flat. Should I thin out the end of the cork nearest the bell with an emery board or fine sandpaper?
Seeing this is a common problem I decided to make my response more public:
The neck is tapered. The hole in the mouthpiece is cylindrical. But it is a thorough nuisance to have a tapered cork going into a cylindrical hole, because the further you push the mouthpiece on, the more ridiculously difficult it becomes to push it on.
So IMO, anybody installing a new cork, should use some method, eg say 120 or 150 grit sandpaper, to remove quite a lot of cork from the non-tip end of the cork, until the entire cork tube is cylindrical on the outside.
(My own process involves quite a bit of thinning before the cork is even glued on, using a sandpaper file - http://www.cws.au.com/shop/category/-sandpaper-file-quick-change - then more adjustment with the same tool after gluing and trimming with a knife.)
If the cork tapers at all, then it should probably taper in the opposite direction, i.e. fatter at the tip end. Allow me to explain... It is important that the tip end, say for the first cm, seals well. Otherwise, low notes can warble and other odd acoustic effects can occur. So this end must be a firm fit. The other end has no other function other than to stop the mouthpiece wobbling, so it does not need to be a very firm fit, even though it usually is, I suppose partly simply to reduce sand-papering time.
All this cork removal should be done before any cork grease is applied, because sandpaper barely works on greased cork, and quickly clogs with grease, even if the surface grease has been removed using naphtha (lighter fluid).
Just how much needs to be removed so that the mouthpiece eventually fits well with grease, is an educated guess based on experience. I suppose if the mouthpiece can be forced tightly on an ungreased cork about 1 cm, and no more, and the cork has been sanded cylindrical, then it will be somewhere near right once greased.
Unfortunately, it is rare for sax manufacturers to taper this cork. I suppose some technicians do; some don't.