I suspect that this will be of interest mainly to horn players, because I have never heard of players of other instruments wanting different mouthpieces for different music, but I may be surprised by a player of trumpet, trombone or tuba.
For the last 5 months, I have been using a Dennis Wick No 4 mouthpiece. This has the largest rim I have ever used (18 mm inside diameter) and a fairly deep cup. It has improved my bottom register a great deal, and I can now make a reasonable sound over three and a half octaves, from C below bass clef (F concert) to G at top of treble (C space concert). At the same time, I made my Hoyer full double horn my main practice instrument, and this has also contributed to the better sound. I knew that the top register would become more difficult, and indeed it has, though after a fair amount of practice I can now work my way up to top C. It is not yet reliable enough to be attempted in performance, so I must consider myself a second and fourth player for the present.
One of the excellent professional players on the horn mailing list that I take has pointed out that when one of his pupils moves to a mouthpiece with a larger rim, he suggests that they should combine it with a shallower cup, so as to minimise the greater difficulty in the upper register, and this makes acoustical sense. Also, another experienced player suggests buying a screw rim onto which you can put cups of different depth, so that the lips always feel the same rim, but the playing characteristics of the mouthpiece can be varied according to choice or to suit the register that is most important at the time. This would be ideal if I could find a screw rim with which I was comfortable, but the one I have tried, a 17 mm rim with a 22 mm cup from the Halstead-Paxman-Chidell system, started to give me pain in the lips, so I had to abandon it.
Next best would be to have two mouthpieces from the same maker, with the same rim and different cups. Unfortunately, although there are two mouthpieces in the Dennis Wick range with 18 mm rim inner diameter, they both have the same cup and differ only in the width of the rim.
To start discussion with a some questions: does any of you know of other screw-rim mouthpiece systems? do any of you change mouthpieces according to the music you are playing? Do other brass players fuss about mouthpieces the way horn players do?