Then you should rethink that. In my view, the rim is more important than the cup. The cup width is also important, as it defines how much muscle you can put into play, and the depth has a major impact on 1) pitch in the high range, and 2) tonal qualities in the low range. Still, the best cup in the world does me no good if it is attached to a rim not to my liking. I have probably bought and sold $1200 worth of mouthpieces for that reason. Some of them had great cups, but I didn't like the rim. The rim (width and flatness) has a huge impact on 1) accuracy of attacks, 2) flexibility, 3) access to the high range, and 4) the focal point of the tone. Wider and/or rounder rims tend to get mushy (to some ears), while narrower and/or flatter rims will seem too hard-edged (to some), and there are infinite variations in between.I never really bought into the rim affecting the the horns sound and overall playability
At the same time, you have to consider the instrument you are playing. A Conn Helleberg on a Miraphone 184 CC will not sound like a BAT York, but that Helleberg on an Alexander or Holton will tend to offset the natural darkness of the instrument. Many players have to compromise between the needs of their face and the sound and technique they want out of the tuba.
In the old days, players used whatever mouthpiece came with the tuba. Today there are many more choices, but one should not be quick to dismiss the many combinations of rim width, rim contour, cup width, cup depth, bowl shape, and throat that are now available.
I gave up on finding that right combination for my Yamaha F, and finally made my own. I took a Laskey 30F, flattened the rim myself and buffed it, and sent it off to Dillons for goldplating. It is now, for me, the perfect mouthpiece for that tuba.
