There is little doubt about it, many brass players are fixated with embouchure and mouthpieces. Trumpet players in particular tend to collect them in order to get the very different nuances that different characteristics provide. So what is it about a brass player that feeds the obsession with the mouthpiece?
Well the first undeniable fact is the comfort of the individual. The variable nature of human physiology means that different shaped embouchures require mouthpieces with different characteristics to ensure the player is comfortable playing with it. This is particularly important with the width of the cup, players with a larger lip structure will often need a larger cup diameter to feel comfortable but this may have implications for the higher register. The other aspects of the mouthpiece, cup depth,Shank diameter and style are not really relevant when it comes to comfort
Then there is range and power. Different cup size will almost always affect the range of a brass player, a wider cup will make it more difficult to get higher notes because more tension is needed in the embouchure to get the same vibration and similarly a narrower cup will make it slightly easier to play in the higher register. Rim shape size is more about personal comfort but the shape and size of the shank can affect the feel of an instrument, a v shank with a small diameter for example will introduce more resistance into the instrument and for those instruments which have little natural resistance it can be of benefit.
It is all too easy though to forget that by far the biggest factor in producing a good sound in the higher register is air. Air is the fuel of all wind instruments and the real skill in playing a brass instrument is the ability to control the air that is moving through the instrument. Making a good sound is a balance of articulation, embouchure and a stream of air focussed down the centre of the mouthpiece. Yes proper breath control and posture are important but a smooth consistent airfow is what we are really after as that is what will produce a rich smooth sound.
OK, a mouthpiece is an important part of the process but do we really need to be so obsessed with them?