Compare your lips against a mouthpiece to a violin string. Like a shorter string, a shorter lip area (created by a smaller ID of a mouthpiece) will produce a shorter vibrating surface. This shorter surface, as well as the added tension of the length, are what help produce pitches in the upper register. Like the longer string, the larger ID of the mouthpiece adds length to the vibrating surface that helps create a lower pitch with a thicker texture of sound.
Many players will choose a mouthpiece size because the rim feels good. While feel is important, it should not be the main criteria for selection. First, you need to have a realistic understanding of your musical range. Your musical range is not based on how high you can play, but how high you can play without the sound becoming thin and forced.
With this knowledge, play a simple melody or series of scales that takes you into all registers. Listen to the sound when you play. If the sound becomes thin and unfocused in the upper register, then chances are good that when you move into the extreme lower register, the sound will be too spread and "tubby" sounding. Conversely, if the lower register is unfocused and thin sounding, then chances are the upper register will be too strident and harsh when you play loudly. Choose a rim size that helps you produce an even sound in all registers.
Many times players believe that their playing ranges are based on the depth of a mouthpiece cup--if the cup is shallow, then they will be able to play high. The truth is, range is based on your ability to vibrate your lips. What the cup depth will affect is the color of tone you wish to produce. The deeper cups will assist in producing a thicker, darker sound. The more shallow cups will support a register that you already are capable of producing, as well as help create a more brilliant sound.
This article appeared on the original Laskey Website, circa 2018. It remains unedited.