Some of the questions answered below
Brass Mouthpiece Breakdown
Parts of the Mouthpiece
The ease of play, comfort level and sound you're striving for will come a lot easier once you understand the various parts of a mouthpiece, and how they can affect your overall tone. Creating a bridge between the instrument and the player, a mouthpiece is made up of a rim, cup, backbore, and throat. The most essential components of a brass instrument mouthpiece is the rim and cup. The rim must be comfortable and sealed against the lips, for it's the main point of contact. In order to achieve a correct pitch and tone, the seal needs to allow the lips to move inside the cup. As for the backbore and throat, these will certainly affect your sound, feel and performance, but they're less of a concern, and typically, the manufacture will just match them to the rim and cup. The size of the backbore and throat can be changed in size by an expert repair technician, but once it is changed there's no turning back. Players are usually quite content with a medium cup and rim.
The round edge of a mouthpiece that you set your lips on is known as the rim. This is a critical part of the instrument. A rim with a moderately flat contour can make a pleasant, even lip pressure and comfortable air seal, but it will not allow much flexibility. For a lot of flexibility, rounded rim mouthpieces will do the job, granting the lips much freedom to move back and forth within the cup. However, rounded rim mouthpieces also require stronger lips and extra lip pressure. Without the air volume and embouchure needed, a player will wind up pressing too hard against their lips. This will cut off circulation, and limit the flexibility attained from using a rounded rim. For this reason, it's recommended to use a medium-wide rim with a semi-rounded contour and a somewhat sharp inner edge. This rim will give the player lip movement to cover an ample range, and also provide plenty of surface to make lip pressure easier.
This is the bowl-shaped area you blow into inside the rim. Typically, deeper and bigger cups produce more volume and have a darker tone. Bigger cups tend to require more control, which can tire out a player quickly. The smaller and shallower cups however produce brighter tones, and the response and control becomes more improved, making for an easier playability. Try not to go too shallow though. Doing so can hurt the volume and flexibility. There are two things to remember when deciding on cup depth – the instruments pitch and the players individual characteristics, from lung power and embouchure, to their lip and teeth formation. Try and go for the biggest diameter cup that you can use before tiring yourself out too fast. A wider cup diameter will provide extra uniform response over all registers, and smoother lip control.
This is the opening that leads out of the cup. The size and length of the throat doesn't come in many variations. Its job is to simply push air from the cup and concentrate the flow without too little or too much back pressure. You can blow stronger and create more volume with a larger throat, but more air pressure from your lungs will be required, which makes softer passages more difficult to perform. If the throat is too small, your tone will stifle, and too much back pressure will be produced. To ensure an excellent tone and projection, the majority of manufacturers construct the throat to compliment the backbore and cup design.
The job of the backbore is to transfer your breath to the horn. It needs to match the design of the mouthpiece's various components with minor adjustments possible in order to blow smoothly and provide favorable intonation. The tone could be brighter or darker depending on the backbore's shape and size. This can also affect its volume and pitch. The preferred choice in performance and tone is a well-rounded backbore that's not too big or small with a smooth, somewhat curved taper from the throat.