How does cup depth affect sound?

GR: I have received many phone calls regarding this question so I will attempt to address it using a different approach.

I would like to use science to address this question. Depth is only measured in one dimension. The cup ends at the point it enters the cylindrical section of the throat bore. Volume is a more appropriate word to define the mouthpiece cup. It is possible to have a very deep cup with a small volume and a very shallow cup that has a great volume. The key word is volume. The mouthpiece cup is non-linear in shape, furthermore, the volume has an effect on pitch and harmonic series.

There is another factor that we are overlooking and that is lip engagement into the mouthpiece.

If you subtract the volume of lip in the mouthpiece from the cup volume what you have left is the effective cup volume. This effective cup volume determines the playing characteristics of the player. That is why one person can play a double C on a Schilke 14A4a and not a 5C. This is the same reason one sounds laser like and the other round and full. Lip engagement is a major factor as well as the mouthpiece volume. If you tune at low C and they play a high C every .001'' increase of lip engagement will cause you to go 2 cents sharp.

What causes a change in sound bright vs. dark?

Take one mouthpiece with a smaller volume and 2 players. Player one has lots of lip engagement, sound very bright and laser like. Player two with much less lip engagement on the same equipment has a sound that is round and full. Why? Less lip in the cup will allow for a greater effective cup volume. As more lip enters the mouthpiece the peak or dominant overtone mode may rise from mode 5 or 6 to mode 9 or 10 or even higher as more lip protrudes into the cup. (Mode one or fundamental is pedal C, mode 2 is low C, etc.) Therefore, the higher modes become more present in the sound with greater lip engagement.

Some people have the same range with just about any volume mouthpiece. Some people can focus the sound on a smaller volume cup but not a larger volume cup due to their amount of lip engagement. Diameter can effect lip engagement. Some cup designs can increase range depending on individual playing characteristics. A change in resistance can also have an effect on lip engagement. This can have an effect on the lip equilibrium into the cup.

Everyone is different, therefore, a smaller cup volume will affect the range of some players. Certain cup designs will aid people to play in the upper register or we would not make them. Everyone is different and we must find what works for us. Volume of the cup minus lip engagement equals effective cup volume. This volume effects the overtone series and pitch not cup depth.