asymmetrical-mouthpiece
There have been mouthpieces made for people with mouth malformations that were Not symmetrical. The rim was offset to avoid a bad spot.
A second type in an oval shape were made in the mistaken belief that it would help in the upper register. Think about a piano string it can play higher by tension or by making it shorter. Tension works to a point but after that the mouth corners should come in to shorten the vibrating surface. This was suggested by lots of teachers Farkas included. The oval shape works against this.
A third type is an off-shoot of double cups. Instead of the sides being shallow and the center deep they make the top half deep and bottom half shallow. If you don’t place it perfectly on your lips then you end up with the shallow part stopping the buzz by touching the wrong lip. Lots of bad air deflection going on in the cup. Pivoting can be messed up. Plus total cup volume is what determines the tone color factors and here total volume is lacking.
Pops
