So, hopefully this question doesn't sound too vague, maybe I can completely explain it here.
I have been trying to invent a theoretical instrument that has three main parts to it. The middle piece is just a linear tube with keys like any other woodwind instrument. The first piece is the mouth piece, I'm not sure whether I should make the mouth piece an open or closed hole (like a flute has an open mouth piece hole and the clarinet has a closed mouth piece hole). The end is a closed spherical chamber attached to the tube, like a cor anglais's bell, except it is closed and is a larger chamber. I don't know if this makes sense at all, I apologize.
I just want to know, how would this affect the sound of the instrument, having that spherical chamber at the end? How would it affect the pitch? If the instrument was closed at both ends, would it even produce a tone? I want to know the specific mathematics behind all of this, if that is even possible. I am speaking theoretically, so I have no specific measurements on these things.