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What will happen if a tuba is played without one of it's rotary valve back bearing plates?

The short answer is nothing. However, keep in mind that the back plate is there to protect the valve from damage and the elements. Rotary valves are fragile; I would get that plate fixed or replaced.

The rotor is supported between the front bearing and the one in the center of the back bearing plate. The outer part of the rotor should not touch the inside of the housing. Without the back bearing plate... the rotor will drag inside the housing... or at least it should. Once the bearings are no longer supporting the rotor properly you begin to wear the inside of the housing and the outside of the rotor which will eventually cause loss of compression due to too much clearance.

Trying to play the horn without a back bearing plate is not good. Get it fixed.

If you literally mean the back bearing plate, then as others have said, that's very bad and you need to have that fixed.

If you mean the rotor cap that screws onto the back, that's less of a problem. That part is there to hold things together, keep it clean, etc. You could get away without having that for a bit, but again, get it fixed sooner than later.

Thank you all for the responses.