but the sheet I got said Bb cornet / concert pitch / C = C.

This missing part was originally in brackets, and somehow was deleted from the original message.

I’ve been told, innumerable times, that if I’m going to play with others, I’ll need to transpose the music. No, I won’t. I just played in a quintet at church: My Bb cornet, a Bb trumpet, an alto saxophone, a euphonium and a clarinet. When the minister (the Bb trumpet player) was arranging a variation of the standard “O Come All Ye Faithful,” I told him I wanted my sheet to be in concert pitch. He looked at me a little funny, but the sheet I got said . Low and behold, when I played Middle C on the score (valves 1 and 3), my C matched everyone else’s C Middle (261 Hz).

On several occasions I’ve taken a piece of music, lowered it a couple of steps to make it more comfortable to play, given a copy to my pianist, and guess what: my Bb cornet Middle C (valves 1 and 3) sounded just like the Middle C on her piano, 261 Hz.

I’ve also been told this kind of thinking will mess up all of the method books. No, it won’t. Suppose the lesson is a series of quarter note – eighth note – sixteenth note patterns, with notes C – D – E. The fingering will simply be 13 – 12 – 2, rather than the needlessly transposed 0 – 13 – 12.

It is actually a Bb. It doesn’t not appear to be a Bb, it is. Because of the tubing of the instrument and the conveniences of writing sheet music we write it on the page as C.
-Estela