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Why are certain instruments "in a key"

I played tenor sax about grades 5-10 and I knew that this was the case but also could never really understand why. The tenor was a Bb instrument so when the band tuned (to a concert Bb) we played C. At least that's what I think we were doing. Clarinets were doing the same (but theirs sounded an octave higher) while the alto saxes and trumpets were Eb instruments so I guess they played an F to tune to -- I don't remember and this all still gets me confused. Did they really start to do this so that our music wouldn't be written way up high (or down low) off the staff? Wouldn't it be simpler for each instrument to have its own staff, the same way that there are different treble and bass clef staves? Or is there some deeper magic to it?

Altos tuned to G.
 
Actually it is Bb. In higher level groups they will tune to A or F# generally, YMMV.
I think jmatt was talking about the note that altos would need to play to sound a concert Bb (see the post he was responding to). That would be a G (same for other Eb instruments like baritone sax).

For standard trumpets, tenor saxes and clarinets, to sound a concert Bb they would play their C.

This PDF is a scan of a little booklet that my Dad bought back in the 1940s when he was just starting to write arrangements for dance bands and big bands. It was very useful to me when I started arranging, too. There's a lot of information crammed in here. It covers the topic of transposing instruments pretty well.

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