I understand piano. I understand choral music. I simply DO NOT understand the concept of an instrument being attached to a certain pitch – a B-flat clarinet, a D trumpet.
Knowing that piano and voice are my points of reference, can someone explain to me what these designations mean and how, for example, a B-flat clarinet differs from an E-flat clarinet?
Before I even begin to explain, I’ll ask you: Do you understand the concept of the piano as a C instrument?
A transposing instrument is one in which, when your sheet music says C, and you finger a C on your instrument, a different note actually comes out.
With a Bb instrument, like a clarinet or a trumpet, when you play a C that’s written, the note that comes out is actually a Bb. When you’re playing a French horn in F and you see a C in your sheet music, the note that comes out is an F. An Eb clarinet…well, you get the idea.
The reason for this is…tradition. And other stuff. The Wiki article has a good explanation.
Let’s consider the sax family. If you learn the fingering for one sax (soprano, alto, tenor, bass and others) you can play another sax (soprano alto, tenor, bass and others). If your written music says “A”, you finger it for a “A”. Easy.
However, none of the above will sound a “A” (440 Hz for the one above middle C), also known as a “concert A”. But do you care? Not a bitty-bit. If you read printed music, the copyist has transposed everything written so when you play an A, the sound is what the composer intended.
Why are they different? Because as an instrument is made larger or smaller proportionally, all of the pitches change as well.
Sound stupid? But remember the alternative, where a tenor sax player would have to learn an entirely different fingering for ALL pitches from an alto, or a bass, or a soprano.
The whole concept is a compromise.
I admit I don’t really understand the Bb clarinet either. The idea of transposing, and having the same fingering for different instruments in the same family. However, Bb is only one tone away from C. Why couldn’t they just make it a little smaller?
Isn’t the clarinet an offshoot of the (clarino) trumpet? I have a feeling if you traced it from there you may get your answer.
No. You’ve got a pretty big heap-o-ignorance to fight here. 
It would sound different, and every piece of music that has been written for the B flat clarinet would need to be rewritten, or may not work at all. In fact, C clarinets are made, but rarely used, because there’s basically no call for them.
One of the purposes is what Musicat describes - a single set of fingerings for a family of instruments. This doesn’t just work for saxes and clarinets:
Recorders: all play from treble clef. The tenor sounds at pitch, the bass and treble are in F, sounding lower and higher than written, and the soprano sounds an octave higher.
Flutes: the piccolo sounds an octave higher than written. The alto and bass sound a fourth and an octave lower than written.
Strings: the double bass sounds an octave lower than written. (Admittedly, there’s no other common instrument using the same tuning of the strings. While coming from the bass’s lineage outside of the violin family, this one is also very much a matter of practicality.)
Yeah, I played a B flat cornet (like a trumpet but mellower) in high school. They make different cornets—C, E flat, etc.—and probably could engineer anything you wanted. But they wouldn’t all sound as good, which I think is why they settle on certain pitches.
The B-flat cornet or trumpet still has quirks. When using the third-valve pitches tend to go sharp, so there’s a slide (often spring-loaded) that the player is supposed to use, effectively lengthening the tube to “flatten” the pitch and bring it in line with the rest of the scale. But the player holds it in place when not using the third valve. Overall I guess B flat is the best compromise when compared to others. IIRC others tend to be sharp and require a lot more correction.
So if I played a C scale on my horn, it would match a B flat scale on the piano. I would play the fingerings: open (no valves pressed), 13, 12, 1, open, 12, 2, open. If you handed me an E flat trumpet, I would play the C scale by using the same fingerings but it would match an E flat scale on the piano.
IIRC I could also pick up a French horn (which I think is typically in F) and finger the same. Assuming I was able to find the tonic to start, I’d be playing a C scale that corresponded to an F scale on piano. This explains why you’ll sometimes see that someone plays “horns” on an album—once you know one, you can pick up the others pretty quickly, though there are other concerns. My band director told me to stop goofing around with my friend’s French horn. Though I could hit insanely high notes (she couldn’t go that high and she was first chair), my director said it could screw up the muscles in my embouchure.
Besides reasons others have given, tradition continues to be a big factor. While you might be able to tell the difference in sound, some musical [del]nuts[/del] purists probably can, and they would be upset, picket the London Symphony, all music would cease, and the world would be a sadder place. 
To expand a little on my earlier posts…if you are building a family of instruments and want to retain the same fingering for all of them, it would be nice if each larger one could be pitched down an octave (octave transposition is the easiest). But this would be too big a spread and leave a gap inbetween the two ranges. But if you size an instrument body in the middle, you will have to either invent an entirely different fingering plan, or declare the instrument to be a transposing one, probably in F or G. Then the only person who needs to worry much what pitch to write is the copyist.
An interesting note to the sax family…since an octave transposition is added to saxes at the extreme ends of the sound range, the written note range for all saxes is identical and so is the clef, from a written B flat below the treble staff to an F natural three ledger lines and a space above. Copyists love that.
I confess I don’t know why saxes are pitched in either E flat or B flat; they could have been in C and F, like recorders, then only half of the family would be transposing (the F model). Adolphe Sax probably had a reason, and I’m sure someone will be along soon to tell us what that was.
IANASerious trumpet player, but since all instruments in a family are built upon the same acoustic, harmonic properties, I don’t see why a B flat trumpet should differ in that regard from a D trumpet.
Me neither. WAG but by going with a smaller bore to raise the pitch, the tolerances become more critical for the manufacturer.
However, if you were playing in a band with a B flat trumpet, what he plays in C you’ll be playing in A flat. In order to give as many as possible (throughout the band/orchestra) a manageable key with the best sound quality, they chose the best compromises they could.
Having recently read a couple of articles in knitty the online knitting magazine about how a designer can take a sweater she knit for herself or a model and then adapt it for other sizes smaller and larger, it’s interesting to some of the same principles played out in a totally different format.
The first saxophone Adolphe Sax built was a bass instrument in C. He then proceeded to design a whole family of saxophones in various sizes and keys, including both C and B-flat variants in the soprano, tenor, and bass ranges, and F and E-flat variants in the sopranino, alto, baritone, and contrabass. The rest is history–players and composers came to favor the B-flat and E-flat versions, and the others became obsolete. C tenors, known as “C melody” saxophones, are still in very limited use.
That, AFAIK, is how all these transposing things came about–they all started out as instruments in C, different sized versions in various keys were developed, and players and composers found the variants more useful than the originals.