Summary: Music for transposing instruments is not written or read at concert pitch.
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In order to make things run smoothly for composers, performers, piano tuners, and instrument makers, a standard has been developed that assigns a particular pitch to every written note. In other words, to save time and hassle, everybody has already agreed on what a C sounds like. This standard is called concert pitch. Most instruments are C instruments. The music for a C instrument is read and played at concert pitch.
A pianist, a cellist, a trombonist, and a flautist all see a C written in their parts. They may play the C in different octaves, but they will all play a note that the others recognize as a C. This may seem obvious, but if a clarinetist sees a C on the page, she will play a note that does not sound like a C to the other players. This is because the clarinet is a transposing instrument. The music for transposing instruments is not written or read at concert pitch. The clarinetist, for example, seeing a C on the page, will play a note that sounds like a Bb. The clarinet is therefore called a Bb instrument. A horn player, seeing a C on his "horn in F" or "F horn" part, will play a note that sounds like an F. Obviously, not just the C but all the notes are different. For a Bb instrument, for example, not just the C sounds a whole step lower, but every note sounds a whole step lower than written. In order to be read correctly by most players, music for transposing instruments must be properly transposed.
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Since every note of the scale is changed, the result is a different scale. This means that the part for the transposing instrument will also be in a different key and have a different key signature than the parts for C instruments.
Some transposing instruments do not change key, but play an octave higher or lower than written.
Things do run more smoothly when everyone agrees on the same name for the same sound. So why are there transposing instruments? The instruments that transpose an octave have either a very high or very low range. Transposition puts their written parts comfortably in the staff and avoids using too many harder-to-read ledger lines.
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Other transpositions are for the convenience of the player. Someone who has learned to play flute or soprano recorder, for example, associates a particular note with a particular fingering. If he plays an alto flute or alto recorder, he can use the same fingerings for the written notes, as long as the part has been appropriately transposed. If it has not (and some modern composers do not bother with transposition), he must learn to associate the same fingerings with different written notes, which can be confusing.
Other transpositions used to be for the convenience of the player, but are now mostly accidents of history. For example, there was a time when French horns, like harmonicas, came in every key, and could only play in that one key. French horn players could switch between different instruments playing what looked like the same set of notes, but which actually sounded in whatever key was needed. As the horn became capable of playing all notes equally well, the horn in F was the one that was chosen as having the nicest sound, so players still read parts in F.