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 a clarinetist who sees a C on the page 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 French 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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Some transpositions are for the convenience of the player. Someone who has learned to play C trumpet, for example, associates a particular note with a particular fingering. If he switches to a B flat trumpet, 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 well in that key or closely related keys. 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.
There are also instruments that do not transpose but are also not considered C or concert-pitch instruments. Players of these instruments read concert-pitch music, but the instruments are considered to be fundamentally pitched on a note other than C. This is of very little practical importance, but is an issue that confuses some people, so let's take two examples. Soprano and tenor recorders, when all the finger-holes are covered (so that the air must go through the entire instrument), play a C. Alto recorders, when all the finger-holes are covered, play an F. Like B flat trumpets, this would seem to make alto recorder a good candidate to be a transposing instrument. If it were, a player could easily switch from one size recorder to another; a written C would have the same fingering on all instruments. But recorder history and tradition differ from trumpet history and tradition; so, although alto recorder can be considered to be "pitched in F", alto players learn to read at concert pitch, associating the fingerings with different notes than a soprano or tenor player would.
The second example is from brass instruments. The fundamental pitch of a woodwind (the recorder, for example) is considered to be the lowest note it can play when all holes are closed. The fundamental pitch of a brass instrument, on the other hand, is considered to be the fundamental of the harmonic series it plays when no valves are being used. For example, the C trumpet, using no valves, plays a harmonic series based on C, while a B flat (transposing) trumpet plays a B flat harmonic series. Tubas, on the other hand, can be based on several different harmonic series, including C, B flat, F, and E flat. But these are not necessarily transposing instruments. A tuba player playing a B flat instrument may read a transposing B flat part, or may read concert-pitch music and simply use different fingerings for the same note than a player on a C instrument.