A transposing instrument is a musical instrument whose music is notated at a pitch different from the pitch that actually sounds (concert pitch). Playing a written C on a transposing instrument produces a pitch other than C (or a C at a different octave than that indicated by the written C in the case of an instrument "transposing at the octave"), and that pitch identifies the interval of transposition when describing the instrument. For example, a written C on a B♭ clarinet sounds a concert B♭.
Reasons for transposing 
Though writing for transposing instruments entails more work for a composer or arranger, there are several reasons to transpose music for certain instruments.
Making it easier to move between instruments in certain families of instruments 
Many instruments are members of a family of instruments that differ mainly in size, such as the saxophone, clarinet, flute, etc. The instruments in these families have differing ranges, with the members sounding lower as they get larger; but an identical pattern of fingerings on two instruments in the same family produces pitches a fixed interval apart. For example, the fingerings which produce the notes of a C major scale on a flute, a non-transposing instrument, produce a G major scale on an alto flute. As a result these instruments' parts are notated so that the written notes are fingered the same way on each instrument, making it easier for a single instrumentalist to play several instruments in the same family. However, a few instrument families, such as trombones and tubas, are not written transposed.
Instruments that transpose this way are often referred to as being in a certain "key", such as the "A clarinet" or "clarinet in A". The instrument's key tells which pitch will sound when the player plays a note written as C. A player of a B♭ clarinet who reads a written C will sound a B♭ while the player of an A clarinet will read the same note and sound an A.
Horn crooks 
Before valves became common in the 19th century, the horn could play only the notes of the overtone series from a single fundamental pitch. This fundamental could be changed by inserting one of a set of crooks into the instrument, shortening or lengthening the total length of its sounding tube. As a result, all horn music was written as if for a fundamental pitch of C, but the crooks could make a single instrument a transposing instrument into almost any key. Changing the crooks was a time-consuming process, so it took place only between pieces or movements. The introduction of valves made this process unnecessary (though Richard Wagner wrote horn parts as if crooks were still in use, evoking a tradition that was quickly becoming archaic). While an F transposition became standard in the early 19th century, composers differed in whether they expected the instruments to transpose down a fifth or up a fourth, especially when written in bass clef.
Reconciling pitch standards 
In music of Germany during the Baroque period, and notably in the music of J.S. Bach, instruments used for different purposes were often tuned to different pitch standards, called Chorton ("choir pitch") and Kammerton ("chamber [music] pitch"). When they played together in an ensemble, the parts of some instruments would then have to be transposed to compensate. In many of Bach's cantatas the organ part is notated a full step lower than the other instruments. See Pitch inflation.
A few early-music ensembles of the present day must do something similar if they comprise some instruments tuned to A415 and others to A440, approximately a semitone apart. Modern builders of continuo instruments sometimes include moveable keyboards which can play with either pitch standard.
Transposition at the octave 
If an instrument has a range too high or too low for composers to easily write its music on bass or treble clef, the music may be written either an octave higher or an octave lower than it sounds, in order to reduce the use of ledger lines. Instruments that "transpose at the octave" are not playing in a different key from concert pitch instruments, but sound an octave higher or lower than written. Some instruments with extremely high or low ranges use a two-, or even three-octave transposition.
What determines what key a wind instrument is in? Mechanical and physical considerations 
Woodwind instruments each have one major scale whose execution involves lifting the fingers more or less sequentially from the bottom to top. This scale is usually the one notated as a C scale (from C to C, with no sharps or flats) for that instrument. The note written as C sounds as the note of the instrument's transposition — on an E♭ alto saxophone, that note sounds as a concert E♭, on an A clarinet, that note sounds as a concert A. The bassoon is an exception; it is not a transposing instrument, yet its "home" scale is F. For tin whistles the "home" scale is notated as D major rather than C major — the most common whistle, pitched in D, is therefore not a transposing instrument.
Brass instruments, when played with no valves engaged (or, for trombones, with the slide all the way in), play a series of notes that form the overtone series based on some fundamental pitch, e.g., the B♭ trumpet, when played with no valves engaged, can play the overtones based on B♭. Usually, that pitch is the note that indicates the transposition of that brass instrument. Trombones are an exception — they read at concert pitch, although tenor and bass trombones are pitched in B♭, alto trombone in E♭. Music for baritone or euphonium is sometimes written in treble clef, transposed to B♭, but sometimes in bass clef at concert pitch.
In general, for these instruments there is some reason to consider a certain pitch the "home" note of an instrument, and that pitch is usually written as C for that instrument. The concert pitch of that note is what determines how we refer to the transposition of that instrument.
Conductor's score 
In conductors' scores and other full scores, music for transposing instruments is generally written in transposed form, just as in the players' parts. A few publishers, however, especially of modern music, provide conductors with scores written entirely in concert pitch, making the pitch relationships of the entire score easier for the conductor to see.
Transposing instruments names always use flat, never sharp (thus there are instruments in E-flat or in B-flat never in D-sharp or in A-sharp). In practice the actual transposition in the score may (for the convenience of the player) depend on the key of the music. For example in a section in C major an E-flat alto saxophone part will of course appear in A major (three sharps). But in a section in B major the saxophone player will not normally be inflicted by a G-sharp major part (a key with eight sharps, i.e. six sharps and one double-sharp). Instead their part will appear in A-flat major (four flats), just as if they were playing a D-sharp instrument. This makes things easier on the player (and even harder on the conductor and any other reader of the score).
See also 
- Apel, Willi, ed. (1972). "Harvard Dictionary of Music". (2nd ed.). Cambridge, MA: Harvard University Press. p. 860. ISBN 0-674-37501-7.
- Del Mar, Norman (1981). Anatomy of the Orchestra. Berkeley and Los Angeles: University of California Press. p. 218-220. ISBN 978-0571250998.
- Dreyfus, Laurence (1987). Bach's Continuo Group. Cambridge, MA: Harvard University Press. p. 11. ISBN 0-674-06030-X.
- Carey Beebe Harpsichords Australia. "CBH Global Harpsichord Technology".
- Willi Apel, ed. (1972). "Transposing instruments". Harvard Dictionary of Music (2 ed.).. According to this article, if an octave-transposing clef is used (with a little 8 above or below, the term "transposition" does not apply.
- Kennan, Kent Wheeler. The Technique of Orchestration, Second Edition. Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1970, 1952; ISBN 0-13-900316-9
- Del Mar, Norman. The Anatomy of the Orchestra. University of California Press, 1981