Trumpet-like instruments have been around for at least 4000 years. Early trumpets, like the modern bugle, had no valves. Thus, a single trumpet could only get the notes of a single harmonic series. The higher you go in a harmonic series, the closer together the notes get. This makes it possible to play many more types of melodies (rather than just "bugle call"-type melodies), but also makes playing trickier. Since the notes are closer together, it's much easier to hit the wrong note! So trumpet players specialized: principal players played in the lower register, and clarino players played in the high register.
In the fifteenth century, there were trumpet-player's guilds, which registered clarino and principal trumpet players and which, like other guilds of the time, ensured that only their members would be allowed to do certain types of work (in this case, playing at feasts, processions, and other official musical events).
As late as the Baroque period, composers such as Bach were still specifying some parts for clarino trumpet. But after the Baroque, the clarino tradition vanished so completely, that modern scholars have been unable to discover exactly what type of instrument Baroque clarino parts were played on.
Useful valved trumpets began to be produced in the early 19th century. The introduction of the valve freed the instrument to play any type of melody in any key in any part of its range. Earlier trumpets came in many different keys, so that players in the lower register could choose an instrument that suited the key of a particular piece of music. With the introduction of valved trumpets, which can play the entire chromatic scale easily, the C and B flat instruments became the most popular, with trumpets in other keys becoming increasingly rare.
The cornet developed from the post horn, a small, valveless instrument which was used by postal carriers to announce the arrival in town of the mail. The flugelhorn evolved from a German bugle used by each wing of beaters during a hunt. (Flugel is German for "wing".)
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