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The trumpet and the Bb cornet are pitched the same and use the same fingering. The cornet, however, is generally considered to be easier for beginners and young musicians to play than the trumpet.
["Typically, no, a trumpet mouthpiece will not fit a cornet as the 2 mouthpieces used in a trumpet and a cornet respectively are somewhat different. "]
["A trumpet has just 3 valves but can play an impressive 45 key combinations. "]