The physics and mathematics involved in the sound production of brass instrument are, surprisingly, quite complicated. However, several models are available to give us a brief overview of the principles behind brass instrument sound generation.
A sounding brass instrument can be understood to have two parts: the lip valve and the body itself. The lip valve basically functions by the player’s lip pressing against the mouthpiece and delivering airflow with controlled magnitude. (Noreland, 2003) Despite its complexity, a simple mass-spring model is successful in modeling the basic principle of the lip valve. Similar to the mechanical reeds commonly seen in other musical instruments, when the player “buzzes” his lips, the air stream is interrupted in a certain pattern. Also, the same type of feedback occurs, with low-pressure portions of the sound wave pulling the lips closed and high-pressure portions forcing the lips open. (Lapp, 2002) This shares similarity with the mass-spring system with harmonic motion. The term, embouchure, describes the use of facial muscles and the shaping of the lips to produce a sound. The performer's use of the air, tightening of cheek and jaw muscles, as well as tongue manipulation can affect how the embouchure works. (Embouchure)
Figure 1. Mouthpiece and simple mass-spring model of the lip valve. (Noreland, 2003)
The second part is the body of the brass instrument, in our case, the trumpet body. It consists of several parts: the leadpipe, the cylindrical section(with valves) and a bell.
Two things determines the pitch of the note, one is the frequency of vibration of the lips, which is directly under the control of the player by adjusting the embouchure. However, when the lips are coupled through a mouthpiece to a piece of tubing, then the tubing itself has its own resonance frequencies. There are certain natural pitches that the tubing of a trumpet, for example, likes to play. Hence, the player has to...
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