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Brass Musical Instruments
To actually play the trumpet, it is convenient to have some- thing on which to place the lips other than the sharp edge of the metal pipe. The small bore makes it extremely diffi- cult to buzz the lips into the pipe, but more importantly, the sharp edges of the pipe ensure that without some intermedi- ate interface, trumpet playing would be a short-lived profes- sion. This interface between the player and the instrument is called the mouthpiece.
The mouthpieces of modern brass instruments are typically turned from a solid piece of brass and have three parts: cup, throat, and backbore (Figure 3). The outside of the mouth- piece generally follows the bore profile, but the point at which the lip touches the mouthpiece is made much thicker to provide a cushion for the lips.
Figure 3. Drawing of a trumpet mouthpiece with the parts labeled. Drawing by D. Bolton, CC SA-BY 2.0.
The cup provides a volume of air that acts as a compliance, the throat introduces an inertance, and the backbore acts to gently expand the bore diameter from the narrow throat passage to the wider diameter of the pipe. There is a section of pipe called the leadpipe, which is approximately 25 cm long in a trumpet, that continues this gradual expansion of the bore until it is equal to the diameter of the cylindrical pipe used for the majority of the instrument. However, most of the bore expansion occurs in the mouthpiece.
The inertance (impedance where pressure leads flow by π/2 radians) and compliance (flow leads pressure by π/2 radi- ans) together with the resistance to the flow that is inherent in any pipe (flow and pressure in phase) creates a resonance just as an inductor, capacitor, and resistor create a resonant electrical circuit. Figure 2b is a graph of the input imped- ance that occurs when the mouthpiece is connected to the cylindrical piping. In the simulation, the length of the pip- ing has been shortened by the length of the mouthpiece to keep the total length constant. Note that the effect of the
32 | Acoustics Today | Winter 2016
mouthpiece is to significantly increase the input impedance between 200 Hz and 1.5 kHz. This frequency range repre- sents the normal playing range of the trumpet. When brass players wish to play above the normal frequency range, they often change the mouthpiece to one with a shallower cup, which raises the resonance frequency of the mouthpiece and makes it easier to play notes that are higher in the musical scale. Professional brass players may have from 3 to 10 dif- ferent mouthpieces. As a general rule, the better the player is, the fewer the number of mouthpieces used, but most players have at least two or three that are regularly used.
If one were to construct an instrument that consisted only of a mouthpiece and a length of piping, the graph in Figure 2b indicates that only the fundamental and the odd harmon- ics could be sounded. There is a less obvious effect of shift- ing the resonances slightly in a manner that is not uniform so that they are no longer exactly harmonic, but construct- ing such an instrument demonstrates that the resonances are still close to the expected relationship. Unfortunately, the radiated sound from this combination is minimal and an audience would hear almost no sound when it is played. To increase the radiation efficiency, the end of the piping is flared into a bell. This flaring increases the radiation into the air so that the audience can hear the instrument, but even with the bell attached, the majority of the sound stays inside the instrument.
The explanation of how the trumpet is constructed is now complete, but one issue remains to be resolved. Musicians know that most brasswind instruments can sound an entire harmonic series and not just the odd harmonics, yet it is common knowledge that the resonances of an open-closed pipe constitute only the odd overtones, as is evident from Figure 2. So how can a quarter-wavelength pipe have reso- nances that include the even overtones? The answer is that it cannot, but by judiciously designing the bell, it is possible to shift the resonance frequencies in such a way that when combined with the small shift attributable to the mouthpiece the overtones become a complete harmonic series.
The design of the bells of brass instruments is still more art than science, but we do understand the importance of the profile of the bore created by the bell. Although it is com- mon to consider the acoustical end of a pipe as occurring near the physical end of the pipe, in the flaring bell, longer wavelengths are reflected in the region well before the ac- tual end of the instrument. Furthermore, as the frequency increases, the final pressure node shifts toward the end of the trumpet. Therefore, acoustically, the instrument appears to