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be shorter than the measured length, but the difference be- tween the actual length and the acoustic length changes with frequency. Near the highest frequencies of the playing range the acoustic length approaches the actual length.
nance in the frequency range that encompasses the normal playing range of the instrument. In contrast, the bell flare and bore shape of brass instruments can vary significantly.
This shortening of the effective length
of the instrument caused by the bell
flare shifts the resonance frequencies
up, but it does not shift them all equal-
ly. The bell flare is designed so that the
resonance frequencies are shifted in
such a way that they become harmoni-
cally related in a different way. The odd-
numbered overtones are still the only
ones that can be produced, but because
the frequencies have been shifted by the
bell flare in a nonlinear way, they pro-
duce a complete harmonic series. The
only resonance that is not part of this
harmonic series is the fundamental,
which in musical parlance is referred to
as the pedal note. The frequency of the
pedal note is much too low to be part of
the harmonic series; therefore, the resonances are referred to as being a harmonic series with a detuned fundamental. This can be seen in Figure 2c where the calculated impedance of a complete trumpet is plotted. The pedal note is indicated in the figure as is the fundamental frequency associated with the harmonic series.
The bell serves another function that helps create the unique sound of each instrument. It preferentially reflects low- frequency sound. The higher reflection causes the low fre- quencies to be more prevalent in the air column because they have a greater effect on the lip motion. However, the bell flare also determines the frequency above which the re- flection becomes negligible; this frequency is known as the cutoff frequency. So although the bell shape is responsible for ensuring that there is more power in the frequencies at the lower resonances than at the higher ones inside the instru- ment, it is also responsible for ensuring that a greater per- centage of the sound is radiated at higher frequencies.
This description of the trumpet can be generally applied to almost all brass instruments. However, there are variations that are unique to each instrument and these variations re- sult in sounds that cover a wide range of timbres. The shapes of the mouthpieces vary between instruments, but they all have a cup, throat, and backbore and they all produce a reso-
Figure 4. Photograph of a modern cornet. The cornet has sections of conical tubing in con- trast to the cylindrical tubing of the trumpet. Photograph from Yamaha Corporation of America.
The Importance of Bore Shape
The bore shape of any instrument is defined by the shape of the internal walls of the tubing that constrains the air. As noted above, the majority of the tubing in the trumpet is cylindrical as is the bore of the trombone. However, not all members of the brass family have cylindrical bores. Some have conical bores and most have a combination of cylin- drical and conical tubing before reaching the bell section. For example, a photograph of a cornet is shown in Figure 4. (As in Figure 1, the black lines on the tubing in Figure 4 are artifacts of the lighting and not part of the instrument.) The cornet has a significant length of nearly conical tubing, which is clearly different from the cylindrical tubing in the trumpet shown in Figure 1.
Few modern instruments are completely conical or com- pletely cylindrical, so the bore shape may include cylindri- cal, conical, and flaring sections. Many modern brass in- struments start with some cylindrical tubing followed by a conical section that may be a significant part of the length. From a musical standpoint, these conical instruments sound more “mellow” than the cylindrical instruments, and it is not possible to get the classically brassy sound from instru- ments having a conical bore. Thus the cornet, which has a large section of conical tubing, does not sound like a trum- pet, especially when played loudly.
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