126
4 After the Lips: Acoustic Resonances and Radiation
in Sect. 4.1.5. In practice r(t) is obtained by deconvolving the reflected signal with
the recorded impulse. The frequency domain reflection coefficient R(ω) of the
instrument is found by Fourier transforming r(t), and the input impedance Z(ω)
can be deduced using Eq. 4.28.
Since the time domain reflection function measured at the entrance of a tube
records the reflections arising from every change in diameter, it is possible to
reconstruct the bore profile of a duct of varying cross-section from an APR
measurement (Amir et al. 1995; Sharp 1996; Li et al. 2005). Bore reconstruction
using APR has found many industrial applications and has been successfully used
to study the bore profiles of musical wind instruments (Campbell and MacGillivray
1999; Buick et al. 2002; Gray 2005; Hendrie 2007; Dalmont et al. 2012).
4.3 Bore Profiles of Brass Instruments
The four brass instruments illustrated in Fig. 4.21a differ significantly in their overall
appearance and also in their playing properties. In timbral terms the two instruments
on the left could be classed as ‘mellow’ instruments, while the two on the right fall
into the ‘bright’ instrument category (see Sect. 6.1). The major difference in tone
colour is not however related to the proportion of straight to coiled tubing, or to
the difference between valves and slides, but rather to the differences between the
internal bore profiles of the instruments as shown in Fig. 4.21b.
In the following sections, we examine in some detail the relationship between the
shape of the bore profile on a brass instrument and the frequencies and strengths of
the acoustic resonances in the contained air column. The nature of these resonances
plays a large role in determining the musical behaviour of the instrument at relatively
low dynamic levels (generally speaking below forte). In loud playing another
effect, known as nonlinear sound propagation, becomes increasingly important, in
some cases giving rise to the very brilliant timbre described as ‘brassy’. Although
nonlinear sound propagation also depends strongly on the nature of the bore profile,
we will defer a discussion of this effect until Sect. 6.1.
4.3.1 Different Parts of the Bore
The most obvious differences between the bores in Fig. 4.21b is that in the two
trombones, the bore remains narrow for most of its length, while in the other two
instruments, the bore grows more gradually before reaching the rapidly flaring bell.
Bore profiles for a number of brass instruments of similar length are shown in
Fig. 7.34. In this figure the bore diameters are plotted against axial distance up to a
maximum diameter of 67 mm, allowing the part of the bore nearest the input to be
seen in more detail. The serpent and ophicleide have an almost completely conical
bore profile; the sackbut and bass trombone are cylindrical for the first 1.5 m and
4 After the Lips: Acoustic Resonances and Radiation
in Sect. 4.1.5. In practice r(t) is obtained by deconvolving the reflected signal with
the recorded impulse. The frequency domain reflection coefficient R(ω) of the
instrument is found by Fourier transforming r(t), and the input impedance Z(ω)
can be deduced using Eq. 4.28.
Since the time domain reflection function measured at the entrance of a tube
records the reflections arising from every change in diameter, it is possible to
reconstruct the bore profile of a duct of varying cross-section from an APR
measurement (Amir et al. 1995; Sharp 1996; Li et al. 2005). Bore reconstruction
using APR has found many industrial applications and has been successfully used
to study the bore profiles of musical wind instruments (Campbell and MacGillivray
1999; Buick et al. 2002; Gray 2005; Hendrie 2007; Dalmont et al. 2012).
4.3 Bore Profiles of Brass Instruments
The four brass instruments illustrated in Fig. 4.21a differ significantly in their overall
appearance and also in their playing properties. In timbral terms the two instruments
on the left could be classed as ‘mellow’ instruments, while the two on the right fall
into the ‘bright’ instrument category (see Sect. 6.1). The major difference in tone
colour is not however related to the proportion of straight to coiled tubing, or to
the difference between valves and slides, but rather to the differences between the
internal bore profiles of the instruments as shown in Fig. 4.21b.
In the following sections, we examine in some detail the relationship between the
shape of the bore profile on a brass instrument and the frequencies and strengths of
the acoustic resonances in the contained air column. The nature of these resonances
plays a large role in determining the musical behaviour of the instrument at relatively
low dynamic levels (generally speaking below forte). In loud playing another
effect, known as nonlinear sound propagation, becomes increasingly important, in
some cases giving rise to the very brilliant timbre described as ‘brassy’. Although
nonlinear sound propagation also depends strongly on the nature of the bore profile,
we will defer a discussion of this effect until Sect. 6.1.
4.3.1 Different Parts of the Bore
The most obvious differences between the bores in Fig. 4.21b is that in the two
trombones, the bore remains narrow for most of its length, while in the other two
instruments, the bore grows more gradually before reaching the rapidly flaring bell.
Bore profiles for a number of brass instruments of similar length are shown in
Fig. 7.34. In this figure the bore diameters are plotted against axial distance up to a
maximum diameter of 67 mm, allowing the part of the bore nearest the input to be
seen in more detail. The serpent and ophicleide have an almost completely conical
bore profile; the sackbut and bass trombone are cylindrical for the first 1.5 m and