5.3 Beyond Pianissimo: Modelling Realistic Playing Amplitudes
233
fact be extended beyond the threshold regime using the mathematical technique of
continuation. These ideas are more fully explained in the Going Further Sect. 5.4.
5.3.1 Analysis of Brass Performance Using Simulations
Many different numerical methods can be used to exhibit the behaviour of the
nonlinear brass instrument model far from threshold. The harmonic balance method,
for example, gives a Fourier series approximation to the steady state of periodic
regimes, including unstable ones (Schumacher 1978; Gilbert et al. 1989; Farner et
al. 2006; Cochelin and Vergez 2009). Following the pioneering work of Schumacher
(1981), McIntyre et al. (1983), and Gazengel et al. (1995), it is possible to carry out
time domain simulations based on reflection functions at moderate computational
cost. These simulations permit the study of transients and also non-periodic solutions; with fast-enough processors, they can generate the real-time output necessary
for synthesisers. The numerical methods available include the finite element and
finite-difference methods which are widely used in many branches of physics and
engineering (Bilbao and Chick 2013). Methods specifically adapted to the acoustics
of ducts include digital waveguides, wave digital filters, impedance-based methods
and those involving impulse responses and reflection functions. It is beyond the
scope of the book to give an extensive view of all the different methods which
have been used in wind instrument simulations; valuable reviews are provided by
Beauchamp (2007), Bilbao (2009) and Smith (2010).
Sound synthesisers based on physical modelling are impressive in their ability
to mimic musical phrasing, even when they are based on the elementary physical
model discussed in this chapter. A critical requirement for musically convincing
output is the control of the slow variation in time of the parameters of the physical
model. One method of acquiring realistic data for the control parameters of a
musician’s embouchure, which has been successfully employed in clarinet and
saxophone synthesis, is to use an instrumented mouthpiece (Guillemain et al. 2005).
Apart from their use in sound synthesis, time domain simulations can provide
interesting and useful insights into particular aspects of wind instrument behaviour.
For example, Velut et al. (2016) have simulated period-doubling and quasi-periodic
oscillations. These unusual regimes are sometimes deliberately employed by performers to create exotic sounds such as multiphonics, but are also familiar traps
for players with tired lips! Simulations have been used to study transients and
unsteady sounds in clarinet-like instruments (Bergeot et al. 2013) and vibrato on
the saxophone (Gilbert et al. 2005), but much work remains to be done on unsteady
regimes in brass instruments.
We present here a few examples of simulations of brass performance using finitedifference time domain (FDTD) methods developed by Reginald Harrison-Harsley
(Harrison et al. 2015, 2016). Figure 5.8 illustrates the attack transient of a brass
instrument: a tenor trombone in first position playing the second natural note B 2.
The attack is controlled by the shape of the control parameter, in this case the
233
fact be extended beyond the threshold regime using the mathematical technique of
continuation. These ideas are more fully explained in the Going Further Sect. 5.4.
5.3.1 Analysis of Brass Performance Using Simulations
Many different numerical methods can be used to exhibit the behaviour of the
nonlinear brass instrument model far from threshold. The harmonic balance method,
for example, gives a Fourier series approximation to the steady state of periodic
regimes, including unstable ones (Schumacher 1978; Gilbert et al. 1989; Farner et
al. 2006; Cochelin and Vergez 2009). Following the pioneering work of Schumacher
(1981), McIntyre et al. (1983), and Gazengel et al. (1995), it is possible to carry out
time domain simulations based on reflection functions at moderate computational
cost. These simulations permit the study of transients and also non-periodic solutions; with fast-enough processors, they can generate the real-time output necessary
for synthesisers. The numerical methods available include the finite element and
finite-difference methods which are widely used in many branches of physics and
engineering (Bilbao and Chick 2013). Methods specifically adapted to the acoustics
of ducts include digital waveguides, wave digital filters, impedance-based methods
and those involving impulse responses and reflection functions. It is beyond the
scope of the book to give an extensive view of all the different methods which
have been used in wind instrument simulations; valuable reviews are provided by
Beauchamp (2007), Bilbao (2009) and Smith (2010).
Sound synthesisers based on physical modelling are impressive in their ability
to mimic musical phrasing, even when they are based on the elementary physical
model discussed in this chapter. A critical requirement for musically convincing
output is the control of the slow variation in time of the parameters of the physical
model. One method of acquiring realistic data for the control parameters of a
musician’s embouchure, which has been successfully employed in clarinet and
saxophone synthesis, is to use an instrumented mouthpiece (Guillemain et al. 2005).
Apart from their use in sound synthesis, time domain simulations can provide
interesting and useful insights into particular aspects of wind instrument behaviour.
For example, Velut et al. (2016) have simulated period-doubling and quasi-periodic
oscillations. These unusual regimes are sometimes deliberately employed by performers to create exotic sounds such as multiphonics, but are also familiar traps
for players with tired lips! Simulations have been used to study transients and
unsteady sounds in clarinet-like instruments (Bergeot et al. 2013) and vibrato on
the saxophone (Gilbert et al. 2005), but much work remains to be done on unsteady
regimes in brass instruments.
We present here a few examples of simulations of brass performance using finitedifference time domain (FDTD) methods developed by Reginald Harrison-Harsley
(Harrison et al. 2015, 2016). Figure 5.8 illustrates the attack transient of a brass
instrument: a tenor trombone in first position playing the second natural note B 2.
The attack is controlled by the shape of the control parameter, in this case the