5.3 Beyond Pianissimo: Modelling Realistic Playing Amplitudes
237
5.3.2 Bifurcation Diagrams
In Sect. 5.3.1 simulations in the time domain have illustrated how approximate
solutions of an elementary model can mimic qualitatively the performance of a
brass instrument in the nonlinear regime corresponding to realistic playing levels.
To progress from qualitative to quantitative agreement between simulations and the
sounds generated by live musicians requires accurate estimations of the parameters
of the embouchure of the musician and how they are slowly evolving in time. This
remains a significant challenge in the physical modelling of brass instruments.
The mouth pressure p m is the most important control parameter in the elementary
model. When p m is raised above its threshold value p thr , the equilibrium state is
destabilised. For a fixed value of p m above the threshold, a permanent periodic
regime is sustained in which the state variables oscillate with a finite amplitude. The
equations describing the system cannot be applied in their linearised form to this
situation, as was done during the linear stability analysis described in Sect. 5.2.2;
information about the permanent oscillatory regimes can only be extracted using the
equations in their original nonlinear form. An example of a periodic solution (after
a transient regime) obtained numerically using the FDTD method was displayed in
Fig. 5.8. If this kind of simulation is repeated for many different values of p m larger
than p thr , an overview of the dynamic behaviour of the nonlinear behaviour of the
brass instrument can be obtained by collecting all the simulated periodic regimes
in the same picture. This can be done by plotting the RMS value of the periodic
solution as a function of the control parameter p m . Such a plot is called a bifurcation
diagram.
An example of a wind instrument bifurcation diagram is shown in the upper curve
of Fig. 5.12. As p m reaches and exceeds p thr , a periodic solution is found with an
oscillation amplitude which increases progressively from zero at p m = p thr . For
very small amplitudes, the oscillation is quasi-sinusoidal. This kind of transition
from a stable equilibrium position (p m < p thr : silence) to a stable periodic regime
(p m > p thr : sound) is called a direct (or supercritical) Hopf bifurcation.
The lower curve in Fig. 5.12 shows the evolution of the playing frequency as the
blowing pressure is increased. The horizontal dashed line indicates the threshold
playing frequency f thr derived from linear stability analysis. For blowing pressures
below p thr , there are no periodic vibration states predicted by the simulation. As
the pressure rises through the threshold, a vibration state appears with a frequency
f thr when p m reaches p thr ; in this case, frequency decreases slowly as the blowing
pressure increases above the threshold. Only at the pianissimo level is the frequency
value f thr found by LSA a reasonable approximation to the playing frequency.
The evolution of the amplitude and frequency of the simulated sound in a slow
crescendo followed by a diminuendo can be seen by following the green arrows in
Fig. 5.12 as the blowing pressure is increased and the red arrows as it is reduced. For
this direct Hopf bifurcation, the path traced as the pressure rises is simply retraced
as it falls again, and the sound is extinguished when the blowing pressure falls below
p thr .
237
5.3.2 Bifurcation Diagrams
In Sect. 5.3.1 simulations in the time domain have illustrated how approximate
solutions of an elementary model can mimic qualitatively the performance of a
brass instrument in the nonlinear regime corresponding to realistic playing levels.
To progress from qualitative to quantitative agreement between simulations and the
sounds generated by live musicians requires accurate estimations of the parameters
of the embouchure of the musician and how they are slowly evolving in time. This
remains a significant challenge in the physical modelling of brass instruments.
The mouth pressure p m is the most important control parameter in the elementary
model. When p m is raised above its threshold value p thr , the equilibrium state is
destabilised. For a fixed value of p m above the threshold, a permanent periodic
regime is sustained in which the state variables oscillate with a finite amplitude. The
equations describing the system cannot be applied in their linearised form to this
situation, as was done during the linear stability analysis described in Sect. 5.2.2;
information about the permanent oscillatory regimes can only be extracted using the
equations in their original nonlinear form. An example of a periodic solution (after
a transient regime) obtained numerically using the FDTD method was displayed in
Fig. 5.8. If this kind of simulation is repeated for many different values of p m larger
than p thr , an overview of the dynamic behaviour of the nonlinear behaviour of the
brass instrument can be obtained by collecting all the simulated periodic regimes
in the same picture. This can be done by plotting the RMS value of the periodic
solution as a function of the control parameter p m . Such a plot is called a bifurcation
diagram.
An example of a wind instrument bifurcation diagram is shown in the upper curve
of Fig. 5.12. As p m reaches and exceeds p thr , a periodic solution is found with an
oscillation amplitude which increases progressively from zero at p m = p thr . For
very small amplitudes, the oscillation is quasi-sinusoidal. This kind of transition
from a stable equilibrium position (p m < p thr : silence) to a stable periodic regime
(p m > p thr : sound) is called a direct (or supercritical) Hopf bifurcation.
The lower curve in Fig. 5.12 shows the evolution of the playing frequency as the
blowing pressure is increased. The horizontal dashed line indicates the threshold
playing frequency f thr derived from linear stability analysis. For blowing pressures
below p thr , there are no periodic vibration states predicted by the simulation. As
the pressure rises through the threshold, a vibration state appears with a frequency
f thr when p m reaches p thr ; in this case, frequency decreases slowly as the blowing
pressure increases above the threshold. Only at the pianissimo level is the frequency
value f thr found by LSA a reasonable approximation to the playing frequency.
The evolution of the amplitude and frequency of the simulated sound in a slow
crescendo followed by a diminuendo can be seen by following the green arrows in
Fig. 5.12 as the blowing pressure is increased and the red arrows as it is reduced. For
this direct Hopf bifurcation, the path traced as the pressure rises is simply retraced
as it falls again, and the sound is extinguished when the blowing pressure falls below
p thr .