5.4 Going Further: From Linear Stability Analysis to Oscillation Regimes
259
The theoretical foundation for the Bouasse-Benade prescription can be explored
by calculating bifurcation diagrams for models describing instruments with different
degrees of resonance inharmonicity. To clarify the basic underlying principles, it
is useful to begin with the simplest possible case: a multimode resonator having
only two quasi-harmonic resonances with frequencies f res1 and f res2 2f res1 . The
deviation from perfect harmonicity is described by the equation
f res2 = 2f res1 (1 +
(5.43)
where is the inharmonicity parameter. To simplify the theoretical investigations,
the sounding mechanism is chosen to be an inward-striking reed operating in the
stiffness-dominated low-frequency regime.
If the resonances are exactly harmonic ( = 0), this theoretical problem can be
analysed analytically (Dalmont et al. 2000). A bifurcation diagram obtained from
such an analysis for the case in which the impedance amplitude of the first resonance
is slightly higher than the amplitude of the second resonance is shown in Fig. 5.21a.
The bifurcation branch marked by circles and labelled ‘standard’ shows the playing
regime with the lowest threshold pressure. This is an inverse Hopf bifurcation of the
type illustrated in Fig. 5.13, which means that the start of a note with infinitesimal
amplitude cannot be achieved with a rising mouth pressure. At the threshold p th1 ,
the oscillation begins with an RMS amplitude more than 30% of the minimum reed
closing pressure p M . Once this sound is initiated, the oscillation amplitude can
be reduced by lowering the pressure below p th1 , until the sound cuts out again at
p sc . The curves marked ‘octave’ and ‘inverted’ are bifurcation branches describing
other possible oscillation states of the model, which are discussed in Dalmont et al.
(2000).
If the inharmonicity is not equal to zero, the problem can no longer be resolved
analytically, and numerical methods must be used. Bifurcation diagrams have been
calculated for the two-mode system with different degrees of inharmonicity using
a continuation method (Sect. 5.4.1). For this purpose the elementary model for the
two-mode system was reformulated using the ‘real mode’ approach (Gilbert et al.
2020). The bifurcation diagram calculated numerically for the case = 0, shown on
the upper graph in Fig. 5.21b, is qualitatively consistent with the analytically derived
diagram in Fig. 5.21a. The continuation method gives information on the stability
nature of the periodic oscillations which is not available from the analytical solution,
showing that only the standard branch from its point of inflexion upwards represents
a stable oscillatory state. On this branch the oscillation frequency f osc is locked at
the value f res1 for all values of p m , as shown on the lower graph in Fig. 5.21b.
Figure 5.22a and b illustrate how the bifurcation diagram changes as the
inharmonicity of the two resonances increases. Figure 5.22a shows the case for
= 0.02, corresponding to a widening of the octave relationship between the two
resonances by 34 cents. The bifurcation diagram is quite similar to the diagram for
= 0 shown in Fig. 5.21b, but there are some changes which have implications
for ease of playing. At the threshold p m = p th1 , the green curve representing the
fundamental regime has the appearance of a direct Hopf bifurcation, but just above
259
The theoretical foundation for the Bouasse-Benade prescription can be explored
by calculating bifurcation diagrams for models describing instruments with different
degrees of resonance inharmonicity. To clarify the basic underlying principles, it
is useful to begin with the simplest possible case: a multimode resonator having
only two quasi-harmonic resonances with frequencies f res1 and f res2 2f res1 . The
deviation from perfect harmonicity is described by the equation
f res2 = 2f res1 (1 +
(5.43)
where is the inharmonicity parameter. To simplify the theoretical investigations,
the sounding mechanism is chosen to be an inward-striking reed operating in the
stiffness-dominated low-frequency regime.
If the resonances are exactly harmonic ( = 0), this theoretical problem can be
analysed analytically (Dalmont et al. 2000). A bifurcation diagram obtained from
such an analysis for the case in which the impedance amplitude of the first resonance
is slightly higher than the amplitude of the second resonance is shown in Fig. 5.21a.
The bifurcation branch marked by circles and labelled ‘standard’ shows the playing
regime with the lowest threshold pressure. This is an inverse Hopf bifurcation of the
type illustrated in Fig. 5.13, which means that the start of a note with infinitesimal
amplitude cannot be achieved with a rising mouth pressure. At the threshold p th1 ,
the oscillation begins with an RMS amplitude more than 30% of the minimum reed
closing pressure p M . Once this sound is initiated, the oscillation amplitude can
be reduced by lowering the pressure below p th1 , until the sound cuts out again at
p sc . The curves marked ‘octave’ and ‘inverted’ are bifurcation branches describing
other possible oscillation states of the model, which are discussed in Dalmont et al.
(2000).
If the inharmonicity is not equal to zero, the problem can no longer be resolved
analytically, and numerical methods must be used. Bifurcation diagrams have been
calculated for the two-mode system with different degrees of inharmonicity using
a continuation method (Sect. 5.4.1). For this purpose the elementary model for the
two-mode system was reformulated using the ‘real mode’ approach (Gilbert et al.
2020). The bifurcation diagram calculated numerically for the case = 0, shown on
the upper graph in Fig. 5.21b, is qualitatively consistent with the analytically derived
diagram in Fig. 5.21a. The continuation method gives information on the stability
nature of the periodic oscillations which is not available from the analytical solution,
showing that only the standard branch from its point of inflexion upwards represents
a stable oscillatory state. On this branch the oscillation frequency f osc is locked at
the value f res1 for all values of p m , as shown on the lower graph in Fig. 5.21b.
Figure 5.22a and b illustrate how the bifurcation diagram changes as the
inharmonicity of the two resonances increases. Figure 5.22a shows the case for
= 0.02, corresponding to a widening of the octave relationship between the two
resonances by 34 cents. The bifurcation diagram is quite similar to the diagram for
= 0 shown in Fig. 5.21b, but there are some changes which have implications
for ease of playing. At the threshold p m = p th1 , the green curve representing the
fundamental regime has the appearance of a direct Hopf bifurcation, but just above