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5 Blow That Horn: An Elementary Model of Brass Playing
Fig. 5.12 Bifurcation
diagram showing a direct
Hopf bifurcation. Upper
curve: RMS value of the
periodic solution p(t) as a
function of the control
parameter p m . Lower curve:
playing frequency as a
function of p m
While a direct Hopf bifurcation corresponds well with what might be expected
intuitively, it is by no means the only way in which a wind instrument valve can
destabilise. Another common process is described as an inverse (or subcritical)
Hopf bifurcation. In this case, the amplitude of the oscillation does not increase
progressively from zero when the control parameter exceeds the threshold value,
but jumps discontinuously to a finite amplitude. This inverse Hopf bifurcation case
is illustrated by the bifurcation diagram shown in Fig. 5.13.
The upper curve in Fig. 5.13 illustrates the dependence of the oscillation
amplitude on the control parameter p m . When p m rises from zero and reaches p thr ,
the equilibrium state becomes unstable, but there is no stable periodic regime with
infinitely small amplitude which can replace it. Instead, there is a transition to a
stable oscillation state with finite amplitude, indicated by the letter F on the upper
curve in Fig. 5.13. This type of behaviour makes it very difficult to start a note
pianissimo. As the blowing pressure is further increased, the amplitude grows, as
shown by the branch from F to the right in Fig. 5.13.
The branch to the left from the threshold point p m = p thr represents a set of
periodic solutions which are unstable, and therefore not observable in practice.
At the inflection point I , corresponding to a blowing pressure p subthr , this branch
turns to the right: it now represents stable periodic solutions and therefore playable
sounds. This behaviour has two main consequences. The first is that periodic
regimes exist for values of p m lower than the threshold of oscillation found with
a rising control parameter p m : the minimum blowing pressure at which periodic
oscillations are possible is p subthr . The second is that there are two possible
5 Blow That Horn: An Elementary Model of Brass Playing
Fig. 5.12 Bifurcation
diagram showing a direct
Hopf bifurcation. Upper
curve: RMS value of the
periodic solution p(t) as a
function of the control
parameter p m . Lower curve:
playing frequency as a
function of p m
While a direct Hopf bifurcation corresponds well with what might be expected
intuitively, it is by no means the only way in which a wind instrument valve can
destabilise. Another common process is described as an inverse (or subcritical)
Hopf bifurcation. In this case, the amplitude of the oscillation does not increase
progressively from zero when the control parameter exceeds the threshold value,
but jumps discontinuously to a finite amplitude. This inverse Hopf bifurcation case
is illustrated by the bifurcation diagram shown in Fig. 5.13.
The upper curve in Fig. 5.13 illustrates the dependence of the oscillation
amplitude on the control parameter p m . When p m rises from zero and reaches p thr ,
the equilibrium state becomes unstable, but there is no stable periodic regime with
infinitely small amplitude which can replace it. Instead, there is a transition to a
stable oscillation state with finite amplitude, indicated by the letter F on the upper
curve in Fig. 5.13. This type of behaviour makes it very difficult to start a note
pianissimo. As the blowing pressure is further increased, the amplitude grows, as
shown by the branch from F to the right in Fig. 5.13.
The branch to the left from the threshold point p m = p thr represents a set of
periodic solutions which are unstable, and therefore not observable in practice.
At the inflection point I , corresponding to a blowing pressure p subthr , this branch
turns to the right: it now represents stable periodic solutions and therefore playable
sounds. This behaviour has two main consequences. The first is that periodic
regimes exist for values of p m lower than the threshold of oscillation found with
a rising control parameter p m : the minimum blowing pressure at which periodic
oscillations are possible is p subthr . The second is that there are two possible