5.4 Going Further: From Linear Stability Analysis to Oscillation Regimes
257
Would a pedal note at the nominal fundamental pitch of the instrument still
be playable if the outward-striking lip valve was replaced by an inward-striking
reed? To answer this question, the ingenious acoustician Henri Bouasse devised an
experiment in which a B tenor trombone was played with a saxophone mouthpiece
(Bouasse 1929). The result was an instrument whose first natural note was E 1, with
an oscillation frequency just under f ac,1 = 38 Hz. A playing frequency below the
acoustic resonance frequency is the behaviour expected from an inward-striking
reed (see Sect. 3.2.4). In this case the phase relationship between reed and air
column resonance makes a feedback loop with positive regeneration impossible
above f ac , and there was no evidence of a playable note at 58 Hz. The results of
the Bouasse experiment have been confirmed by physical modelling simulation of a
trombone with a saxophone mouthpiece (Velut et al. 2017a).
The fact that an essentially linear description can be given of the threshold
sounding of the pedal note does not mean that nonlinear coupling of harmonic
upper modes is irrelevant to pedal note behaviour. The player’s experience is that
the pitch of a pedal B 1 is relatively uncentred when played very quietly, but gains
stability at higher dynamic levels. The increasing participation of the upper modes
during a crescendo such as that illustrated in Fig. 7.67 helps the player to keep the
pitch constant while also contributing to the spectral enrichment which is such a
spectacular characteristic of a fortissimo pedal note (Sect. 7.8.5).
In Sect. 5.4.5 bifurcation diagrams will be used as maps to explore the nonlinear
landscape which unfolds once we leave behind the foothills of threshold behaviour.
5.4.5 Bifurcation Diagrams of Reed and Brass Instruments
In the quest for a theory of wind instrument behaviour which is both musically and
scientifically convincing, the holy grail is the understanding of the principles which
determine the intonation and ease of playing of the instrument. We have seen in
Sect. 5.4.3 that linear stability analysis of the physical model of a brass instrument
can supply information about the oscillation threshold frequency f thr which is
relevant to intonation, while data on the corresponding threshold pressure p thr can
be related to the ease of playing of the instrument. However this information is
only directly applicable to the periodic oscillations obtained in musical performance
when p m is near p thr , and cannot be used to explain how intonation and playability
evolve as the amplitude of oscillation increases. Instruments which display an
inverse Hopf bifurcation behaviour at the initiation of a note present a particular
problem for LSA, since the oscillation begins immediately at a finite amplitude
(Sect. 5.3.2).
In Sect. 5.3.1 examples were given of time domain solutions of the physical
model equations using numerical methods. Such simulations can be carried out
many times with different sets of control parameters, allowing the investigation of
periodic solutions over a wide range of amplitudes. Since a fresh simulation has
to be carried out for each change in a control parameter, this method is relatively
257
Would a pedal note at the nominal fundamental pitch of the instrument still
be playable if the outward-striking lip valve was replaced by an inward-striking
reed? To answer this question, the ingenious acoustician Henri Bouasse devised an
experiment in which a B tenor trombone was played with a saxophone mouthpiece
(Bouasse 1929). The result was an instrument whose first natural note was E 1, with
an oscillation frequency just under f ac,1 = 38 Hz. A playing frequency below the
acoustic resonance frequency is the behaviour expected from an inward-striking
reed (see Sect. 3.2.4). In this case the phase relationship between reed and air
column resonance makes a feedback loop with positive regeneration impossible
above f ac , and there was no evidence of a playable note at 58 Hz. The results of
the Bouasse experiment have been confirmed by physical modelling simulation of a
trombone with a saxophone mouthpiece (Velut et al. 2017a).
The fact that an essentially linear description can be given of the threshold
sounding of the pedal note does not mean that nonlinear coupling of harmonic
upper modes is irrelevant to pedal note behaviour. The player’s experience is that
the pitch of a pedal B 1 is relatively uncentred when played very quietly, but gains
stability at higher dynamic levels. The increasing participation of the upper modes
during a crescendo such as that illustrated in Fig. 7.67 helps the player to keep the
pitch constant while also contributing to the spectral enrichment which is such a
spectacular characteristic of a fortissimo pedal note (Sect. 7.8.5).
In Sect. 5.4.5 bifurcation diagrams will be used as maps to explore the nonlinear
landscape which unfolds once we leave behind the foothills of threshold behaviour.
5.4.5 Bifurcation Diagrams of Reed and Brass Instruments
In the quest for a theory of wind instrument behaviour which is both musically and
scientifically convincing, the holy grail is the understanding of the principles which
determine the intonation and ease of playing of the instrument. We have seen in
Sect. 5.4.3 that linear stability analysis of the physical model of a brass instrument
can supply information about the oscillation threshold frequency f thr which is
relevant to intonation, while data on the corresponding threshold pressure p thr can
be related to the ease of playing of the instrument. However this information is
only directly applicable to the periodic oscillations obtained in musical performance
when p m is near p thr , and cannot be used to explain how intonation and playability
evolve as the amplitude of oscillation increases. Instruments which display an
inverse Hopf bifurcation behaviour at the initiation of a note present a particular
problem for LSA, since the oscillation begins immediately at a finite amplitude
(Sect. 5.3.2).
In Sect. 5.3.1 examples were given of time domain solutions of the physical
model equations using numerical methods. Such simulations can be carried out
many times with different sets of control parameters, allowing the investigation of
periodic solutions over a wide range of amplitudes. Since a fresh simulation has
to be carried out for each change in a control parameter, this method is relatively