6.4 Improving the Lip Model
307
Looking at the first graph of Fig. 6.21 for the 1.31 m pipe length, it can be seen
that in the case of no mouth overpressure, there is only one visible resonance of the
lips at 120 Hz, with an outward-striking behaviour. When the mouth overpressure
is increased, another resonant peak appears at 160 Hz. This resonance is also close
to the playing frequency of the lips and has an inward-striking behaviour. Although
this peak cannot be seen in the amplitude response curve for the case of no mouth
overpressure, its presence can be predicted by the phase response at this frequency
which has a value of −π/2 at 152 Hz. The lack of an amplitude peak at this
frequency can be attributed to the fact that it corresponds to an impedance maximum
of the pipe; this additional load on the lips reduces the amplitude response. Both
peaks are seen in the amplitude response curve when the measurement is carried out
with the pipe removed.
As the mouth pressure is increased, the 160 Hz resonance approaches destabilisation, the height of the amplitude response peak growing much more dramatically
than that of the 120 Hz resonance. This gives the expected behaviour that the lips
must behave as an inward-striking reed if the playing frequency is below the acoustic
resonance frequency of the instrument. A contrasting case can be seen in the graph
for the 2.11 m pipe length. Here the resonance responsible for the playing of the
instrument (135 Hz) has an outward-striking character, and the playing frequency is
above the acoustic resonance of the instrument.
In the intervening lengths between these two cases, there is a much more
complicated interaction between the different lip resonances, and there is clearly
some coupling between the resonant modes of the lips. Despite this change in the
dominant lip resonance, there is a smooth change in playing frequency as the pipe is
extended. Although the change in frequency is continuous, some interesting features
of the coupling between lip resonances can be found in the region in which neither
lip resonance is clearly dominant. As the pipe is extended, the playing frequency
does not always show a monotonic drop in frequency; for some embouchures there
is a small region in which the playing frequency is found to rise as the telescopic
pipe is extended. This anomalous behaviour occurs when the playing frequency is
very close to the acoustic resonance frequency.
The work of Cullen (2000) and Neal (2002) shows that the near-threshold
behaviour of the lips cannot be completely explained using a lip model with only
a single degree of freedom. In a 1DOF model, the threshold playing frequency
for a fixed embouchure should be always above the coupled acoustic resonance
frequency in the outward-striking case and always below it in the inward-striking
case; however Fig. 6.21 shows a continuous transition from one case to the other.
In general, mechanical response measurements of the lips show several distinct
mechanical resonances. However, the experimentally observed behaviour described
above suggests that important features of the lip motion may be reproduced by a
model involving only two mechanical modes, the lower-frequency mode having
outward character and the higher-frequency mode having inward character.
307
Looking at the first graph of Fig. 6.21 for the 1.31 m pipe length, it can be seen
that in the case of no mouth overpressure, there is only one visible resonance of the
lips at 120 Hz, with an outward-striking behaviour. When the mouth overpressure
is increased, another resonant peak appears at 160 Hz. This resonance is also close
to the playing frequency of the lips and has an inward-striking behaviour. Although
this peak cannot be seen in the amplitude response curve for the case of no mouth
overpressure, its presence can be predicted by the phase response at this frequency
which has a value of −π/2 at 152 Hz. The lack of an amplitude peak at this
frequency can be attributed to the fact that it corresponds to an impedance maximum
of the pipe; this additional load on the lips reduces the amplitude response. Both
peaks are seen in the amplitude response curve when the measurement is carried out
with the pipe removed.
As the mouth pressure is increased, the 160 Hz resonance approaches destabilisation, the height of the amplitude response peak growing much more dramatically
than that of the 120 Hz resonance. This gives the expected behaviour that the lips
must behave as an inward-striking reed if the playing frequency is below the acoustic
resonance frequency of the instrument. A contrasting case can be seen in the graph
for the 2.11 m pipe length. Here the resonance responsible for the playing of the
instrument (135 Hz) has an outward-striking character, and the playing frequency is
above the acoustic resonance of the instrument.
In the intervening lengths between these two cases, there is a much more
complicated interaction between the different lip resonances, and there is clearly
some coupling between the resonant modes of the lips. Despite this change in the
dominant lip resonance, there is a smooth change in playing frequency as the pipe is
extended. Although the change in frequency is continuous, some interesting features
of the coupling between lip resonances can be found in the region in which neither
lip resonance is clearly dominant. As the pipe is extended, the playing frequency
does not always show a monotonic drop in frequency; for some embouchures there
is a small region in which the playing frequency is found to rise as the telescopic
pipe is extended. This anomalous behaviour occurs when the playing frequency is
very close to the acoustic resonance frequency.
The work of Cullen (2000) and Neal (2002) shows that the near-threshold
behaviour of the lips cannot be completely explained using a lip model with only
a single degree of freedom. In a 1DOF model, the threshold playing frequency
for a fixed embouchure should be always above the coupled acoustic resonance
frequency in the outward-striking case and always below it in the inward-striking
case; however Fig. 6.21 shows a continuous transition from one case to the other.
In general, mechanical response measurements of the lips show several distinct
mechanical resonances. However, the experimentally observed behaviour described
above suggests that important features of the lip motion may be reproduced by a
model involving only two mechanical modes, the lower-frequency mode having
outward character and the higher-frequency mode having inward character.