We consider now the problem of the residue as a multiperiodic forcing of a generic nonlinear
system which roughly represents the auditive periphery. The more simple case for the stimulus
is a complex sound consisting of only two partials, say k and k+1, lying in the vicinity of
successive multiples of some missing fundamental 
.
Now we search for structurally stable solutions which could be associated with the residue. As
we have seen, periodic solutions are structurally unstable to perturbations of the external
stimulus. Two-frequency quasiperiodic solutions (difference combination tones) are unable to
reproduce residue behaviour. Three-frequency quasiperiodic solutions are structurally
unstable to perturbation of the system's parameters (Ruelle--Takens--Newhouse theorem). There
remain only two possibilities, three-frequency resonant solutions and chaotic ones. Bearing in
mind the results of section 3.2.2, we propose that the residue is associated with the third
frequency in a three-frequency resonance formed by a frequency generated in the auditive
system itself (in the vicinity of the missing fundamental 
) and two external frequencies (in
the vicinity of 
and 
, respectively).
The vicinity of the external frequencies to successive multiples of some missing fundamental
ensures that 
is a good rational approximation to their frequency ratio. Consequently,
from the results of section 3.2.2, 
and 
are adjacents. With the aid of eq.6 we
obtain the value of the third frequency in the three-frequency resonance of greatest width
between them
Since the external frequencies can be written as (equal detuning):
the shift of the third frequency with respect to the missing fundamental is:
This equation gives a linear dependence of the shift on the detuning 
,
in accordance with the
first pitch shift effect (see figure 2).The predicted slope is 
, just in the middle between
and 
. In figure 5 we have superimposed the behaviour of the corresponding
three-frequency resonances on the data of figure 2. The fit is very good, explaining the first aspect
of the second pitch shift effect (section 2). The second aspect can be interpreted as follows: the
term 
in eq.9 arises from two equal contributions 
obtained by means of a uniform
shift in the two forcing frequencies. If now, maintaining 
fixed we increase the distance to
(we enlarge the spacing between successive partials) the first contribution remains constant
and equal to 
while the second diminishes, determining a decrease in the third frequency
of the resonance and thus in the residue (see eq.9).
Figure 5:
Plot of the predicted pitch shift effect (eq.9) on the data of figure 2.