The primary goal of this study was to investigate how upstream resonances might influence the resulting instrument sound and the oscillations of the reed valve.
With this in mind, simple vocal tract characterizations having control parameters directly tied to resonance peak and bandwidth features were explored.
Note that this is not equivalent to modeling the mouth cavity as a Helmholtz resonator, which has an impedance minimum at resonance.
An upstream system with a single resonance is represented by the electrical circuit analog of Fig. 24.
Electrical circuit analog for upstream windway with a single resonance.
The impedance seen from the reed is characterized by peaks at DC (set with Cv) and at the resonance frequency, which is determined by the components L1, C1, and R1.
Despite the extreme simplicity of this characterization, wind instrument performers are typically making use of just a single resonance in their windway to influence the response of the reed.
Within the digital waveguide context, the lumped impedance representation of the upstream system is converted to a traveling-wave scattering junction expressed in terms of reflectances and transmittances.
Figure 25 shows a representative reflectance characteristic when the lung and trachea impedance is assumed infinite.
Upstream reflectance derived from the circuit of Fig. 24.
The complete system of Fig. 24 can be transformed to a traveling-wave scattering characterization using a transmission-matrix approach.
If the series combination of the resonant circuit and volume capacitance are represented by an impedance Zs and the upstream resistance by Za = Ru, the following matrix approach can be followed:
To render these relationships in the digital waveguide domain, it is necessary to transform the plane-wave physical variables of pressure and volume velocity to traveling-wave variables as
where Z0 is the characteristic impedance of the section.
Waveguide pressure variables on both sides of the upstream system are then related by an expression of the form
The process of deriving appropriate discrete-time reflectance and transmittance filters is detailed elsewhere with respect to woodwind tonehole modeling Scavone (1997).
For the system of Fig. 24, the resulting implementation requires four third-order digital filters.
A simplified, intuitive approach is illustrated by the block diagram of Fig. 26.
A simplified upstream resonance block diagram.
A single second-order digital resonator is used to model the upstream resonance while the lung pressure component of the model is extracted and simply added to the reflected upstream pressure component.
A coupling constant g is included to control the relative level of upstream influence.
The unit delay shown in this signal path is necessary to avoid a delay-free loop through the digital resonance filter and reed scattering junction.
From this structure, it should be obvious that second-order digital resonators can be cascaded in parallel to simulate multiple upstream resonances.
However, because vocal tract resonances will not typically have harmonic relationships, it is unlikely that a performer would be able to manipulate the upstream system in such a way that multiple upstream resonances could be used to reinforce multiple downstream resonances.