- 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.
- The impedance seen from the reed is characterized by peaks at DC (set with
*C*_{v}) and at the resonance frequency, which is determined by the components*L*_{1},*C*_{1}, and*R*_{1}. - 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.
- 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
*Z*_{s}and the upstream resistance by*Z*_{a}=*R*_{u}, the following matrix approach can be followed:

= (43) =

- 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

(44) *Z*_{0}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

(45) - 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 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.

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