A Three-Port Tonehole Model

• For the purpose of real-time modeling, the two-port implementation has a particular disadvantage: the two lumped characterizations of the tonehole as either closed or open cannot be efficiently unified into a single tonehole model.

• It is preferable to have one model with adjustable parameters to simulate the various states of the tonehole, from closed to open and all states in between.

• To this end, it is best to consider a distributed model of the tonehole, such that “fixed” portions of the tonehole structure are separated from the “variable” component.

• The junction of the tonehole branch with the main air column of the instrument can be modeled in the DW domain using a three-port scattering junction, as described in Scavone (1997).

• This method inherently models only the shunt impedance term of the Keefe tonehole characterization, however, the negative length correction terms implied by the series impedances can be approximated by adjusting the delay line lengths on either side of the three-port scattering junction.

• The other “fixed” portion of the tonehole is the short branch segment itself, which is modeled in the DW domain by appropriately sized delay lines.

• This leaves only the characterization of the open/closed tonehole end.

• A simple inertance model of the open hole end offers the most computationally efficient solution. The impedance of the open end is then given by
    $$Z_{e}^{(o)}(s) = \frac{\rho t}{A_{e}} s,$$ (11)
  where $\rho$ is the density of air, $A_{e}$ is the cross-sectional area of the end hole, $t$ is the effective length of the opening ( $\approx A_{e}^{1/2}$), and $s$ is the Laplace transform frequency variable.

• The open-end reflectance is
    $$\mathcal{R}_{e}^{(o)}(s) \ensuremath{\stackrel{\Delta}{=}}\frac{P... ...- Z_{cb}}{Z_{e}^{(o)}(s) + Z_{cb}} = \frac{t A_b s - c A_e}{t A_b s + c A_e },$$ (12)
  where $Z_{cb} = \rho c / A_{b}$ is the characteristic impedance of the tonehole branch waveguide, $A_{b}$ is the cross-sectional area of the branch and $c$ is the speed of sound.

• An appropriate discrete-time filter implementation for $\mathcal{R}_{e}^{(o)}$ can be obtained using the conformal bilinear transform from the $s$-plane to the $z$-plane (Oppenheim and Schafer, 1989, pp. 415-430), with the result
    $$\mathcal{R}_{e}^{(o)}(z) = \frac{a - z^{-1}}{1 - a z^{-1}},$$ (13)
  where
    $$a = \frac{t A_b \alpha - c A_e}{t A_b \alpha + c A_e}$$ (14)
  and $\alpha$ is the bilinear transform constant that controls frequency warping.

• The discrete-time reflectance $\mathcal{R}_{e}^{(o)}(z)$ is a first-order allpass filter, which is consistent with reflection from a “masslike” impedance.

• It is possible to simulate the closing of the tonehole end by taking the end hole radius (or $A_{e}$) smoothly to zero.

• In the above implementation, this is accomplished simply by varying the allpass coefficient between its fully open value and a value nearly equal to one.

• With $a \approx 1,$ the reflectance phase delay is nearly zero for all frequencies, which corresponds well to pressure reflection at a rigid termination.

• A complete implementation scheme is diagrammed in Figure 5.

  Figure 5: “Distributed” digital waveguide tonehole implementation.

  

• Figure 6 shows the reflection functions obtained using this model in comparison to the Keefe transmission-line results.

  Figure 6: Calculated reflection functions for a simple flute air column [see (Keefe, 1990a)]. Transmissionf line model vs. DW “distributed” tonehole model with one hole closed (top), three holes closed (middle), and six holes closed (bottom).

  

• This efficient model of the tonehole produces results very much in accord with the more rigorous model.

• A more accurate model of the tonehole branch end, which is not pursued here, would include a frequency-dependent resistance term and require the variation of three first-order filter coefficients.