Tonehole Models

A method was previously reported [8] for converting the continuous-time woodwind tonehole model of Keefe [2] to a discrete-time two-port scattering junction model for implementation in the digital waveguide (DW) domain. The results using this technique are shown in Figure 1 and compared with reproduced results using the technique of Keefe [3] for a simple flute air column with six toneholes. Discrepancies between the two methods are mainly evident in early closed hole reflections. Keefe's results were calculated for a frequency range of 10 kHz and subsequently smoothed in the time-domain with a hamming window [5]. By trial and error, a lowpass filter with a cutoff frequency around 4 kHz was found to best reproduce Keefe's results. The DW results were obtained at a sampling rate of 44.1 kHz and then lowpass filtered to a 10 kHz bandwidth to correspond with the calculations of [3]. Further lowpass filtering is inherent from the Lagrangian, delay-line length interpolation technique used in this model [6]. Because such filtering is applied at different locations along the air column and is dependent on the particular fractional delay length modeled, a cumulative effect is difficult to accurately determine. As diagrammed in Figure 2, that portion of the signal reflected at the first tonehole is affected by only two interpolation filters, that at the second tonehole reflection is affected by four filtering operations, etc. Thus, early reflections in the DW model results are less lowpass filtered than the results of [3]. It should be noted that each fractional delay interpolation filter in this implementation can be combined with a lossy propagation filter, which models lumped thermoviscous losses along its corresponding segment of the air column and which is also given by a lowpass frequency response. In this way, the inaccuracies inherent in low-order delay length interpolation filters can often be minimized. Alternately, higher-order interpolation filters can be used which introduce minimal frequency magnitude distortion.

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. While it is possible to develop a cross-fading/interpolation scheme to simulate ``half-holing'', this would require that two simultaneous models be run to simulate just one tonehole. 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 [8]. 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

$$Z_{e}^{(o)}(s) = \frac{\rho t}{S_{e}} s,$$ (1)

where $\rho$ is the density of air, $S_{e}$ is the cross-sectional area of the end hole, $t$ is the effective length of the opening ( $\approx S_{e}^{1/2}$), and $s$ is the Laplace transform frequency variable. The open-end reflectance is

$$\mathcal{R}_{e}^{(o)}(s) \mathrel{\stackrel{\mathrm{\Delta}}... ...- Z_{0b}}{Z_{e}^{(o)}(s) + Z_{0b}} = \frac{t s - c}{t s + c },$$ (2)

where $Z_{0b}$ is the characteristic impedance of the tonehole branch waveguide 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 [7, pp. 415-430], with the result

$$\mathcal{R}_{e}^{(o)}(z) = \frac{a - z^{-1}}{1 - a z^{-1}},$$ (3)

where

$$a = \frac{t \alpha - c}{t \alpha + c}$$ (4)

and $\alpha$ is the bilinear transform constant which controls frequency warping. A good low-frequency discrete-time fit is achieved for $\alpha = 2 f_{s}.$ 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 $S_{e}$) 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 3. Figure 4 shows the reflection functions obtained using this model in comparison to the Keefe transmission-line results. 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.

Figure 3: ``Distributed'' digital waveguide tonehole implementation.

Figure 4: Calculated reflection functions for a simple flute air column (see [3]). Transmission line model vs. DW ``distributed'' tonehole model with one hole closed (top), three holes closed (middle), and six holes closed (bottom).