Register Hole Models

• Woodwind register holes are designed to discourage oscillations based on the fundamental air column mode and thus to indirectly force a vibratory regime based on higher, more stable resonance frequencies.

• A register vent functions both as an acoustic inertance and an acoustic resistance (Benade, 1976). It is ideally placed about one-third of the distance from the excitation mechanism of a cylindrical-bored instrument to its first open hole.

• Sound radiation from a register hole is negligible.

• The DW implementation of a register hole can proceed in a manner similar to that for the tonehole. The series impedance terms associated with toneholes are insignificant for register holes and can be neglected.

• Modeling the open register hole as an acoustic inertance in series with a constant resistance, its input impedance as seen from the main bore is given by
    $$Z_{rh}^{(o)}(s) = \frac{\rho t}{A_{rh}} s + \xi,$$ (15)
  where $\rho$ is the density of air, $t$ is the effective height, $A_{rh}$ is the cross-sectional area of the hole, $\xi$ is the acoustic resistance, and $s$ is the Laplace transform frequency variable.

• Proceeding with a two-port DW implementation, the register hole is represented in matrix form by
  

  $$\left[\begin{array}{c} P_{1}^{-} \\ P_{2}^{+} \end{array}\right] = \left[\begin{array}{cc} \mathcal{R}^{-} & \mathcal{T}^{-} \\ \ma... ...array}\right] \left[\begin{array}{c} P_{1}^{+} \\ P_{2}^{-} \end{array}\right]$$

  $$= \frac{1}{Z_{c} + 2 Z_{rh}} \left[\begin{array}{cc} -Z_{c} & 2 Z_{... ...rray}\right] \left[\begin{array}{c} P_{1}^{+} \\ P_{2}^{-} \end{array}\right],$$ (16)

  where the open register hole shunt impedance is given by $Z_{rh}^{(o)}$ and $Z_{c}$ is the characteristic impedance of the main air column.

• The reflectances and transmittances are equivalent at this junction for wave components traveling to the right or to the left. As $\mathcal{T} = 1+\mathcal{R},$ a one-filter form of the junction is possible.

• Using the bilinear transform, an appropriate discrete-time implementation for $\mathcal{R}_{rh}$ is given by
    $$\mathcal{R}_{rh}^{-}(z) = \mathcal{R}_{rh}^{+}(z) = \frac{-c \lef... ...eft( \zeta + \alpha \psi \right) + \left( \zeta - \alpha \psi \right) z^{-1}},$$ (17)
  where
    $$\zeta = c + 2 A_{0} \xi/\rho \hspace{0.3in} \mbox{and} \hspace{0.3in} \psi = 2 A_{0} t/A_{rh},$$ (18)
  $A_{0}$ is the cross-sectional area of the main air column, and $\alpha$ is the bilinear transform constant that controls frequency warping.

• Assuming the closed register hole has neglible effect in the acoustic model, simulated closure of the register hole in this implementation is achieved by ramping the reflectance filter gain to zero.

• This implementation is similar to that of Välimäki et al. (1993), though resistance effects were not accounted for in that study.

• As discussed by Benade (Benade, 1976, p. 459), a misplaced register hole will raise the frequency of the second air column mode by an amount proportional to its displacement from the ideal location (in either direction).

• Such behavior is well demonstrated when this register hole implementation is added to the real-time clarinet model.