The “Cyclone”

• The “cyclone” conical bore model is based on a compound cylindrical-conical segment air column model as illustrated at the top of Fig. 19.

  Figure 19: “Cyclone” physical structure (top) and digital waveguide block diagram (bottom).

  

• The input cylindrical section roughly models the instrument mouthpiece cavity and its use avoids the complications previously discussed with respect to the non-linear driver.

• In addition, the cylindrical section can be designed to have an equivalent volume equal to the missing conic section volume. Assuming no diameter discontinuity at the cylinder-cone junction, this constraint is met using a cylindrical section length equal to $x_{o}/3$.

• It should be noted that Benade distinguishes between a cavity's physical and equivalent volumes under playing conditions, which are typically not the same. For the simplified reed function used in this implementation, however, it is reasonable to ignore this difference.

• The cylinder-cone junction filter is derived assuming continuity of pressure and conservation of volume velocity and then discretized using the bilinear transform as:
  $$\mathcal{R}_{j}^{-}(z) = \mathcal{R}_{j}^{+}(z) = - \frac{\gamma}{\alpha + \gamma} \left(\frac{1 + z^{-1}}{1 - a_{1}z^{-1}} \right),$$
  where $\alpha$ is the bilinear transform constant,
  $$\gamma = \frac{c}{2 x_{o}}, \hspace{0.3in} a_{1} = \frac{\alpha - \gamma}{\alpha + \gamma},$$
  $c$ is the speed of wave propagation in the structure, and $x_{0}$ is the length of the truncated conic section.

• This expression could just as well have been derived from the parallel combination of the input inertance $M_{o}$ and the wave impedance of the input cylindrical section.

• The junction transmittance magnitude response ( $\vert 1 + \mathcal{R}\vert$) is shown in Fig. 20 for various values of $x_{o}$.

  Figure 20: The cylinder-cone junction transmittance for various values of truncation $x_{o}$.

  

• The “high-pass” filter characteristic associated with the conical waveguide input inertance term can vary significantly depending on the frustum dimensions.

• Shorter values of $x_{o}$ correspond to steeper flare rates, which produce greater wave discontinuity at the junction and greater low-frequency attenuation.

• While this might appear to imply a preference for less steeply flared conic sections, it should be remembered that larger values of $x_{0}$ correspond to larger values of $\beta$ in Fig. 15 and thus greater mode inharmonicity. The result is a design conflict between junction discontinuity, which destabilizes the lower air column modes, and mode harmonicity.

• The cylinder-cone junction can be implemented using a single first-order digital filter, as discussed by Välimäki and Karjalainen (1994); Smith (1991) and others. A block diagram of the resulting digital waveguide model is shown in Fig. 19.

• Figure 21 displays the input impedance and sound spectrum produced by an example “cyclone” waveguide model.

  Figure 21: Example “cyclone” model input impedance and synthesized sound spectrum.

  

• Despite significant inharmonicity of the input impedance peaks, the resulting synthesis spectrum is harmonic and exhibits contributions from “misaligned” peaks as a result of the non-linear regenerative process.

• The sounds produced by the “cyclone” model have a distinctive saxophone quality, though the instabilities associated with truncated conic frusta as outlined in earlier sections are present and the functional parameter space can be difficult to assess.