The Cylindrical Saxophone

• With the help of Fig. 14, we can develop an alternative interpretation of the “cyclone” model as two uniform transmission-line elements separated by a shunt inertance component.

• If the inertance is implemented as a small, mass-like “register hole”, we have the physical structure shown in Fig. 22 and a model that is equivalent to Benade's “cylindrical saxophone” (Benade, 1988).

  Figure 22: The “cylindrical saxophone” model.

  

• The impedance of an open-hole shunt inertance, given in terms of a Laplace transform, is:
  $$Z(s) = \frac{\rho t}{A_{h}} s,$$
  where $t$ is the effective height of the hole and $A_{h}$ is its cross-sectional area.

• The junction filter derived for the register hole is (Scavone and Cook, 1998)
  $$\mathcal{R}_{j}^{-}(z) = \mathcal{R}_{j}^{+}(z) = - \frac{\gamma}{\alpha + \gamma} \left(\frac{1 + z^{-1}}{1 - a_{1}z^{-1}} \right),$$
  where $\alpha$ is the bilinear transform constant,
  $$\gamma = \frac{c A_{h}}{2 t A_{o}}, \hspace{0.3in} a_{1} = \frac{\alpha - \gamma}{\alpha + \gamma},$$
  and $A_{o}$ is the cross-sectional area of the cylindrical pipe.

• The “cyclone” and “cylindrical” circuits are equivalent with parameters related by:
  $$x_{o} = \frac{A_{o} t}{A_{h}} \hspace{0.2in} \mbox{or} \hspace{0.2in} t = \frac{x_{o} A_{h}}{A_{o}}.$$

• It is interesting to compare the corresponding conic frustum and register hole parameters. We see that the length of the truncated conic section is proportional to register hole height and inversely proportional to register hole radius.

• The “cylindrical” model parametrization scheme has the added benefit of allowing synthesis of either a clarinet or a saxophone-like system via control of the filter gain parameter $\gamma$.

• The register hole is effectively closed when its radius is zero, which results in $\gamma = 0$ and $\mathcal{R}_{j}(z) = 0 \; \forall \; z$. Since there is no junction discontinuity, the system reduces to a single continuous cylindrical section.

• The “cylindrical saxophone” model presents a slightly more complex parameter space (parameters $t$ and $A_{h}$ versus $x_{o}$), though its response is similar to the “cyclone” model.