Time-domain synthesis of conical bore instrument sounds

Model Approaches: The ``Cylindrical Saxophone''

• The junction filter is derived as for a register hole:
  
  $$\mathcal{R}^{-}(z) = \mathcal{R}^{+}(z) = - \frac{\gamma}{\alpha + \gamma} \left(\frac{1 + z^{-1}}{1 - a_{1}z^{-1}} \right),$$
  
  
  where $\alpha$ is the bilinear transform constant and
  
  $$\gamma = \frac{c S_{h}}{2 t S_{o}}, \hspace{0.3in} a_{1} = \frac{\alpha - \gamma}{\alpha + \gamma}.$$
  
  

• The ``cyclone'' and ``cylindrical'' circuits are equivalent with parameters related by:
  
  $$x_{o} = \frac{S_{o} t}{S_{h}} \hspace{0.2in} \mbox{or} \hspace{0.2in} t = \frac{x_{o} S_{h}}{S_{o}}.$$