The “Reed” Model

• For this study, we will explore digital waveguide techniques for achieving realistic conical bore instrument sounds.

• A simple memory-less, non-linear reed function is used for the realtime implementations discussed here.

• The reed-channel volume flow and pressure difference characteristic, shown in Fig. 12, is solved in terms of a non-linear traveling-wave reflection function (Smith, 1986) as:
  $$p_{d}^{+} = r(p_{\Delta}) \left[ p_{d}^{-} - \frac{p_{u}}{2} \right] + \frac{p_{u}}{2},$$
  where
  $$r(p_{\Delta}) = \frac{Z_{r}(p_{\Delta}) - Z_{o}}{Z_{r}(p_{\Delta}) + Z_{o}},$$
  $p_{d}$ is the pressure on the downstream side of the reed, $p_{u}$ is the player's breath pressure (upstream), $p_{\Delta} = p_{u} - p_{d}$ is the pressure difference across the reed, $Z_{r}$ is a time-varying reed “impedance”, and $Z_{o}$ is the real wave impedance at the air column input.

  Figure 12: Bernoulli static flow expression (top); Simplified reed reflection function (bottom).

  

• Good results have been achieved using the simplified reed reflection function shown at the bottom of Fig. 12 in clarinet synthesis algorithms.

• While the behavior of the reed function plays an important role in the overall quality of a synthesis model, the memory-less system is adequate for the purposes of evaluating the various air column models explored here.