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Digital Waveguide Single-Reed Implementation

A diagram of the basic clarinet model is shown in Fig.9.39. The delay-lines carry left-going and right-going pressure samples $ p_b^{+}$ and $ p_b^{-}$ (respectively) which sample the traveling pressure-wave components within the bore.

Figure 9.39: Waveguide model of a single-reed, cylindrical-bore woodwind, such as a clarinet.
\includegraphics[width=\twidth]{eps/fSingleReedWGM}

The reflection filter at the right implements the bell or tone-hole losses as well as the round-trip attenuation losses from traveling back and forth in the bore. The bell output filter is highpass, and power complementary with respect to the bell reflection filter [502]. Power complementarity follows from the assumption that the bell itself does not vibrate or otherwise absorb sound. The bell is also amplitude complementary. As a result, given a reflection filter $ H_r(z)$ designed to match measured mode decay-rates in the bore, the transmission filter can be written down simply as $ H_t(z) = 1 - H_r(z)$ for velocity waves, or $ H_t(z) = 1 +
H_r(z)$ for pressure waves. It is easy to show that such amplitude-complementary filters are also power complementary by summing the transmitted and reflected power waves:

\begin{eqnarray*}
P_t U_t + P_r U_r &=& (1+H_r)P \cdot (1-H_r)U + H_r P \cdot (-H_r)(-U)\\
&=& [1-H_r^2 + H_r^2]PU \;=\; PU,
\end{eqnarray*}

where $ P$ denotes the z transform transform of the incident pressure wave, and $ U$ denotes the z transform of the incident volume-velocity. (All z transform have omitted arguments $ (\exp(j\omega T)$ , where $ T$ denotes the sampling interval in seconds.)

At the far left is the reed mouthpiece controlled by mouth pressure $ p_m$ . Another control is embouchure, changed in general by modifying the reflection-coefficient function $ \rho(h_{\Delta}^{+})$ , where $ h_{\Delta}^{+}
\isdeftext p_m/2 - p_b^{+}$ . A simple choice of embouchure control is an offset in the reed-table address. Since the main feature of the reed table is the pressure-drop where the reed begins to open, a simple embouchure offset can implement the effect of biting harder or softer on the reed, or changing the reed stiffness.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA