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Physical Audio Signal Processing
    Woodwinds
       Single-Reed Instruments
          Digital Waveguide Single-Reed Implementation

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Digital Waveguide Single-Reed Implementation

A diagram of the basic clarinet model is shown in Fig.6.2. 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 6.2: 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 [513]. 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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Previous: Single-Reed Instruments
Next: A View of Single-Reed Oscillation
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.