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Single-Reed Instruments

A simplified model for a single-reed woodwind instrument is shown in Fig. 9.38 [431].

Figure 9.38: A schematic model for woodwind instruments.
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If the bore is cylindrical, as in the clarinet, it can be modeled quite simply using a bidirectional delay line. If the bore is conical, such as in a saxophone, it can still be modeled as a bidirectional delay line, but interfacing to it is slightly more complex, especially at the mouthpiece [37,7,160,436,506,507,502,526,406,528], Because the main control variable for the instrument is air pressure in the mouth at the reed, it is convenient to choose pressure wave variables.

To first order, the bell passes high frequencies and reflects low frequencies, where ``high'' and ``low'' frequencies are divided by the wavelength which equals the bell's diameter. Thus, the bell can be regarded as a simple ``cross-over'' network, as is used to split signal energy between a woofer and tweeter in a loudspeaker cabinet. For a clarinet bore, the nominal ``cross-over frequency'' is around $ 1500$ Hz [38]. The flare of the bell lowers the cross-over frequency by decreasing the bore characteristic impedance toward the end in an approximately non-reflecting manner [51]. Bell flare can therefore be considered analogous to a transmission-line transformer.

Since the length of the clarinet bore is only a quarter wavelength at the fundamental frequency, (in the lowest, or ``chalumeau'' register), and since the bell diameter is much smaller than the bore length, most of the sound energy traveling into the bell reflects back into the bore. The low-frequency energy that makes it out of the bore radiates in a fairly omnidirectional pattern. Very high-frequency traveling waves do not ``see'' the enclosing bell and pass right through it, radiating in a more directional beam. The directionality of the beam is proportional to how many wavelengths fit along the bell diameter; in fact, many wavelengths away from the bell, the radiation pattern is proportional to the two-dimensional spatial Fourier transform of the exit aperture (a disk at the end of the bell) [308].

The theory of the single reed is described, e.g., in [102,249,308]. In the digital waveguide clarinet model described below [431], the reed is modeled as a signal- and embouchure-dependent nonlinear reflection coefficient terminating the bore. Such a model is possible because the reed mass is neglected. The player's embouchure controls damping of the reed, reed aperture width, and other parameters, and these can be implemented as parameters on the contents of the lookup table or nonlinear function.

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.
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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 [500]. 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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A View of Single-Reed Oscillation
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