### Single-Reed Instruments

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

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 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 and (respectively) which
sample the traveling pressure-wave components within the bore.

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 designed to match measured mode
decay-rates in the bore, the transmission filter can be written down
simply as
for velocity waves, or
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:

where denotes the *z* transform transform of the incident pressure wave,
and denotes the *z* transform of the incident volume-velocity. (All
*z* transform have omitted arguments
, where
denotes the sampling interval in seconds.)

At the far left is the reed mouthpiece controlled by *mouth
pressure* . Another control is *embouchure,* changed in
general by modifying the *reflection-coefficient* function
, where
. 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.

**Next Section:**

A View of Single-Reed Oscillation

**Previous Section:**

Literature on Piano Acoustics and Synthesis