The first three figures on this page provide a brief review of the model clarinet theory as given in the MSW review paper. Although the model here is the same, the equations are written in a slightly different form for convenience.

The first figure illustrates the pressure and flow variables used in the model. It is assumed that the pressure in the player's mouth is a steady (non-oscillatory) pressure p. The pressure q in the bore of the instrument results from a summation of pressures associated with outgoing and incoming waves.

The volume flow rate of air through the mouthpiece/reed combination is a nonlinear function of the pressure difference p-q across the reed, as is illustrated schematically by the small graph. The flow increases monotonically with p-q for small pressure differences, but then decreases and eventually becomes zero at large pressure differences, as the reed aperture is forced closed.

The volume flow rate through the bore is a linear function of pressure, and is given by the difference in pressure of the outgoing and incoming waves, divided by the impedance of the bore (Eqn. 1). Since the air flow is continuous through the instrument, the flow rate through the mouthpiece is equal to that through the bore (Eqn. 2).

It is convenient to write the flow-continuity equations in a somewhat different form. (See the "scattering theory" approach of Smith, Ref. 2.) First, an "impedance" function for the mouthpiece is defined as in Eqn. 3 in the figure below. The inverse of this impedance is a monotonic function of the pressure difference across the reed as indicated by the upper graph.

Next, the impedances of the mouthpiece and bore are used to formally define "transmission" and "reflection" coefficients in the usual manner (viz., the Fresnel relations in optics). The transmission coefficient T is given by Eqn. 4. From this equation, it is seen that T has the same general shape as the inverse of the mouthpiece impedance 1/z_m. For my experimental system I have approximated T by straight-line segments as shown in the second graph.

Using the definitions of T and of the reflection coefficient R = 1 - T, it is straightforward to show that the flow-continuity equation takes the simple form q_o = q_iR + (p/2)T. Eliminating R yields Eqn. 5, the form of the flow-continuity equation used for the present experiments.

It is easy to show that Eqn. (5) may be written in a re-parameterized form as Eqn. (6), where T' is T expressed as a function of p - 2q_i. [Hint: Subtract p-q_i from both sides and rearrange terms to show that Eqn. (5) expresses a relation between just two independent variables, p - q_i - q_o and p-2q_i.]

It should be noted that Eqn. 5 is an implicit equation that must be solved for q_o as a function of q_i. In previous work the implicit solution of the flow-continuity equation has been obtained by several methods. The oldest method is to obtain the solution graphically by plotting the two sides of the equation separately and finding their intersection. The second method is to find the solution by computer, using a root-finder algorithm. For real-time implementations, more efficient techniques, such as the use of lookup tables or the use of Eqn. 6, have been employed. (See Refs. 2 and 3.)

In the present work, I have chosen to solve Eqn. 5 directly in real time, using an electronic circuit serving as an analog computer. In addition, I have used Eqn. (6) with T' having the same functional shape as shown above for T. Both methods produce stable, clarinet-like oscillations.

Solution of Eqn. 5 gives one relation between the variables q_o and q_i, but another one is needed to actually determine their values. This second relation results from the propagation of the outgoing wave down the bore and back, producing an incoming wave at a later time. This process is illustrated in the next figure below.

It is convenient to think of the outgoing wave as decomposed into a series of narrow pulses. As these pulses travel down the bore and back, they are broadened, as well as being inverted (due to the impedance discontinuity at the end of the tube.) The incoming wave at a given time consists of a superposition of outgoing pulses that were produced over a range of (previous) times corresponding to the width of the broadened pulses.

The mathematical expression of this delayed and broadened response is given by the convolution integral of Eqn. 7. Here r(t) is a function (the "reflection function") describing the shape of the reflected pulses. With the specification of the reflection function, the physical model is complete, and the time-dependent behavior of the model is obtained by solving Eqns. 5 and 7 self-consistently.

Implementation

The two figures below give two slightly different views of how I have implemented the clarinet model described above. The "Schematics" section gives more details of the actual circuits.

The figure immediately below shows how the current values of the incoming and outgoing waves are determined from previous values of the outgoing wave. In my implementation, successive values of the outgoing wave are captured and saved by a bank of sample-and-hold circuits. To compute the current value of q_i (at time t = 2N in the figure), values of q_o from three previous times (t = N-1, t = N and t = N+1) are combined in a weighted average. The weighted averaging acts as a transversal filter and provides an approximation to the convolution integral of Eqn. 7 above. After inversion, this filtered signal becomes the new value of q_i.

The current value of q_o is computed from the mouthpiece pressure p and the current value of q_i by the nonlinear "mouthpiece" circuit, which is an analog computer for either Eqn. 5 or Eqn. 6. The dotted line indicates that an implicit-function solution is being obtained in the case of the circuit for Eqn. 5.

Once the new value of q_o is obtained, it is captured by the sample-and-hold bank, and the calculational cycle starts over, with all times incremented by one sampling unit. The period of oscillation is 2N, just as in an acoustic clarinet, where one wave cycle corresponds to two round-trip transit times.

The figure below shows another view of my model implementation. The left-hand side of the figure shows the mouthpiece algorithm -- represented in flow-diagram form -- for the circuit implementation of Eqn. 6. The remainder of the figure illustrates the frequency tracking, sampling and optional extra filtering of the system.

Timing is based on a standard 1 V / octave synthesizer VCO, referred to as the "pilot wave". The ramp output of this VCO sequentially fires ten trigger pulses (per pilot-wave cycle), each of which enables a sample-and-hold circuit connected to the q_o signal. Another bank of sample-and-hold circuits reads out the three delayed signals that are fed into the transversal filter to implement the convolution function. The output of the sample / delay / convolve unit is the q_i signal, with optional extra filtering provided by a 1 V / octave low-pass VCF. The instrument output can be taken from any of several points in the signal path and further filtered as desired.

Results

The model instrument produces stable oscillations for a wide variety of parameter settings. These oscillations are similar to those reported previously, e. g., in Refs. 2 and 3. At low values of the mouth pressure, oscillations do not build up, because of the loss produced by the convolution filter. Above a certain threshold pressure, nearly sinusoidal, low-amplitude oscillations become stable. At higher values of pressure, the waveform looks like a partially rounded square wave. The oscilloscope photo below shows a typical example of the response for three excitation levels.

Stable oscillations have been obtained with both the Eqns. 5 and 6 analogs. The simplified Eqn. 6 model is easier to set up for clarinet-like oscillations. The Eqn. 5 model generally exhibits a rather high threshold for the onset of oscillations. A similar tendency for high thresholds was noted in Ref. 2 and attributed to a non-optimal choice for the non-linear flow function. Additionally, the Eqn. 5 model tends to be unstable (chaotic) near the oscillation threshold.

The model system also exhibits chaotic oscillations under certain parameter settings, as is to be expected due to the system's non-linear nature. With variation of just one parameter (e. g., the reed circuit's "width" control), the progression to the chaotic state through two period doublings can be observed. This progression is illustrated in the oscilloscope photos below.

Future work on this system will emphasize the development of circuitry for improved representations of the reed response. This work will include implementing a more realistic flow response, and the addition of high-frequency resonance to better simulate the reed's dynamic behavior.