Clarinet-like system modeled with Simscape

Time domain model of the oscillations in a clarinet-like system. Includes coupling of reed vibration and with the tube acoustic resonance.

This submission describes a physical model of a clarinet-like system that provides a physically realistic simulation of the playing behavior of reed woodwind instruments. The model uses the Acoustical Domain for Simscape to calculate the acoustical behavior of the resonant air column in the woodwind instrument. The Simscape Translational Mechanical domain models the motion of the clarinet reed. The present version of the model assumes the blowing pressure in the player's mouth is generated by an acoustical pressure source of (relatively slowly) time varying amplitude. The acoustic pressure difference across the reed drives the motion of the reed. The time varying aperture between the reed and the tip of the mouthpiece allows air flow into the mouthpiece that depends on the instantaneous dimension of the aperture. A good description of the physical interaction between the reed, the pressure inside the instrument and the sound radiated into the room can be found on the web page from UNSW maintained by Joe Wolfe.

These models use the Acoustical Library for Simscape, in addition to the files provided here.

The Clarinet tube models

Two simple clarinet-like tube models are included here. Each is useful for producing only a single note. Both tubes have an internal diameter of 15 mm, which is a reasonable approximation for modern clarinets. The closed length of the simple tube is chosen to produce the highest tone in the chalumeau register. This note has two input impedance resonance modes below cutoff. The impedance at these two mode frequencies is high enough to allow a threshold oscillation to exist. This means that both resonances are relevant to the playing behavior for any blowing pressure above threshold.

The tube first will be called the “simple tube.” Starting at the mouthpiece end, this tube has a section of solid tube (no tone holes), followed by a row of ten tone holes with the same dimensions and separation. The tone hole lattice is designed to have a cutoff frequency of 1500 Hz. The tube terminates with an open end after this tone hole lattice. Each tone hole and the open end are all terminated by an approximation of the radiation impedance for a radiating area the size of the opening.

For the simple tube, the frequencies of the two resonances differ from a true harmonic relationship by several Hertz. It is generally believed that this inharmonicity would produce an instrument that might produce tones, but would not be an acceptable musical instrument. For comparison, the second tube, which will be called the “better tube,” has been modified so that its resonances are much more accurately harmonically related. This is done increasing the inside diameter of the tube for a small segment located at approximately one third the closed length. All other dimensions for the two tubes are the same. The model of the simple tube is shown here.

Transient blowing pressure

One thing that can be investigated with these simple models is the startup transient behavior of the tube depending on the blowing pressure and embouchure/tongue. Some previously published examples of calculated waveforms start the simulation with the blowing pressure at its full value, equivalent to an instantaneous increase in pressure in the mouth. The examples included here use a Simscape code file in the clarinet library that is included with the submission. This code produces a blowing pressure as shown in the figure below. This waveform works well with the Simscape solver because its first derivative is continuous. The curve shown has a 30 ms rise, which has been used for the cases shown below, unless explicitly stated otherwise.

Computational experiments with these files will be available here in later submissions.