Clarinet model

The theory of the single reed is described in [McIntyre et al., 1983]. In the digital waveguide clarinet model described below [Smith, 1986a], the reed is modeled as... 

A diagram of the basic clarinet model is shown in Fig. 10.17. The delay-lines carry left-going and right-going pressure samples p and (respectively) which sample the traveling pressure-wave components within the bore. 

We noted that the ideal string equation and the ideal acoustic tube equation are essentially identical. Just as there are many refinements possible to the plucked-string model to make it more realistic, there are many possible improvements for the clarinet model. Replacing the simple reed model with a variable mass/spring/damper allows the modeling of a lip reed as is found in... 

At the highest level, the clarinet model consists of a source-sensor (SS) component referred to as the player, that exerts an effort (i.e. pressure) on the reed component which is connected to the tube (or pipe) component. The tube imposes a pressure on a second SS component referred to as the listener. As the SS component is predefined by default (Gawthrop, 1995), the system already knows how it works, but the reed and the tube components need to be created. 

The input and output of the reed component can be thought of as the pressure applied to the mouthpiece by the player and the pressure inside of the mouthpiece. These correspond to the two ports of the reed component of the clarinet model and are specified as SS components. As the flow of air will be the same for both SS components, they are linked by a common flow function, that is, a V junction, in bond graphs parlance. As explained earlier, the effort carried by the bond entering the junction will equal the sum of the effort carried by the bonds exiting the junction. This means that it is the difference in pressure inside and outside the mouthpiece that drives the movement of the reed. 

This signal representation with only odd harmonics is an approximate model for a clarinet as with a uniform tube closed at one end and open at the other. In order to capture the time-varying envelope and bandwidth, one applies a A(n ) with a fast attack and slow release, and also makes the modulation index I(n ) inversely proportional to this envelope, thus emulating the decreasing bandwidth as a function of time. 

If the bore is cylindrical, as in the clarinet, it can be modeled quite simply using a bidirectional delay line [Smith, 1986a, Hirschman, 1991], 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 [Benade, 1988, Gilbert et al., 1990, Smith, 1991, Valimaki and Karjalainen, 1994a, Scavone, 1997] 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. 

Figure 10.17 Waveguide model of a single-reed, cylindrical-bore woodwind, such as a clarinet.

This chapter extends our modeling techniques from solids to tubes and chambers of air (or other gasses). First we will look at the simple ideal acoustic tube. Next, we will enhance the model to create musical instruments like a clarinet. Then we ll look at our first three-dimensional acoustical systems— air cavities— but we ll discover that many such systems can be treated as simpler (essentially lumped resonance) models. 

All we need is a model of the reed to build a clarinet. A simple model of a clarinet reed is shown in Figure 11.3. We can assume that the reed is nearly massless, that is, the mass is so small that the only thing that must be considered is the instantaneous force on the reed (spring). The pressure inside the bore, Pj is the calculated pressure in our waveguide model the mouth pressure, P is an external control parameter representing the breath pressure inside the... 

Clarinet Pretty good physical model of the clarinet... 

BlowHole Clarinet physical model with tonehole and register vent... 

Chapter 11 looks at sounds produced by air in tubes and cavities. We develop some simple, but amazingly rich, models of a clarinet and a blown pop bottle. Appendix C has derivations and proofs related to acoustic tubes. 

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