A Physical Model of a Single-Reed Wind Instrument Including Actions of the Player

Today, the musical interest of physical modeling synthesis is well defined (e.g., Keefe 1992; Smith 1996). In particular, time-domain simulation of the physical operation of musical instruments makes it possible to create transitory and perceptive phenomena that are difficult to obtain by other data-processing methods. The design of the complete model (including the simplified modeling of the main actions of the player) of a single-reed instrument presented here illustrates the fact that "a simplified model of a wind instrument produces sound exceedingly similar to that of the actual system" (Keefe 1992). Contrary to many approaches where musical application of sound synthesis is the principal objective, the objectives of physical modeling are first to "test the accuracy of the implicit assumptions and structure of the underlying physical model" with "simulation[s] of . . . known instrument-player system[s]," as Keefe (1992) remarks. The second objective is to allow the instrument maker to have an idea of the sound produced by an instrument in a performance situation before the construction of a prototype. The final objective is to be able to design physically unrealizable virtual instruments—for example, a transverse saxophone (Bouasse 1929)—while remaining in a particular timbre space and keeping the aspect of membership in a family of instruments. The primary emphasis is placed on roughly modeling the components of the instrument as well as significant performance controls.

This article first presents a model of a single-reed instrument based on the principle of modularity: the instrument is built with interconnected elements. This is made possible by an adapted representation of tubes. Next, the model of the single-reed mouthpiece is detailed including the action of the player, including breath, tongue, and lips. The article then presents the tone-hole model including the key or finger motion. Each important element of the model is illustrated by sound examples included in the 2003 Computer Music Journal Sound Anthology.

A Modular Representation of the Instrument

The postulate of modularity is necessary for various reasons. First, it is perfectly adapted to object-oriented programming, of which one of the most important characteristics is flexibility. Second, modularity can be easily applied to the modeling of wind instruments owing to the existence of anticipative elements that connect to lumped elements. (These are both described later.) Finally, modularity holds much potential for evolution and a progressive extension to the entire wind instrument family.

A Representation Adapted to Object-Oriented Programming

In an object-oriented programming environment, each object or class has associated attributes characterizing its structure as well as methods defining its behavior. This object communicates with others by its parameters and public properties, namely, properties seen by the other objects, which must be defined according to a communications standard. On the other hand, the internal working procedure of the object, characterized by its parameters and private properties, can be modified without disturbing the entire model. This allows great flexibility, because one element of an instrument can be changed while maintaining the integrity of the whole model.

For modeling single-reed wind instruments, four objects are distinguished: the mouthpiece and reed, the tube, the tone hole with possible closing by a finger or a key, and the bell (see Figure 1). Each element has one or two ports, each one being characterized by an input pin and an output pout. According to the plane wave approximation at any port, the average pressure p on the cross section S and the volume velocity u entering through this section are given by

where q0 is the density of the air and c the speed of sound.

Figure 1.

Four basic modules for building a single-reed wind instrument.

Each connection of two or more elements is realized by writing the continuity of pressure and conservation of volume velocity. Thus, this constitutes the communication standard between the various elements.

Decoupling of Lumped Elements Via the Anticipative Property of Tubes

A causal discrete-time system is called "anticipative" if the outputs can be calculated at least one time step in advance...