Commuted Synthesis of Strings

In a complete stringed musical instrument, such as a guitar, the string couples via the bridge into a resonating ``body'' which is needed for coupling to the surrounding air, and which imposes a frequency response of its own on the radiated sound. In addition, spectral characteristics of the string excitation affect the radiated sound. Thus, we have the components shown in Fig. 4.29.

Figure 4.29: Schematic diagram of a stringed musical instrument.

Because the string and body are approximately linear and time-invariant, we may commute the string and resonator, as shown in Fig. 4.30.

Figure 4.30: Equivalent diagram in the linear, time-invariant case.

The excitation can now be convolved with the resonator impulse response to provide a single, aggregate, excitation table, as depicted in Fig. 4.31. This is the basic idea behind commuted synthesis, and it greatly reduces the complexity of stringed instrument implementations, since the body filter is replaced by an inexpensive lookup table [449,233]. These simplifications are presently important in single-processor polyphonic synthesizers such as multimedia DSP chips.

Figure 4.31: Use of an aggregate excitation given by the convolution of original excitation with the resonator impulse response.

In the simplest case, the string is ``plucked'' using the (half-windowed) impulse response of the body.

An example of an excitation is the force applied by a pick or a finger at some point, or set of points, along the string. The input force per sample at each point divided by gives the velocity to inject additively at that point in both traveling-wave directions. (The factor of comes from splitting the injected velocity into two traveling-wave components, and from the fact that two string end-points are being driven.) Equal injection in the left- and right-going directions corresponds to an excitation force which is stationary with respect to the string.

Figure 4.32: Possible components of a guitar resonator.

In a practical instrument, the ``resonator'' is determined by the choice of output