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Commuted Synthesis

In acoustic stringed musical instruments such as guitars and pianos, the strings couple via a ``bridge'' to some resonating acoustic structure (typically made of wood) that is required for efficient transduction of string vibration to acoustic propagation in the surrounding air. The resonator also imposes its own characteristic frequency response on the radiated sound. Spectral characteristics of the string excitation, the string resonances, and body/soundboard/enclosure resonator, are thus combined multiplicatively in the radiated sound, as depicted in Fig. 8.9.

Figure 8.9: Schematic diagram of a stringed musical instrument.
\includegraphics[width=\twidth]{eps/guitar}

The idea of commuted synthesis is that, because the string and body are close to linear and time-invariant, we may commute the string and resonator, as shown in Fig. 8.10.

Figure 8.10: Equivalent diagram in the linear, time-invariant case.
\includegraphics[width=\twidth]{eps/commuted_guitar}

The excitation can now be convolved with the resonator impulse response to provide a single, aggregate, excitation table, as depicted in Fig. 8.11. 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 [443,231]. These simplifications are presently important in single-processor polyphonic synthesizers such as multimedia DSP chips.

Figure 8.11: Use of an aggregate excitation given by the convolution of original excitation with the resonator impulse response.
\includegraphics[scale=0.9]{eps/fexcitation}

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 $ 4R$ gives the velocity to inject additively at that point in both traveling-wave directions. (The factor of $ 4$ 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 8.12: Possible components of a guitar resonator.
\includegraphics[width=\twidth]{eps/guitar_resonator_components}

In a practical instrument, the ``resonator'' is determined by the choice of output signal in the physical scenario, and it generally includes filtering downstream of the body itself, as shown in Fig. 8.12. A typical example for the guitar or violin would be to choose the output signal at a point a few feet away from the top plate of the body. In practice, such a signal can be measured using a microphone held at the desired output point and recording the response at that point to the striking of the bridge with a force hammer. It is useful to record simultaneously the output of an accelerometer mounted on the bridge in order to also obtain experimentally the driving-point impedance at the bridge. In general, it is desirable to choose the output close to the instrument so as to keep the resonator response as short as possible. The resonator components need to be linear and time invariant, so they will be commutative with the string and combinable with the string excitation signal via convolution.

The string should also be linear and time invariant in order to be able to commute it with the generalized resonator. However, the string is actually the least linear element of most stringed musical instruments, with the main effect of nonlinearity being often a slight increase of the fundamental vibration frequency with amplitude. A secondary effect is to introduce coupling between the two polarizations of vibration along the length of the string. In practice, however, the string can be considered sufficiently close to linear to permit commuting with the body. The string is also time varying in the presence of vibrato, but this too can be neglected in practice. While commuting a live string and resonator may not be identical mathematically, the sound is substantially the same.

There are various options when combining the excitation and resonator into an aggregate excitation, as shown in Fig. 8.11. For example, a wave-table can be prepared which contains the convolution of a particular point excitation with a particular choice of resonator. Perhaps the simplest choice of excitation is the impulse signal. Physically, this would be natural when the wave variables in the string are taken to be acceleration waves for a plucked string; in this case, an ideal pluck gives rise to an impulse of acceleration input to the left and right in the string at the pluck point. If loss of perceived pick position is unimportant, the impulse injection need only be in a single direction. (The comb filtering which gives rise to the pick-position illusion can be restored by injecting a second, negated impulse at a delay equal to the travel time to and from the bridge.) In this simple case of a single impulse to pluck the string, the aggregate excitation is simply the impulse response of the resonator. Many excitation and resonator variations can be simulated using a collection of aggregate excitation tables. It is useful to provide for interpolation of excitation tables so as to provide intermediate points along a parameter dimension. In fact, all the issues normally associated with sampling synthesis arise in the context of the string excitation table. A disadvantage of combining excitation and resonator is the loss of multiple output signals from the body simulation, but the timbral effects arising from the mixing together of multiple body outputs can be obtained via a mixing of corresponding excitation tables.

If the aggregate excitation is too long, it may be shortened by a variety of techniques. It is good to first convert the signal $ a(n)$ to minimum phase so as to provide the maximum shortening consistent with the original magnitude spectrum. Secondly, $ a(n)$ can be windowed using the right wing of any window function typically used in spectrum analysis. An interesting choice is the exponential window, since it has the interpretation of increasing the resonator damping in a uniform manner, i.e., all the poles and zeros of the resonator are contracted radially in the $ z$ plane by the same factor.



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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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