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Related Forms

We see from the preceding example that a filtered-delay loop (a feedback comb-filter using filtered feedback, with delay-line length $ 2M+2N$ in the above example) simulates a vibrating string in a manner that is independent of where the excitation is applied. To simulate the effect of a particular excitation point, a feedforward comb-filter may be placed in series with the filtered delay loop. Such a ``pluck position'' illusion may be applied to any basic string synthesis algorithm, such as the EKS [432,208].

By an exactly analogous derivation, a single feedforward comb filter can be used to simulate the location of a linearized magnetic pickup [201] on a simulated electric guitar string. An ideal pickup is formally the transpose of an excitation. For a discussion of filter transposition (using Mason's gain theorem [303,304]), see, e.g., [336,452].7.9

The comb filtering can of course also be implemented after the filtered delay loop, again by commutativity. This may be desirable in situations in which comb filtering is one of many options provided for in the ``effects section'' of a synthesizer. Post-processing comb filters are often used in reverberator design and in virtual pickup simulation.

Figure 6.18: Use of a second delay-line tap to implement comb filtering corresponding to hammer-strike echoes returning from the far end of the string.
\includegraphics[width=\twidth]{eps/pianoSecondStringTap}

The comb-filtering can also be conveniently implemented using a second tap from the appropriate delay element in the filtered delay loop simulation of the string, as depicted in Fig.6.18. The new tap output is simply summed (or differenced, depending on loop implementation) with the filtered delay loop output. Note that making the new tap a moving, interpolating tap (e.g., using linear interpolation), a flanging effect is available. The tap-gain $ c$ can be brought out as a musically useful timbre control that goes beyond precise physical simulation (e.g., it can be made negative). Adding more moving taps and summing/differencing their outputs, with optional scale factors, provides an economical chorus or Leslie effect. These extra delay effects cost no extra memory since they utilize the memory that's already needed for the string simulation. While such effects are not traditionally applied to piano sounds, they are applied to electric piano sounds which can also be simulated using the same basic technique.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-06-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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