Commuted Waveguide Synthesis

Plucked string instruments are well modeled as linear systems and this allows great flexibility in the way they are implemented. Commuted synthesis takes advantage of this fact and provides a highly efficient way of producing high quality plucked-string sounds.

The model components discussed thus far only account for vibrations of the string. In order to convincingly simulate a complete string instrument system, we must also account for the bridge and body response, as well as radiation patterns into the surrounding environment.
Assuming we could develop representative filters for the various components of a string instrument, the output response of the complete system would be diagrammed and computed as shown below:

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For plucked or ideally struck strings, each component of the system is linear and time-invariant. Therefore, the system components can be rearranged without affecting the overall output.
The body response of a string instrument is complex and generally requires a high-order digital filter to accurately simulate. However, by taking advantage of system commutativity, it is possible to record a body “impulse response” and use it as the string excitation. This technique is referred to as commuted waveguide synthesis. As shown, it is also possible to commute the radiation directivity response.

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When using commuted synthesis, body size can be roughly simulated with the body response playback rate, which shifts the body resonances higher or lower in frequency. This technique is demonstrated in the STK Mandolin class, which implements commuted synthesis of a two string mandolin instrument.
For commuted synthesis, it is possible to pre-filter the input signal with a given pluck position filter. To maintain synthesis parameter flexibility, this is typically done during the synthesis computation.