12.3 Summary#
Physical modeling synthesis models the acoustics of an instrument directly, by simulating its physics, rather than synthesizing the properties that emerge from them.
Waves propagate through a medium at a speed \(v\), pass through each other by superposition, and reflect with opposite polarity at rigid boundaries. A sinusoidal wave has a wavelength \(\lambda = \frac{v}{f}\).
On a string of length \(L\) with fixed ends, waves traveling in opposite directions superimpose into standing waves, with nodes every half wavelength.
Each standing wave is a mode of oscillation. The modes of a string have wavelengths \(\lambda_i = \frac{2L}{i}\) and frequencies \(f_i = i \cdot \frac{v}{2L}\), which are the harmonics of \(f_1 = \frac{v}{2L}\). A tube closed at one end has only odd harmonics, and the modes of two- and three-dimensional objects are generally not harmonic.
A feedback delay line, \(y[n] = x[n] + y[n - M]\), simulates propagation with a delay line and reflection with feedback. A delay line is implemented as a circular buffer, which takes constant work per sample and constant memory.
Tuning a feedback delay line to a frequency \(f\) requires \(M = \mathrm{round}(\frac{f_s}{f})\), whose rounding error grows with frequency. Its impulse response is an impulse train, so it is a comb filter, with teeth at the harmonics of \(\frac{f_s}{M}\).
A gain \(g < 1\) in the loop simulates damping. To decay by \(60\) dB in \(T\) seconds, \(g = 10^{-3M/(f_s T)}\). A low-pass filter in the loop, like a two-point average, makes the higher harmonics decay faster, as in real strings.
White noise has independent, identically distributed random samples with mean zero, and its spectrum is flat on average. A burst of white noise is a simple model of the broadband excitation of a pluck.
The Karplus-Strong algorithm excites a feedback delay line, with a low-pass filter and attenuation in its loop, with a burst of white noise.
A digital waveguide models the right-going and left-going waves on a string explicitly, so we can listen at any point along the string. Since delays and losses are LTI filters, we can commute them, and lump the losses together wherever we don’t listen. This is the idea behind commuted synthesis, and Karplus-Strong is a fully commuted digital waveguide.