The Karplus-Strong Algorithm

The Karplus-Strong algorithm is not exactly a physical model, but it can be considered a preliminary stage to waveguides. The algorithm is based on a ringbuffer, filled with (white) noise, which is then manipulated. With very simple means, Karplus-Strong can synthesize sounds with the characteristics of plucked strings. Although not entirely realistic, the result has a intriguing individual character.

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The Ringbuffer¶

Ringbuffers are the central element of the Karplus-Strong algorithm. As the name suggests, they are FIFO (first in - first out) buffers, with beginning and end connected. A ringbuffer with N samples can be visualized as follows:

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White Tone from White Noise¶

If a ringbuffer is filled with a sequence of white noise, it can be used for creating a white tone - a harmonic sound with a strong overtone structure. Without resampling, the ring buffer can be shifted by one sample each $1/f_s$ seconds. The resulting pitch of the sound is then determined by the buffer size:

$$f_0 = \frac{f_s}{N}$$

For a sampling rate of $48000$ Hz, a ringbuffer with a length of $N=200$ samples, results in the following pitch:

The sound of this harmonic signal is similar to a buzzer:

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Spectrum and Individual Timbre¶

The spectrum of the white tone includes all harmonics up to the Nyquist frequency with a random amplitude. The overtone structure is individual for every white noise sequence, as is the timbre. These are three versions, started with an individual noise sequence of $N=400$ samples.

Version 1¶

Version 2¶

Version 3¶

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Karplus-Strong¶

Karplus-Strong makes use of the white-tone buffer and combines it with a moving average filter with the impulse response $h[n]$ and an additional gain $g$:

In the original (and most basic form), two samples are read from the buffer $b$, starting from index $i$ (the playhead), and the average of both samples is written to the buffer. An additional gain factor - set to $0.95$ in this example, results in a faster decay:

$$ b[i] = 0.5 (b[i] + b[i+1])$$

$b[i]$ is directly sent to the output $y$ in each step:

$$y[i] = b[i]$$

$i$ is increased, until it reaches $N$ and continues from the beginning. The following image shows this for the step $i=3$:

The result of this circular smoothing is a gradual decrease in high frequencies. The sound begins with a metallic transient:

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Increasing the Filter Size¶

A faster decay of high frequencies can be achieved by larger moving average filters. This example uses a moving average of $10$ samples and a gain of $0.99$.

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Exercise¶