In this section, the creation of a complex waveform, like a square wave via an additive process and the application of the Karplus-Strong model of a plucked string to subtractive synthesis are analysed and compared.
Additive synthesis
If you take a sinusoidal sound, and start adding odd partials to the fundamental, you obtain a square wave with one component, two, three and so on. To obtain that, you need to make sure that the amplitudes of each of the harmonics are inversely proportional to their partial number, the third partial is 1/3 as strong as the first, the fifth partial is 1/5 as strong, and so on.
You can try this out by downloading the Partials patch by Jean-Francois Charles available here.
In that patcher, at the top left, you can set your preferred frequency. Still on the left hand side but below, in the amplitude section, please click on the button labelled square: this will set the correct amplitudes proportions. Then, in the top left section, labelled frequency, type in the correspondent value for each of the harmonics (3rd, 5th, 7th…) and you will be able to see in the waveform display (to the right, in blue) the gradual forming of the square wave.
From bottom to top: starting from a sinusoidal sound at 400 Hz, increasing the amplitude for 3rd, 5th, and 7th harmonic, respectively at 1200 Hz, 2000 Hz, 2400 Hz
This brief recall of additive synthesis should evidence once more how a complex timbre is achieved by increasing the number of harmonics, being careful to adjust adequately their reciprocal amplitudes, thus producing a richer sounding waveform.
If amplitudes are not just fixed over time, but they vary and have different shapes, the timbre complexity can expand further.
This picture shows three sinusoidal waveforms set at different frequencies, duration, and shape, and triggered at the same time to create a more complex waveform
By working on the shape of sound, on the complexity of its transient attack, its body, and decay times and shape, and on the number and amplitudes of the forming harmonics, it is possible to obtain numerous sounds just with the simplest of the waveforms, the sinusoidal (cycle~ object).
Subtractive synthesis: Karplus-Strong
A reverse idea to that of additive synthesis is subtractive synthesis. Starting from a complex sound, most often a full spectrum broadband white noise (noise~ object in Max), and reducing its harmonic and amplitude content with different methods, we can achieve a variety of very interesting sounds. One of the most remarkable methods is the one obtained using the Karplus-Strong model.
This algorithm is a physical model of the acoustics of a plucked string. Just by triggering a burst of noise and applying to that a feedback delay effect, to reduce at each delay the sound’s frequency content, the acoustic model of a plucked string is represented.
The basic idea is, in fact, that the transient attack of a plucked string can be as rich in frequency as a burst of noise (recall the Lesson 3 reading by M. Schafer). As soon as the attack time has passed, friction between the air and the string help to normalise the string motion, leading to a less rich sound, because it loses harmonics and amplitudes on the way. The feedback delay effect applied to the burst of noise has this specific purpose: the sound passing through the delay, and feeding back infinitely until complete reduction to silence of its amplitude, simulate exactly the progressive simplification of the string motion of a plucked string. Thus the sound passes from a very complex attack transient to a simpler body and decay.
To try the effect in Max, please select and copy to your clipboard all the text in green here below. Once you have done it, open Max and from the menu bar select File>New From Clipboard. This should create the patch for you. Have fun!
ps.: if you can’t copy the text or something wrong happens in the visualisation of the patcher, please download it from here: karlplus_explained-maxpat.
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