Sliced lemon (openclipart.org, public domain)
To better understand sound synthesis and synth waveforms it is very useful to learn more about Fourier synthesis, the reverse process of Fourier analysis. In the following sections, I’ll try to condense the necessary knowledge as much as possible. I hope this simplification does not produce mistakable results. Otherwise please drop me a line.
A Sinewave
The most simple tone is a sinewave. None of the natural instruments is capable to create one, but the sound of a tuning fork is very close to it. In synthesizers, a sinewave usually is available directly as the output of an oszillator. We do recognize the frequency of this single sinewave as the pitch of this tone.
More harmonic Sinewaves
To change the timbre of the tone, further sinewaves are added, where their frequencies are integer multitudes of the base frequency. The second partial, for example, just is the sinewave at the double frequency of the base sinewave – the octave. The third partial is the sinewave at the triple frequency – a fifth. The fourth again is an octave, the fifth is a third, the sixth again is a fifth, the seventh is a minor seventh, the eighth is an octave again and so forth. You may be surprised that there are so many “inharmonic” frequencies in a tone, but this model works very well for synthesizing sounds. Those harmonic partials are called the “harmonic series”. An instrument that uses this approach to create different sounds is the Hammond organ:
Drawbar Partials of a Hammond Organ
The nine partials (called drawbars) of a Hammond organ, however, are not enough to simulate any natural instrument or to really create a huge set of different sounds. It is necessary to use even more partials, at least until the upper limit of the audible area was passed. Here’s another example. It’s a sawtooth wave from a Korg Z1 synthesizer. The first 32 harmonic partials are emphasized one after another, so they are clearly audible. Note that the emphasized partials are not added but already contained (and only emphasized):
Partials emphasized by a resonance oscillator
The Amplitudes of Sinewaves
The characteristic of a particular musical instrument is the result of the amplitudes of the partials. Some partials may be nonexistent (zero amplitude), while others may be dominant. Additionally, the tone of natural instruments changes over time. Imagine a piano or mallet tone slowly fading away. The amplitude of the partials decreases, and some will decrease faster than others. Considering the stack of individual sinewaves, both aforementioned requirements mean that it was desirable to somehow influence the amplitudes of the individual sinewaves over time. The Hammond does this by fading out the tone of one of the drawbars. It’s called “percussion”:
Here’s what it sounds like when used to play the organ:
Using a low pass filter (available in almost any synthesizer), one can damp the topmost partials of the aforementioned sawtooth wave:
A Sawtooth wave filtered by a lowpass filter
Inharmonic Frequencies
Until now, it was assumed that the partial tones are integer multitudes of the base sine frequency. Most instruments, however, also produce inharmonic partials – some less, some more. A violin (whose timbre is very similar to a sawtooth wave BTW) has a clear pitch and inharmonic partials will play a minor (but nevertheless important) role. If it comes to bells, drums or cymbals, however, the inharmonic partials will play a more significant role.
Synthesizer sounds will also benefit from inharmonic partials, either applied softly to add some “dirt” to the sound, or to create drum and cymbal like sounds.
Many Inharmonic Frequencies
Noise can be imagined as a tone with many frequencies constantly changing over time. By filtering some frequencies out of the noise, it is easily possible to imitate wind or the roar of the surf. Added to a triangle wave, it is possible to imitate the blowing noise of a real flute. Noise alone can be used to imitate drum like sounds.
Additive vs. subtractive Synthesis
Besides others, two synthesis systems have been described in this posting:
- The Hammond organ provides a (very basic) additive synthesis system, where the amplitude of the first 9 harmonic partials can be altered by dedicated drawbars.
- Subtractive synthesizers provide waveforms which contain lots of partials. Filters are used to remove the unwanted partials.
Further synthesis systems, such as frequency modulation or granular synthesis, are more difficult to understand and use and will thus not be described further in this posting.
Resume
I’d like to repeat some of the most essential portions of this posting:
- The most basic tone we know of is a sinewave.
- Adding more harmonic (or inharmonic) sinewaves alters the characteristics of the tone.
- Dynamic sounds consist of partials whose amplitudes change over time.
- Most synthesizers provide a noise generator to create inharmonic frequencies.
- Most synthesizers provide subtractive synthesis capabilities, where filters are used to alter the amplitudes of the partials. A Hammond organ is a very simple example for additive synthesis.
I hope this posting will help to better understand some of the following postings I plan to write. If not, I did something wrong. Please drop me a line.
