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Synth Patches Video Demo Lessons Forum Community - Synthonia by SynthCloud
Saturday, 10 January 2015 11:53

Acoustic physics: complex sound

Last time we talked about the simple sound and its features.

The sound we usually hear is not a simple one (sinusoidal oscillation), but a complex sound.

In order to treat this issue, it is necessary to clarify the concept of envelope (we will study this argument in deep later) and harmonics.

The sound envelope is the variation of its amplitude over time.

Let's consider now an instrumental sound: his complexity is evident. If you listen to just a single note, you will find out how the sound is made up, not only of an oscillation, but also of a changeable number of "elements". Those elements are the harmonics we are talking about.

Harmonics show well defined relationship between themselves, and with the fundamental oscillation.

Our definition of a complex sound will thus be:

A periodic oscillation constituted by a fundamental frequency and a series of other frequencies which are integer multiples of the fundamental, and are named higher order harmonics.

This definition naturally leads to the Fourier theorem: “Every periodic function, anyhow complex, can be represented as a (weighted) sum of simple sinusoidal functions whose frequencies are integer multiples of a fundamental frequency”.

So, every harmonics has its own frequency, but also its own amplitude and phase, which are completely independent with respect to the fundamental component. It's indeed the number of harmonics and the parameters of each of them which determines the character of a complex sound that is its timbre.

Complex sounds may be composed of a varying number and kind of harmonics. They can have more or less, even and/or odd harmonics.

Let's look at the harmonics series starting from C2 (about 66Hz). As one can see from the example, the harmonics which are generated have well defined frequency intervals. Starting from 66Hz (C2) we find 132Hz (C3 - octave), 198Hz (G3), 264Hz (C4) and so on.

We may now define the interval which is mainly a musical concept, but may be defined also in a physical framework. The interval is the pitch difference between two sounds, whose value is represented by a ratio between their frequencies.

We may say, at this point, that both the kind and the number of harmonics contained in an instrumental sound depend on the shape of the instrument and the materials it is built of: thus we may give a first definition of a sound TIMBRE as depending (from a physical point of view) on the kind and number of harmonics added to the fundamental one, on their amplitude and phase.

Let's see now some practical examples of complex sounds. 

The Hammond organ works exactly in this way, by generating a fundamental sound plus even and odd harmonics which can be managed through the Drawbars.

The "classical" waves of our synthesizers (triangle, sawtooth, pulse) are nothing else than complex sounds formed by different harmonics. Let's look at them one at the time:

Square Wave

Only odd harmonics are present, whose amplitude is proportional to their order, so the second component (third harmonics) has an amplitude which is one third of the fundamental, and so on. They are all in phase.

Triangle Wave

Only odd harmonics are present, and they are in counter-phase. Their amplitude is proportional to the square of their order. This means that the second component (third harmonics) has an amplitude which is one ninth of the fundamental, and so on.

Sawtooth Wave

There are both even and odd harmonics, they are in phase and their amplitude is proportional to the harmonics order, so that the second harmonics has an amplitude which is half the amplitude of the fundamental and so on.

In order to obtain a reversed sawtooth one should simply change the harmonics phase (counter-phase)

This seems very complex, but one simply has to sum up all the components and the fundamental to obtain the resulting wave.

The acoustic instruments sound is a complex sound itself. But, compared to the sounds seen above, they also show a number of inharmonic components, due to the instrument features. For example, the noise of a grand piano hammer beating up a string, or that of a bow sliding on a violin strings are examples od inharmonic components.

We will treat all these things in the next issues.

Published in Tutorials