Sound theory: Harmonic content of a waveform

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What we've said so far regarding sinusoids are the foundations upon which we will build the rest of our sound world. Pure sinusoids don't exist in real life. Real sounds are enriched by their so called harmonics.

Let's take a practical example considering what happens when the 5th string of a guitar is plucked by a guitarist. Clearly the answer is that the A note plays:

A note played by a guitar  [Track 10]

Sound theory - A note played by a guitar [Track 10]
Fine. But what actually happens?

The string begins to oscillate at a 440 Hz frequency. So then why doesn't it emit a simple sinusoid sound rather than the sound we have just listened to? There are many reasons for this which we will clarify one at a time. What's for sure is that the harmonic content of the note played by the guitarist isn't the same as that of a simple sinusoid. When a note is played on an instrument, the frequency that corresponds to the note is generated and this is called the fundamental. Together with this fundamental other harmonics are generated, which are all the integer multiples of this frequency with a progressively decreasing amplitude. Together with the A note, the following sinusoids are generated:

  • 440 Hz Fundamental (first harmonic)

  • 880 Hz Second harmonic

  • 1320 Hz Third harmonic

  • ... ... ...

  • n*440 Hz nth harmonic

This behaviour depends on the fact that the string plucked by the guitarist doesn't oscillate only at the fundamental frequency but also at the harmonic frequencies, as shown in the following figure:

Sound theory - Harmonic oscillations of a vibrating string

Harmonic oscillations of a vibrating string

As we have said, the first harmonic is called "fundamental" and it characterizes the note that we effectively perceive. This is because it is the harmonic with the greatest amplitude. The second harmonic has double the frequency of the fundamental, which means that the string is vibrating as illustrated in the figure, overlapping with the fundamental.

If you are reading this course you probably have some basic musical notions and so you probably know that if you add an octave to a note you get to the same note you started off from, in our case the A note, clearly a more acute A (try it on a piano if you're not convinced). So, the second harmonic is the same note as the fundamental and adds warmth to the sound. The third harmonic is no longer an A and therefore enriches the sound differently.

In the figure you can see how the successive harmonics are generated and how the their amplitudes diminish as frequency increases. In other words, if a guitar string is plucked, the harmonics that relevantly contribute to the generated sound are about 10. Beyond this point, the amplitude of the rest of the harmonics become irrelevant if compared to the amplitude of the fundamental. We can also see how at the centre of the string low frequencies prevail, whilst on the sides high ones are more present. This is very important when placing microphones: if we wanted a high frequency sound to come out of a snare drum we'd point the microphone towards the edge, whilst if we wanted to have a sound with more low frequencies we'd point the microphone towards its centre.

Guitar amplifiers can be made with both valves and transistors and you'll find strong supporters of both because of the different sounds they produce. Transistors tend to emphasize the third harmonic whilst the valves emphasize the second harmonic. Now you know why this has such a strong influence on sound.


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