Sound theory: Harmonic content of a waveform

What we've said up to now with regards to sinusoids is a fundamental truth upon which we will base the rest of our sound reality. An aspect which contributes a lot to the characteristics of sound is its harmonics. To illustrate this concept we will use a practical example, that way we can start to leave the purely theoretical examples behind and begin to see the practical meaning of what we're saying. Let's consider what happens when the 5th string of a guitar is plucked by a guitarist. Clearly the answer is that the A note plays, but what is it that actually happens?

The string has begun to oscillate at a frequency of 440Hz. So then why is it that it doesn't emit a sound that is that of a simple sinusoid but with the sound of a guitar? The reason involves different reasons that 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 called the fundamental harmonic, and together with this fundamental harmonic are generated the other harmonics, in other words all the whole multiples of this frequency with a progressively decreasing amplitude. With the A note, the following sinusoids are generated:

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

Harmonic oscilations of a vibrating string

The first harmonic is called "fundamental" and that characterizes the note that we effectively perceive, is the harmonic with the greatest amplitude. The second harmonic has double the frequency of the fundamental, this means that the string is vibrating as shown in the figure, overlapping this vibration with the fundamental harmonic.

If you are reading this course you probably have some basic musical notions and thus 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 harmonic and adds warmth to the sound. The third harmonic is no longer an A and therefore contributes to enrichening the sound.

In the figure you can see how the following harmonics are generated and the amplitude of these diminishes as the harmonic's frequency increases. In other words, if a guitar string is plucked, the harmonics that relevantly contribute to the generated sound are about 10. The amplitude of the following harmonics compared to the amplitude of the fundamental harmonic become irrelevant. 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 in placing microphones for example: if we wanted to have a high frequency sound to come out of a drum's snare drum we'd direct the microphone towards the edge, whilst if we wanted to have a sound with more low frequencies we'd direct the micorphone towards its centre.

Guitar amplifiers can be both valve and transistor and you'll find strong fans of both the first and the second type because of the different sounds they produce. Transistors tend to emphasize the third harmonic whilst the valve emphasize the second. Now you know why this has such a strong influence on sound.