Harmonics in sound

Frequency Facts

As you know, all sound can be represented by a wave. Waves have a property called ‘Frequency’ which determines how high or low the pitch of the sound is. Generally, the human ear can hear sound that ranges from 20Hz (Hertz) all the way up to 20kHz (20,000Hz).

Most sounds you hear are a lot more complicated than a simple wave and are made up of a multitude of simple waves at many different frequencies. You can see this in the graph below:

Picture1

The graph shows a snippet of a song being played – the purple area being all the frequencies that make up the sound. Along the x axis is frequency, lower numbers are lower, or more bass-y, frequencies, and higher numbers are higher, or more treble-y, frequencies. The y axis is how loud the frequencies are. The scale is measured in negative decibels (-dB), so -90dB is very quiet (practically inaudible), and -11dB is rather loud. The loudest you can go on this scale is 0dB.

You may notice that the scale of the x axis (Hz) is not even. The numbers are ‘closer together’ than those on the right – a jump from 30 to 100, and 3000, and 10,000 are represented as the same size. This is because of how we hear sound. A jump from 100 to 200 sounds like the same sized jump as 200 to 400. It’s worth knowing that doubling the frequency of a sound is the same sized jump as a musical octave; like going from the note A, at 440Hz, all the way up through all 12 notes until you get to the next A, at 880Hz.

Harmonics are harmless

Okay, so now we understand roughly how frequency works. The next part is understanding harmonics. Harmonics are more easily understood if we only look at very simple sounds – like a sine wave for example:

Picture2

The peak above is a sine wave with a frequency of 400Hz. This peak is the fundamental – the main, and lowest, part of the sound. We perceive the fundamental frequency, or fundamental harmonic, as a sound’s pitch. The fundamental is often also the loudest frequency in the sound.

Let’s introduce another harmonic, the second harmonic. To work out what frequency 2nd harmonic of a sound should be at, we take the fundamental, then multiply it by 2. This is the same as adding another sine wave with half the wavelength of the fundamental. It’s called the 2nd harmonic because you can fit two full wavelengths into one wavelength of the fundamental:

Picture3

Here’s what the 2nd harmonic looks like when added to our original 400Hz fundamental:

Picture4

We can work out the frequency of any harmonic with some simple maths:

Fundamental Frequency x n = nth harmonic

So for example, the 7th harmonic of 400Hz would be:

400Hz (fundamental) x 7 = 2800Hz (7th harmonic)

Here’s what our graph looks like when we add all the whole-number harmonics up to the 7th:

Picture5

How this affects timbre

If you take a piano and a guitar and play the exact same note on each instrument, the sound they produce will be different. The fundamental frequency of each sound will be the same, however the harmonics that resonate above will be completely different. In the world of music, this is known as a sound’s timbre. You can see this quite clearly in the two pictures below:

Piano playing the note C3 (261.6Hz)

Picture6

Guitar playing the note C3 (261.6Hz)

Picture7

Both the sounds have a fundamental of 261.6Hz (middle C on a piano keyboard). The strange parts, like the low frequencies (less that 100Hz) and the peak just below the fundamental on the guitar are not worth paying attention to because they are produced by the mechanical action of pressing a piano key or plucking a guitar string.

A piano tends to sound softer than a guitar string, which you can see in the pictures above. There are less high pitched frequencies (or harmonics) in a piano which gives the sound a softer or less harsh quality. The guitar string is a far more metallic sound, with lots of high frequency content. In short, the sonic identity of an instrument depends on the harmonics it produces when played.

Taking it further

The harmonics we’ve been dealing with are simple examples and in reality, things get far, far more complex. Most sounds have harmonics that extend even higher than we can hear, and they may not be at nice even intervals like our 2nd, 3rd, 4th, etc. It’s possible to have a harmonic equal to ‘0.7th’ or some other non-whole number. It’s also fairly common for sounds to have no harmonic content at all! For example, what pitch does a wave crashing have? Or a rustling of leaves in the wind?

If you have ever had any involvement with audio synthesis, you will find understanding harmonics incredibly useful. Producing sounds via Additive synthesis is quite similar to what we did earlier when adding more and more harmonics to a fundamental frequency. You start with a fundamental, then decide on a method for adding frequencies. You could decide to add every whole-number harmonic and synthesise a warm sounding saw-tooth wave, or only the odd numbered harmonics and get a spooky, hollow square wave. There are plenty more advanced additive synthesis techniques that make a huge range of sound qualities possible – but writing about all of that will mean a blog post ten times the length of this one at least!

In summary

The basics of harmonics helps us pull apart a sound into it individual building blocks. It makes them easier to understand and visualise. Harmonics are everywhere, and they give every single sound it’s unique character. If you ever get the chance, look at a recording of your own voice on a  Spectral Analyser – like the graphs above. Nobody else produces the same series of harmonics as you. Your harmonics are your fingerprint!

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