We hear music all the time, but has it ever crossed your mind how it works? Why do pianos and guitars sound different, and some notes work together particularly well with others?

This might strike some of you as being somewhat irrelevant to this blog, but the connections with Maths and Physics are very deep-rooted; the very bases of sound and harmony are founded on Mathematics, and possibly the very essence of the Universe is founded upon Music.

So, the first question that I’ll answer is what actually makes sound? When we hear sound, what we’re actually hearing are the patterns of air particles hitting our eardrum and different speeds and timings. This is caused by an object vibrating, after being struck, plucked, or blown, as seen in the diagram below. For the purposes of this post, I am going to use strings as an example, as they are the easiest to represent diagrammatically.

Credit to http://bit.ly/zewsEg for image.

When the string is pulled back at a, it creates an area of low pressure just below a on this diagram, this is called rarefaction. When the string gets to i though, it has compressed all the air particles below it on the diagram, creating an area of high pressure, which is called compression.

So, a diagram of the position of air particles after a sound has been made would look like this:

Credit to http://bit.ly/zVAnZC for image.

When graphed against time, the pressure due to a sound wave at any point looks like this.

Credit to http://bit.ly/zVAnZC for image.

The diagram above is where we get the idea of “sound waves,” it is these repeating patterns that make up sound. Different notes are made by varying the time difference between the compression and rarefaction (changing the length of the wave), and the volume varies depending on how great the pressure difference is between compression and rarefaction (changing the height of the wave).

This all seems fairly simple so far, but if this was all there was to it, everything would sound the same! A guitar and a piano can play the same note with the same volume, which would create an identical sound wave. But how come the same note played on different instruments sounds different? Why do people’s voices sound different?

This is all because of a mathematical series known as the “Harmonic Series,” which is as follows:

Essentially, this series is basically the sum of 1 divided by the integers, where the nth term is 1+12+13+⋯+1n. Unlike some series of this form (e.g. 1+12+14+⋯), it doesn’t tend towards any particular value. Whilst 1+12+14+⋯+12n  gets closer and closer to 2 the higher n is, and eventually equals 2 exactly when n = ∞, the Harmonic Series doesn’t converge on any number. When n = ∞, Hn = ∞ (where Hn means the nth Harmonic number, or simply the nth term of the series). This is what is known as a divergent series.
This is all very well, but what does this have to do with music?

Well, when a string vibrates, it doesn’t just move back and forth like the first diagram shows. The string is actually vibrating in an infinite amount of ways, all simultaneously! The diagram below shows how:

This looks familiar doesn’t it? It doesn’t stop at 17 though, theoretically the pattern continues indefinitely.

An important thing to note here is that the string is actually doing all of the vibrations show in the diagram at the same time, and the sound you are hearing is the sum of all these frequencies. When a note is played on the guitar, for the open B string, it is the first frequency (at the top of the diagram) that you notice the most: it is that that makes the note a B. This is called the fundamental frequency of the note, or just f, and this is what is meant when the ‘frequency’ or ‘pitch’ of a note is used in everyday language, even though we now know that there are actually infinite different frequencies.

The frequencies of the 12, 13 and other vibrations are known as harmonics, or overtones, so the fundamental frequency is known as the 1st harmonic, the 12 vibration as the 2nd harmonic and so on. These harmonics have pitches themselves. Here are the first few:

However, most of the upper harmonics are much quieter than the fundamental. This shows that not all the harmonics are played at an equal volume, indeed it varies greatly. An expression for the sound that we hear then, would be this:

x0(1f) + x1(2f) + x2(3f) + …

Where f is the fundamental frequency, and xn is a variable.

In other words, we hear all of the overtones (at least all the ones within our hearing range), and they’re all at different volumes.
Certain instruments or playing techniques emphasise different harmonics (by having different values of xn), which creates the different tones of instruments. Having a larger body to a guitar may emphasise the 7th harmonic for example, which gives the whole guitar a different sound.
The values of xn don’t remain constant though. The expression above only describes the xn values at that particular point, when in reality, the values change, and they change at different rates for different harmonics on different instruments. For example, on a guitar, the 2nd harmonic may take a very long time to ‘decay,’ or get quieter, but on a violin it may be one of the first to decay.
So, it is a combination of the volumes of the various harmonics, and the rate of change in this volume that gives an electric guitar that twang, that gives a saxophone that mellow tone, and gives a bass that warm sound.
Pythagoras was one of the first to look into the mathematical properties of sound, and only really discovered the first harmonic: the octave. He realised that when you halve the length of a string (thereby doubling the frequency), the note becomes exactly one octave higher. However, it was not until the 14th century and Nicole Oresme that the entirety of the Harmonic Series was discovered.

Recently, String Theory and related theories have stated that the entirety of the universe, all the sub-atomic particles that make up our world are themselves made up of vibrating strings. Michio Kaku explains this in this video:

If String Theory is correct, then harmony is emanating all throughout our universe: the sub-atomic particles are the notes, the atoms and molecules are the melodies, and our entire universe combines into one glorious piece.

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Theo.

Check out our last two posts:
What are the Subatomic Particles? – What are the most basic things that make up everything we see, hear and know?
What is Time? – It is one of the most debated phenomenon of the universe so, what is time? Is time travel possible? Does time even exist?