The connection between maths and music

Matt Martin, Manager

While Archimedes was having his eureka moment by splashing some water out of his bath, Pythagoras realised that hammers of different sizes striking the anvil produced different ringing notes, and that there was a correlation between the size and weight of the hammer and the note that was produced, with hammers of twice the size creating the same note, just an octave higher. This story, however exciting, is apparently not true at all, and is not scientifically correct either, but it IS true that a string that is half the size of another will play the same note as a string double it’s length, just an octave higher. So old Pythag wasn’t a million miles away.

But numbers and music are about so much more than hitting things to make a note. The other major component of music is rhythm, and whether it’s a drummer or a bassist you’ll hear them talk of playing in 4/4 or 3/4 which not only tells us how many times a drummer should hit the thing so we can all make the noise at the same time, but also has a fundamental impact on how the music feels. You only need to look at one of my favourite pieces of music – Kashmir by Led Zeppelin to feel how much these numbers actually matter while John Bonham plays the traditional 4 beats per bar 4/4 rhythm while his band mates are playing 3 notes to every bar in a 3/4 rhythm which basically creates, well, just listen to the song!

So, we’ve talked about the notes themselves as hammers ring against anvils and the rhythm to bring it all together, but that’s not all. Some musicians go their whole lives playing an instrument without ever delving into the mysterious world of musical theory. A musician can know how it works, but knowing musical theory is when you learn WHY it works, and this is where the maths really kicks in. There are many forms of music around the world but focusing on what is often termed “western music” we can learn much from looking at, or at least envisaging a piano. If you know nothing of how to play music then looking at the keys of a piano is pretty meaningless. But there are in fact a total of 12 notes, after which they repeat themselves just higher or lower, these higher or lower notes are called octaves but let’s just focus on 12 of them in isolation for now. What brings order to the chaos is a scale, where 7 notes out of the 12 available to you are selected and played, pretty much ignoring the others. Most songs are in a particular scale, choosing 7 notes that sound pretty good together and completely ignoring the other 5. So, there are 12 notes in total, 7 notes in a scale and within that scale of 7 notes are the chords that you will actually play in your song. The likes of C Major, G Minor, and so on. Chords are generally made up of 3 notes that are next to each other within the 7-note scale, starting with each of the 7 notes and playing the next two notes above to give you a chord, and once you know where those notes are you can play pretty much anything, and it’ll sound pretty good. Which is the foundation of most manufactured pop music that we hear today.

But why 12 notes? Why 7 notes in a scale? And why 3 notes in a chord? The answer really comes down to wavelengths. This is where my GCSE Science comes into its own. When a note is played your ear is picking up a soundwave, a soundwave that has a frequency. That frequency is what determines what that note actually is, a high-pitched violin, or a deep and booming bass note. But when two notes play together and interfere with each other, that’s where the maths kicks in. If two notes are the same and have the same frequency they sound rather nice together, but if you play two notes that are slightly different, such as two keys that are next to each other on the piano, the two frequencies will completely ruin each other as the two wave lengths get tangled up, and we as humans don’t like that, we want things to be in harmony. But what happens when a wave length is half the size of another, so two can fit inside another, or 3, or 4? That’s where harmony comes from and forms the basis of the chords that sound nice to our ear and make us feel good. This is called resonance and dissonance. If a musical composer wants you to feel uncomfortable while watching a scary film you can bet they are playing a bunch of notes that are close together and interfering with each other which will stress you out on a fundamental level, whereas the soundtrack to Love Actually will be filled with sound waves that complement each other, unless they want you to feel sad, of course.

So, as you can see, music is ultimately about patterns, and those patterns can almost always be expressed in numbers, which is why musicians are often number people, and vice versa. So, what does this have to do with accountancy? Absolutely nothing – you’re welcome!