The relationship between musical theory and mathematics is no secret. While there is no defined axiomatic system of music theory in modern mathematics (unlike group theory for example) the basis of musical sound can be described mathematically, and displays many familiar numerical properties. Starting with Pythagoras over 2,500 years ago, this has been a popular area of study that has led to some beautiful discoveries: from the rich complexities of Bach Fugues, to translating the Fibonacci’s sequence into a tune, to why The Beatles are so catchy. To understand these concepts it’s vital to have an appreciation of the fundamentals of the science of music, something that’s not always included in an academic paper. And so, to introduce you to the world of intervals and octaves, here’s a reminder of some of the basics of sound!
It all begins with physics. Plucking the first string on your standard (in tune) guitar will cause it to vibrate 330 times each second (this is quite difficult to count, so just take our word for it). The surrounding air will then vibrate at the same frequency, causing a domino effect of further vibrations to ripple out from the string – we call this a sound wave. Once it reaches the ear, the brain interprets this wave as sound, with the frequency of the wave translating to the pitch that you hear. For most of us these 330 vibrations per second will just sound relatively high, but if you’re blessed with a musical brain, you will recognise it as the note E.
Whether musical or not, it’s common knowledge that playing the same string again but with a finger pressed between two of the frets will produce a different, higher, note. By trapping the string, we are effectively shortening it and causing it to vibrate more frequently. When this new sound wave reaches the ear, the brain acknowledges this difference in frequency, which it will translate to us as hearing a higher pitch.
Mathematics comes into play when we start to define the frequencies of different musical notes. Julie Andrews famously sings a scale in ‘The Sound of Music’, “Do-Re-Mi-Fa-So-La-Te-Do” however, in the UK, it is more common practise to use letters, “C, D, E, F, G, A, B, C” for example. By definition, the ratio of the frequencies of the two C notes must be 2:1, a property that we label as being one ‘octave’ apart. The frequency of every single musical note is also defined by such a ratio, normally with respective to the note 𝐴4 whose frequency is fixed somewhere close to 440 hertz.
What’s interesting is that some of these ratios produce sounds that our brains like and some we cannot stand. What’s more fascinating is how we can predict this. The explanation is beautiful – and deserving of its own blog post at the very least… you’d better come back next week!