Perhaps Pythagoras’ most valuable and far-reaching discovery was the link between maths and music. The story goes that one day he heard a blacksmith hitting two anvils, which emitted two distinctly different notes. Intrigued, Pythagoras started to explore noises and the things that make them. By hitting. Blowing and plucking different objects, he eventually discovered that all the sound they make are governed by the same mathematical rules (see plate section).

Pick up a guitar and pluck a string. Now find the halfway mark on the string, press the string to the neck of the guitar at that point, and pluck again. The second note is the same note as the first, but an octave higher.

Let’s assume the string is tuned to C: open, it emits a low C and, at the halfway point, C one octave higher. Now, from the halfway point, add another third of the string. The new note is G. Add another half and you get an F, and so on. Those are natural spacings, but then it gets a little more complex. hHere is a full octave from C down to lower C. To C (half the string) add 1/1 5 for B, 1/5 for G,1/2 for F,3/5 for E,1/9 for D and 1/1, or the same distance again, for the lower C.

Pythagoras discovered that the same rules govern the size of bells, and the volume of air above a liquid in a row of identical flasks. Just a few years BT (before television), when I was a lad, people often played the ‘bottle phone’  — a xylophone made of bottles or glasses on the kitchen table. Try it, by filling glasses with water measured out using the factions mentioned above.

For all musical instruments- whether strung, blown or bashed-simple maths defines the notes. Pythagoras even suggested that notes rise by an octave if you square the tension applied to them. So a String attached to a 1Kg weight will produce a particular note, but to raise the note by an octave would take not 2Kg but 4Kg of weight. The maths of the modern music we know today can be very complex, with sharps and flats and subtle tone variances. But because computers can handle complex number crunching with ease, synthesizers can now produce any sound, pitch or timber electronically, simply by applying the maths.

For Pythagoras it was much more straightforward. The sound emitted by bells or chiming glasses depends on the volume of the reverberating material; with stringed instruments it is the tension and length of the vibrating string that counts; with wind instrument (such as a simple reed pipe, or the opening of a bottle when you blow over it), it is the volume of the same simple mathematical proportions.

Pythagoras even suggested that musical sounds might govern the stars and planets, with each celestial body emitting a particular note- a concept known as the Music of the Spheres. Some 2,000 years later, a chap called Johan Bode discovered that the distances of the various planets from the Sun do have a mathematical pattern perhaps as beautiful as the maths that produces music.

This is extract from Johnny Ball's Wonders Beyond Numbers published by Bloomsbury Sigma