How do guitar strings make different notes?

In The Murderous Maths of Everything we see how you can cut organ pipes to different lengths and so make different musical notes. For example:

Pythagoras is supposed to be the first person who worked out that neat fractions make music. One day he was passing a blacksmiths and heard some bits of metal being hammered and they happened to strike a nice chord. He loved simple numbers and fractions so when he realised that one bit of metal was exactly 2/3 the length of the other he was utterly delighted. He made a very simple sort of guitar so he could twang the strings and change their length. (The strings had to be the same thickness and they had to be equally tight for his experiments to work.)

Modern guitars work in exactly the same way. If you have a guitar and count up from the end of the neck to the 12th fret, you should find that’s exactly halfway along the string. The 7th fret should be 2/3, the 5th fret should be 3/4 and the second fret should be 8/9.

You might also be interested in: Why do some musical notes sound AWFUL together?

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