Maths and music

How is an instrument tuned?

To tune any two instruments, the sound waves from both of them must be at the same frequency. If the frequencies differ very slightly, the two sound waves interfere, making another sound wave that undulates in volume or 'beats'.

The beat frequency is the number of volume undulations heard per second. It is found by subtracting the lower frequency from the higher one. When two instruments are in tune, the beat frequency should be zero. This is an extremely useful tool for tuning instruments accurately by ear.

Most people can usually tell whether an instrument is in tune or not without knowing the underlying science.

Three types of instrument

Stringed instruments 
The pitch of a stringed instrument depends on the tension and the length of the string. In most stringed instruments the pitch gets higher when the player moves their hand closer to the bottom of the string making the vibrating area shorter. However, Mike's double bass depended on changing the tension of the string to obtain each note.

In many stringed instruments, the strings themselves only produce a small fraction of the sound that is heard. The rest is due to resonance from the body of the instrument vibrating in sympathy with the strings. Mike's double base had a huge box and a long string which gave it a very low pitch.