Any sound, any music, is, before it hits the brain, a mechanical vibration. An instrument vibrates when plucked, blown or struck and those vibrations are transmitted to the ear as waves of varying air pressure. The air between instrument and ear vibrates in sympathy with the instrument, the ear drum picks up the vibrating pressure waves and transmits them to the inner ear.
The miracle is what the brain then does with this. The musician creates vibration; the brain transforms this into music. Essentially, you hear not the instrument, but what is going on in your head in response.
What is in a sound wave that can make the brain respond so beautifully? The Pythagorians were not interested in finding out by analysing vibrations other than in terms of the length of vibrating strings. That's not good enough for discovering what is going on in detail.
The vibrations of a string are too small and quick to see in any detail. The air itself is invisible. Seeing vibrations would be a revealing help.
Any simple vibration can be represented visually by a waveform and this is described by two properties: its frequency, which is how fast the vibration repeats its cycle (the higher the frequency, the higher the pitch); and the amplitude, which is a measure of its energy (the higher the energy, the louder is the sound).
What on earth has trigonometry to do with music? Indeed, what is trigonometry? It is a study of the relationships between the angles and sides of triangles and can be used to understand and visualise the motion of a wave.
The angles and sides of triangles are related to oneanother and are expressed in terms of "cosines", "sines" and "tangents", which, hopefully, you came across at school and then, probably, forgot all about.
However, they reappear again as soon as you look into how waves behave, hence, for example, the term a "sine wave", which is the simplest graphical representation of a pure tone, a wave caused by an object vibrating at a single frequency.
Fourier Analysis is great. Its basis is quite straightforward. Any complex waveform, for example, the sound made by a musical instrument, can be broken down into a series of simple sine waves, each vibrating at a particular frequency and with a particular amplitude. Combine these back again and you reconstruct the original sound.
The technique was developed by French mathematician, Joseph Fourier (1768-1830). Fourier was also credited with discovering the greenhouse effect.
The ability to analyse sound waves into pure tones and reconstruct them leads in the 20th century to the development of synthesised sound and the electronic synthesiser.
Applied science in the twentieth century enabled high quality sound systems which can recreate original sound recordings at any place or time. One element in the chain of sound reproduction involves a combination of electrical, mechanical and acoustical properties and this is the loudspeaker.
The input to a loudspeaker is an electrical signal which varies in a way that follows the outline of the original acoustic wave pattern. This electric current is transformed back into sound by wrapping the wiring around a magnet. An electric current coiled around a magnet in this way will cause it to vibrate in sympathy with the current. The vibrating magnet pushes against the surrounding air, creating pressure waves and, bingo, you have recreated the original sound.
The study of sound and vibration made visible is called "cymatics". Typically, a pattern can be formed by vibrating a metal "Chladni" plate on which a thin layer of particles like sand or salt are allowed to jump and bump to form areas of maximum and minimum displacement. The resulting patterns are aesthetically pleasing and physically complex. Similar evocative forms can be created by vibrating liquids.
A practical application of cymatics is in understanding the acoustic response of violin backs and bellies.