Musical Synthesis 101
1.4 Harmonics and Sine waves
Jean-Baptiste Joseph Fourier, born in 1768, was a French mathematician renowned for introducing the Fourier Series. This series represents a periodic function as an infinite sum of sines and cosines. Interestingly enough, any periodic function can be broken down into a Fourier Series. Therefore, any waveform can be represented with a set of harmonics. This ultimately means that every sound you here is a infinite composition of the most basic wave type; the sine wave. While it is a bit of an overlook to state that all sounds are just sine waves, it’s not technically incorrect.
Harmonics are useful for more than just determining the shape out the sound. They are what help us audibly differentiate sounds from one another. For example while a piano and a guitar can both play the same note, one sounds like a piano, and one sounds like a guitar. This is because the harmonic composition of each sound is unique to the instrument that made it. This is also how we can tell the difference between peoples voices.
Each harmonic is part of a set of harmonics called the harmonic series. The harmonic series is an infinite series defined by the reciprocals of the natural numbers. This means that each harmonic completes one more full cycle than the last.