The Harmonic Series

Originally, I intended to make an interactive Core Learning for this topic but Josh Comeau has already done it very nicely.

In short, a harmonic series is a sequence of frequencies where each frequency is an integer multiple of the lowest frequency in the series, called the "fundamental frequency".  So if you have a fundamental frequency of 440Hz, its harmonics are 880Hz, 1760Hz, 3520Hz, ... and so on.  In some musical education material, harmonics are also called "overtones".

Tune & Timbre

Instruments will produce sounds which include the fundamental frequency in a addition to the harmonics and we perceive the note being played to be that of the fundamental.  The amplitude of the harmonics are almost always lower than the fundamental frequency.  However, it is the amplitude characteristics of the harmonics that provide what is called "timbre"; these harmonic amplitudes are what allow us to differentiate one instrument – or person's voice – from another.  For those of you with some background ion signal processing, Human brains are effectively doing a Fourier transform of incoming sounds to characterize them!

Psychoacoustics

Humans perceive equally spaced frequencies a smaller intervals the higher the frequencies are.  That is, we will perceive a difference of 100Hz as "larger" from 400Hz to 500Hz than we will from 13000Hz to 13100Hz.  We perceive changes in frequency on a logarithmic scale; there have been attempts to model this (mel scale, Bark scale) but I have not found anything that is widely accepted.

Manipulation

There are a few ways to manipulate the harmonic characteristics of a sound.  Since my DAW is Logic Pro, I'll be linking to reference material for it but every DAW I have glanced at seems to include similar functionality.

Equalization

The simplest form of harmonic manipulation is to use an equalizer (EQ) to boost the frequencies above the fundamental.  This does not, however, add or remove harmonics.

Distortion

Distortion is achieved primarily by simply increasing the gain of an audio signal such that it exceeds the upper amplitude threshold of the device producing the sound.  The result is that the audio signal is "clipped" at the amplitude ceiling.  This has the effect of introducing new frequencies in the resulting sounds but these can be both harmonic and inharmonic.  For this reason, the resulting sound can seem harsher.  Further core learnings could be written about the effects of hard clipping and soft clipping; the names of the circuits that perform these are referred to in some DAWs (e.g. clean, valve).  Distortion overview for Logic Pro.

Saturation

A less aggressive form of distortion, saturation seeks to emulate the characteristics of analog equipment whose circuitry result in the addition or amplification of harmonics.  Saturation is often used for more controlled harmonic manipulation and is generally more musical in nature.  ChromaGlow saturation documentation for Logic Pro.

Excitement

Excitement add or amplify harmonics in specific frequency bands.  The result is, seemingly, a more controlled approach to adding distortion and help to increase the higher frequency components of a sound that may be weak in them.  Exciter documentation for Logic Pro.