Overtones, Harmonics and Additive Synthesis

For some upcoming projects I wanted to review some sound theory basics, and post some useful learning resources. I like to post links to videos (and tutorials), and write short summaries for future reference.

In this post, a quick intro to overtones, harmonics and additive synthesis, using a video lovingly prepared by Synth School:

And here are the notes:

Oscilloscope vs. Frequency Analysis view of a sine wave
One of the first things in the video is a comparison of the representation of a sine wave in an oscilloscope, with a sine wave in a frequency analysis (FFT) view:

A sine wave represented by an oscilloscope

Sine - Oscilloscope view

A sine wave represented by frequency analysis

Sine - FFT view

This is a useful way to get to know the two views.

They both represent a number of samples from a waveform, say 512 or 1024 samples. The oscilloscope view is the result of taking those samples and displaying them fairly literally, one after the other, and plotting a line through them. The FFT view is the result of applying a Fourier transform which is designed to produce information about whichever frequencie are present across those samples.

FFT analysis can be thought of as the opposite of synthesis. In synthesis you might take a series of tones at different frequencies and combine them to produce a waveform which contains all of those frequencies. Fourier analysis in this context is designed to decompose that waveform and describe the individual frequencies that were originally combined to produce the waveform.

The oscilloscope can be thought of as a close-up version of the representation of a waveform you get in any DAW. Amplitude is plotted in the Y axis, and time is plotted in the X axis. The amplitude displayed here can be thought of as the position of a the surface of a loudspeaker, moving back and forth over a central 'resting position'.

In the FFT view, amplitude is plotted in the Y axis, and frequency is plotted in the X axis. By convention, lower-pitched frequencies are represented on the left, and higher-pitched frequencies are represented on the right.

The 'amplitude' in the FFT view is different - but related - to the 'amplitude' in the oscilloscope view. They are related because they both refer to loudness in the end, but where the oscilloscope describes amplitude in a way closer to the underlying physics, in a range from -1 to 1, the FFT describes amplitude in more of a human-understandable range (one we are familiar with from TV and stereo volume controls) in a range from 0 - 1.

Fundamental and overtones
A sine wave is called the 'pure' waveform because it is the only waveform type to contain only a single frequency. This frequency is called the fundamental frequency.

Other wave types always have additional frequencies on top of the fundamental. The lowest frequency of the sound is the basics on which the sound is built, and is called the fundamental frequency. The additional frequencies are called overtones.

Saw - Oscilloscope view

Saw - FFT view

Overtones are what constructs the sound's timbre. Overtones can be harmonic, or non-harmonic. Harmonic overtones support the fundamental frequency and keep it's tonality intact. Non-harmonic overtones result in noise, or sounds with ambiguous pitch.

Additive synthesis
A saw wave can be constructed by combining multiple sine waves, as demonstrated in the video. For a 'classic' saw tone, the amplitude of each added harmonic should be divided by it's harmonic count, i.e. the fundamental is divided by 1, the second harmonic is divided by 2, the third by 3 and so on.

Fundamental + 2nd harmonic

Fundamental + 2nd and 3rd harmonics

So a saw tone can be thought of as the combination of infinite harmonics, each divided by the harmonic count. Though in practice doing so would require infinite CPU power!

Square wave
A square wave is also constructed by adding harmonics, but this time only odd-numbered harmonics, i.e. by skipping the 2nd, 4th, 6th (etc) harmonics, and including those in between.

This is illustrated quite nicely in this old-school training video:

That's a nice intro to overtones, harmonics and additive synthesis. I'll be posting some more sound theory stuff soon.