Harmonic and Inharmonic overtones

Question

What is the difference between harmonic and inharmonic overtones? What are their uses in sound design ?

Answer 1

The frequencies of harmonic overtones are an integer multiple of the fundamental frequency. If the fundamental is at f0 then these overtones will be at 2f0, 3f0 and so one. A periodic wave (which shape is repeating in time), only has harmonic overtones.

Inharmonic ones are not integer multiple of the fundamental, for instance sqrt(2)*f0. They are generated for instance in FM synthesis and useful to generate bell sounds for example.

Answer 2

First we need to understand the motivation behind harmonicity: Many things producing pitched sounds work by having a medium (a string, an air column, ...) that have certain limitations on how a wave can be shaped. In a string for example the end points are fixed, so you can only have wave forms where the ends do not move (which also leads to standing waves). If you then try to find simple waveforms that fit this criterium you’ll find this is the case when you have a single bend and a node at each side (wave length 2L), when you have two bends and a node at each side and in the middle (wl L), when you have three bends and two nodes in the middle (wl 2L/3) and so on.

You see that the possible wavelengths are integral division of 2L (where L is the length of the string). Now, under some strong theoretical assumptions this would translate to frequency as integral multiples of a base frequency.

This is the concept we call harmonic. Now if we take as base waveform a (potentially shifted) sine mathematics can prove that any period continuous signal can be written as an (potentially infinite) sum of such sine waves of such harmonic frequencies (with the base frequency being a single sine wave in the whole period). This is what we call the Fourier decomposition of the signal.

Note that the human ear does something similar to a Fourier decomposition, just that it does not use exact sines as base functions. So the human brain has learned this concept of harmonicity: Integral multiples of frequencies often occur together, so the brain has learned to recognize them as "belonging together" or in other words really consonant.

Now the problem is that while this harmonic property is something that happens with a lot of things approximately, it does often not work exactly. For example the wavelength of a wave in a string cannot be directly translated to the frequency, but it depends on other things (for example amplitude will affect tension, which means the pitch goes down as the note gets softer), and this does not affect each overtone the same way.

So suddenly we get a composition of overtones that are not exact integral multiples of a base frequency, but that deviate a bit from that. These are then inharmonic overtones, and a measure of how much they deviated is the inharmonicity (in fact more generally any overtone that is not a harmonic overtone, that is, an integral multiple of the base frequency would be inharmonic).

The inharmonicity depends on lots of factors and it strongly affects how "exciting" we find a tone. So use a harmonic signal to get a very clean sound, and increase the inharmonicity to make it more exciting and intense.

If we have inharmonic overtones the signal will not be periodic with the same period of the base frequency. If we have a finite number of inharmonic overtones with rational factors we have a period signal with a much higher period length. This can have an effect on nonlinear effects such as distortion.