[jsenn]

> It will have overtones that are integer multiples of the fundamental that give it its characteristic sound.

What I’m wondering is why would the overtones go in integer multiples (I.e. be harmonic) for a fan? A flute and a saxophone have harmonic(ish) overtones because of the physics of a vibrating column of air

[amluto]

This is just math, not physics. Suppose you have a thing vibrating in a periodic manner. You might imagine that it will vibrate the air so that the sound pressure is some periodic function of time. Fourier transform that function to get a spectrum, and it will have discrete peaks at the fundamental frequency (1 / period) and at integer multiples of that frequency. You don’t even need to decompose into sine and cosine functions for this to work — all that’s really going on is that you have f(t) = f(t + period), and you’re turning f into the sum of a bunch of other functions g_1, g_2, etc, all of which have the same property that g_i(t) = g_i(t + period). Of course, if a function g has the property that, for all t, g(t) = g(t + period/n) for any integer n, then you can iterate that property n times and you’ll also have g(t) = g(t + period). And these functions with the fundamental period, half the fundamental period, one third the fundamental period, etc, are the fundamental tone and its overtones. You could decompose into square waves or just about anything else and you would get the same result.

(In any discussion of Fourier transforms complete with equations, you’ll usually see a bunch of factors of 2π because the frequencies are angular frequencies. This is done for mathematical convenience and has no effect on any of this.)

[DoctorOetker]

Your question displays sufficient knowledgeability of the phenomenon of strictly integer harmonics, that you can basically disregard the replies explaining how they could arise, whereas your question is more about given that it shouldn't arise inevitably, how do we know they would be harmonic given the case of a fan.

Some of the replies pontificate and assume sounds are periodic, and hence their harmonics must have been perfectly integral, which is of course totally bonkers.

Yes, some instruments are harmonic (i.e. integral harmonics down to ~ ppm frequency ratio errors) like violins, but only because those are bowed strings, resulting in phase locking.

Plucked strings are much further from integral harmonics, due to dispersion: yes standing waves for a frequency-independent wavespeed c on the string would give perfectly harmonic partials. Real strings show dispersion (a frequency dependent wavespeed) resulting in inharmonic partials.

Nothing indicates fan noise to be strongly harmonic. Their composite sound may have structured and repeatable (in)harmonic components in many ways, harmonics would be easiest to explain. The part that sounds "white" would presumably be hard to characterize and cancel.

[munificent]

It's the nature of resonance and vibration.

If the fan has any recognizable pitch at all, it's because something periodic is happening. If it's loud enough to be annoying, there's probably some resonance going on to amplify it.

For example, maybe the motor spins at 120 Hz, and it's slight asymmetry shakes the chassis of the fan. That shaking will send waves across the body of the fan. Any of those waves whose wavelength is not an integer multiple of the size of the body will bounce around and end up destructively cancelling out. But the wavelengths that are at are close to integer multiples of the resonating frequency of the body will reinforce themselves as the bounce back and forth across the chassis and get amplified.

If you do an image search for "string overtones", you can get a picture of what I mean. Random physical objects aren't all strings, but many of them have at least a little plasticity and rigidity such that they can vibrate and resonate. When they do, the result will be harmonics at the object's fundamental frequency and integer multiples.

Other frequencies occur too. If you strike a bell, for example, that impulse will produce waves at basically all frequencies. It's just that the ones that don't resonate with the bell's fundamental will cancel themselves out and fade out nearly instantly (that's the clanky part of the very beginning of a bell sound). The multiples of the resonance frequency will ring out (the bell-like peal that decays slowly).

[ejoso]

So are you saying the only difference between a woodwind instrument and a fan is the column? What about a stringed instrument then?

Every sound found in nature contains multiple frequency components. When these align as integer multiples of the fundamental, they are harmonics; when they do not, they are inharmonic partials. Only a pure sine wave lacks them, and such signals don’t occur naturally.

[jsenn]

A string fixed at both ends produces harmonic sounds because of its particular structure. In order to have a non-integer overtone the ends would have to move up and down, which by construction they can't. Similarly for wind instruments: the air stops at either end and is reflected back, and a non-integer overtone would require changing the length of the tube (or sticking holes in it to allow the pressure to go to zero at the hole, effectively creating an artificial "end" of the tube).

By contrast, a freely vibrating bar (not fixed at the ends) does not have harmonic overtones. To make the bars of a xylophone, marimba, or vibraphone sound nice, you have to cut out a little "scoop" shape from the bottom of the bar to force it to vibrate such that its overtones match up with integer multiples of the fundamental frequency of the bar.

As you say, most sounds in nature do not have a harmonic spectrum, so if a fan did I would find that surprising and interesting.

[ssivark]

> why would the overtones go in integer multiples (I.e. be harmonic) for a fan?

The fan noise is from its own vibrations -- presumably driven by the motor. These vibrations will correspond to natural vibrating modes on the body of the vibrating object -- which could be the motor, or the chassis, or even possibly the fan blades. Whatever the shape, the natural modes will be naturally quantized into "harmonics". Those vibrating modes could have more nuanced spatial forms (eg. Bessel functions) but their temporal pattern would likely be sinusoid.