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Comment by cmer | Hacker News Reader

Let's make things super simple. Let's say you record 4 sine waves at a 192kHz sampling rate: 15kHz, 30kHz, 45kHz and 60kHz. All 4 frequencies will be captured and the 15kHz frequency will sound different to your hear because its harmonics.

If you take this recording and master it for a CD (44.1kHz), you'll effectively get up to ~20kHz (since they're a low pass filter starting at around 16-18kHz). This means that only our first frequency will be captured: 15kHz. It will be exactly the same as if you only recorded 15kHz alone. The harmonics don't modify the fundamental frequency, they just trick the human hear. But when they're gone, they have no effect whatsoever.

Hope this helps!

EDIT: the frequency numbers I used are actually somewhat of a bad example. Harmonics are never exactly double, triple the fundamental. Those would be mostly inaudible. But you get the idea.

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I don't think I understand how it could sound different to my ear. My understanding is that my ear doesn't have the sensory equipment to detect signals above ~20kHz - this is what I was told at university, and a decent trawl of the web suggests this is still true. If there is any sound that is in the range 20Hz-20kHz then why doesn't the microphone pick it up?

Or am I wrong, and the ear is able to detect frequencies above 20kHz?

> Harmonics are never exactly double, triple the fundamental. Those would be mostly inaudible. But you get the idea.

Actually, they are: https://en.wikipedia.org/wiki/Harmonics

The second half of the statement is wrong, but the first half is right. Harmonics in real-world instruments are not usually exact multiples of the fundamental. A simple diffeq model of a rigid oscillator will show you this mathematically.

An extreme example is present on modern pianos, where the high rigidity of the loud, heavy piano strings can cause tuners to stretch the lowest and highest notes as much as a half-semitone so that their harmonics are in tune with the note the next octave down or up. In other words, the first harmonic on the lowest note of a piano can be as much as 1/2 of a note sharp.

And when your oscillator is no longer one-dimensional, most harmonics aren't even close to integer multiples. The harmonics of bells, cymbals and drums are all over the place. That's what gives them their percussive sound. (Edit: some of these modes of vibration aren't harmonics in the linear sense.)

Harmonics in real-world instruments are not usually exact multiples of the fundamental. A simple diffeq model of a rigid oscillator will show you this mathematically.

That is absolutely incorrect, mathematically speaking, harmonics are by definition "integral multiples of the fundamental." (Fundamentals of Acoustics, Kinsler & Frey).

People from a musical, non-signals background tend to use 'harmonics' as a synonym for 'overtones' or 'partial tones', which is where the confusion arises, I suspect.

There's a measure -- inharmonicity[1] -- of how far the actual overtones of a particular instrument differ from their theoretical fundamental multiples.

[I suspect you already know this. This reply is probably for others' benefit]

[1] https://en.wikipedia.org/wiki/Inharmonicity

Then one would be forced to conclude that many instruments have no harmonics at all, which is obviously not what 'harmonic' is referring to in this thread of discussion. Why be pedantic when it's obvious what everyone is talking about?

Anyway, it's not as though mathematical literature requires you to use a term exactly one way. I had a diff eqs textbook that used the word 'harmonic' in exactly the way I used it above when I made reference to diff eqs...

> I had a diff eqs textbook that used the word 'harmonic' in exactly the [incorrect] way I used it above

So you had a textbook with a mistake in it. What book was it?

> And when your oscillator is no longer one-dimensional, most harmonics aren't even close to integer multiples. The harmonics of bells, cymbals and drums are all over the place. That's what gives them their percussive sound.

But those aren't harmonics, they're inharmonic partials.

> The harmonics don't modify the fundamental frequency, they just trick the human hear. But when they're gone, they have no effect whatsoever.

This is the part I really do not understand... either my ear CAN pick up those frequencies, maybe the harmonics are "tickling" the little hairs inside my cochlea and ultimately the frequencies I can actually hear were altered in my perception that way - or I can not hear or sense the harmonics and they physically alter the "original" wave that I end up actually hearing.

Either way, pretty much the exact same thing should happen in a studio microphone. Those all do have frequency limitations and AKG, Royer, Rode, Shure, Sennheiser, Audio Tech, what-have-you pretty much all go up to 15kHz or 20kHz according to specs, if I understand them correctly, but not further than that. If it isn't even recorded, those frequencies I also cannot hear can NOT alter my perception so they HAVE to somehow change the frequencies I can hear and are being recorded... on top of that you are making "room" for frequencies up to, say, 60kHz but I very strongly doubt your mics can go even remotely that high.

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