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We know the harmonic series increases, f*1, f*2, f*3 etc... where f = fundamental. We also know there is a drop in amplitude roughly corresponding to 1/h, where h = harmonic.

So if 440 is our fundamental, 880 is our second harmonic. Does this mean the amplitude of the second harmonic =440 (880*1/2), or =220 (440*1/2)?

As the human ear can hear from 20hz > 20,000hz, if the former is true, it would mean in this instance we can hear up to the 46th harmonic (440*46 = 20,240), however if the latter is true, it suggests we can hear only up to the 22nd harmonic (440*1/22 = 20).

Which is correct here?

ItHertz
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    The drop in amplitude must be greater than that, because the sum over all positive integer h of 1/h diverges. – Rosie F Jan 18 '20 at 08:26
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    what do you mean by "drop in amplitude"? That the ear loses sensitivity? The harmonics are independent and can be of any relative amplitude - that's what determines the shape of the overall waveform. – danmcb Jan 18 '20 at 10:45
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    @RosieF not relevant here, because the harmonics don't have all the same phase. Effectively you get an alternating harmonic series, and that does converge. (But anyway it's moot, because no overtone series is actually infinite – it only goes until something like 20 or perhaps 100.) – leftaroundabout Jan 18 '20 at 19:13
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    "roughly corresponding to 1/h": but the actual relative strength of each harmonic correlates with the tone you are analyzing. The relative strength of higher harmonics is wildly different for a flute, an oboe, and a clarinet. The last of these has no (or negligible) even harmonics above the fundamental. So whatever use you can make of that "rough" relationship between harmonic and relative amplitude, do keep in mind that it is limited because the relationship is indeed (very) rough. – phoog Jan 18 '20 at 19:27
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    You can never hear above the threshold of hearing. The harmonics sequence does not allow that to happen. –  Jan 18 '20 at 23:28
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    'We also know there is a drop in amplitude roughly corresponding to 1/h, where h = harmonic': no we don't. Where did you read that? Some instruments produce more harmonic than fundamental. – user207421 Jan 19 '20 at 08:52
  • I'm not sure that precise drop in amplitude is true in all real-world cases. I know for certain musical instruments, like violin or guitar, the lowest notes often have a fundamental with a lower amplitude than some of the harmonics. This is due to how the resonate cavity more efficiently transmits the higher frequencies, but not the lower frequencies. – Cerin Jun 01 '21 at 02:55

2 Answers2

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We also know there is a drop in amplitude roughly corresponding to 1/h, where h = harmonic.

Even qualified by the word 'roughly', I'm not sure how useful that is as rule of thumb. Some waveforms (e.g. a sawtooth wave) have that property, but many waveforms that can be produced both by real and electronic instruments have different relative strengths of their harmonics. Also, the relative strengths typically vary over time, according to playing technique, and so on.

So if 440 is our fundamental, 880 is our second harmonic. Does this mean the amplitude of the second harmonic =440 (880*1/2), or =220 (440*1/2)?

I think you are mixing up amplitude with frequency. If Your first Harmonic has a frequency of 440, the second harmonic has a frequency (not amplitude) of 880.

If assume that we have a sawtooth wave, so the relative amplitudes of harmonics are 1/harmonic number, and we assume that the amplitude of a fundamental at 440Hz is 1 (in some arbitrary unit), then we'd have

amplitude of 2nd harmonic = 1 * (1/2) = 0.5
amplitude of 3rd harmonic = 1 * (1/3) = 0.333...
amplitude of 4th harmonic = 1 * (1/4) = 0.25

and so on.

As the human ear can hear from 20hz > 20,000hz, if the former is true, it would mean in this instance we can hear up to the 46th harmonic (440*46 = 20,240), however if the latter is true, it suggests we can hear only up to the 22nd harmonic (440*1/22 = 20).

Again, we have to be careful not to mix up frequency with amplitude.

A simple way to calculate the highest harmonic you can hear is

20 000 / (fundamental frequency of note)

If your note is at 440 Hz, then 20 000 / 440 = 45.45.... so simplistically speaking, it's true that you can in theory hear up to the 45th or 46th harmonic for a note at 440Hz. For lower notes, you might be able to hear higher harmonics. But remember that all this depends on how loud the harmonics are. The human ear is less sensitive at very high frequencies, so if the amplitude of the harmonic were only 1/46th of that of the fundamental, it might be hard to hear.

Нет войне
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    Thanks very much for the detailed reply. You're right, I did mix up frequency and amplitude. I am only looking for a rough guide here, an approximate theoretical limit to how many harmonics we can we hear. Being just outside our range, and only 2% (1/46) amplitude of the fundamental, 46th harmonic seems like a good candidate for that. – ItHertz Jan 18 '20 at 10:06
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    @ltHertz but remember it totally depends on the fundamental frequency. for a note at 50Hz, our simple calculation would tell us that you'd be able to hear up to the 400th Harmonic. – Нет войне Jan 18 '20 at 10:10
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Your statement about the "drop" in harmonic amplitude is simply not true. It may be true for some instrument or long time reverb in a room but not true in general. The amount of each harmonic produced by an instrument is due to the attack, initial conditions of whatever drives the instrument. It is entirely possible to attack an instrument in such a way that the second harmonic is louder than the fundamental. In general the effect of attack on the harmonic sequence is not frequency dependent or harmonic dependent. If that seems to be the case it is a coincidence. As a simple example consider a guitar string plucked in the center. By plucked I mean specifically that the string is pulled away from equilibrium and released from rest. For this attack ALL odd harmonics will be absent, or amplitude = 0. Only even harmonics will be present.

Your question about hearing beyond 20,000Hz is a red herring. The threshold of hearing is what it is. Just because a frequency might be present in a wave form does not mean you will hear it ever. It will be "invisible" to you. A dog might hear it but you won't. Conversely, just because you cannot hear it does not mean it is not there. The physics of the source determines what harmonics might be in the wave form emitted. They propagate to your ear and get detected. The threshold of hearing is a cut off beyond which your ear+brain system will not respond. There is no contradiction in this.