Why is saxophone growling produced by modulation of the sound waves?

[sophiecentaur]

But in the end the sound you hear is in fact a sum of sine waves.

Of course but they're not harmonically related, necessarily - is my point.

does your body not form a resonant cavity?

At what resonant frequency? If the Q factor is low then it's not relevant and will not affect the amplitude.

 

[Daniel Petka]

I'm singing while playing, but it's not obvious to me at all why that should do anything. Take a flute for example - it doesn't work with a flute..

Edit: it totally works on a flute. Guess I didn't record a spectrogram when I made that claim

 

[tech99]

Edit: it totally works on a flute. Guess I didn't record a spectrogram when I made that claim

The reed is a relaxation oscillator, so the frequency is pressure sensitive. The second tone apples pressure waves to the reed. So my suggestion is that the second tone frequency modulates the original tone, giving rise to additional side frequencies.

 

[Daniel Petka]

The reed is a relaxation oscillator, so the frequency is pressure sensitive. The second tone apples pressure waves to the reed. So my suggestion is that the second tone frequency modulates the original tone, giving rise to additional side frequencies.

Yes, it's modulation, as I have mentioned in my post and it's caused by the reed's nonlinearity. More pressure inside the mouth means more are comes out, which it turn decreases the pressure and closes the reed. The cycle repeats because the pressure wave gets backreflected at the end of the pipe. The pressure vs air plot shows a curved slope, and when you sing a note you basically move the operating point around which you oscillate. I have yet to figure out how the flutes linearity gets created. Another rabbithole, yay

 

[sophiecentaur]

The reed is a relaxation oscillator, so the frequency is pressure sensitive.

Yes, exactly. The "resonances" that have been mentioned provide broad band matching rather than frequency selection. The bell on the end of many wind instruments is for matching and, at the same time, it pulls the frequencies of the overtones away from where they'd be with a straight cut end.

The reed provides the non-linear switching required for sub-harmonic generation.
The reed is situated between two cavities, and so is subjected to a differential pressure.
The energy storage can then be in the body, with the vocal cords that produce the tone.
The combined beat passes through the reed, moving energy to the sub-harmonic.

I don't see how low frequency product generation is any more relevant than high frequency products. If we're discussing steady state situations, any relative delay in formation of products doesn't count. During the initial formation of a note, the attack waveform will always be different because of the initial relative delays. A plucked string is a good example of where the attack portion is often the most important bit.

Many discussions about musical instruments spend too much time dealing with the steady state situation and 'Harmonics'. Far too simple for an understanding of what goes on. Early electric / electronic organs had very distinctive sounds; nothingl lke the suggested instrument names on the key switches. But the Hammond sound is known and loved for its artificiality.

 

[Baluncore]

I don't see how low frequency product generation is any more relevant than high frequency products.

The growling is a sub-harmonic, at a lower frequency than the fundamentals of the two input frequencies.
Harmonics above the fundamental, f⋅n, are generated by amplitude distortion.
Sub-harmonics, below the fundamental, f/n, are generated by energy storage, which can be seen as time distortion.

 

[sophiecentaur]

Sub-harmonics, below the fundamental, f/n, are generated by energy storage,

Calling it "energy storage" seems unnecessary. With any form of modulation there is no distinction between upper or lower sidebands - you put it all in the pot and you get your products out. When the modulator involves a delay then you clearly have to wait for the result until all the products have caught up and that will affect the temporal response (pulse shape etc.). The growl will take time to build up. Our hearing system deals with low frequency sounds a bit differently and the 'tuning' of low strings and organ pipes is less critical. The growl is down there.

I think you may be referring to the way the attack of a note is affected by (or described by) the maximum length of the temporal response of the system.

 

[Daniel Petka]

If you blow a single mouthpiece, is your vocal tract not on one side of that reed, and does your body not form a resonant cavity?

Calling it "energy storage" seems unnecessary. With any form of modulation there is no distinction between upper or lower sidebands - you put it all in the pot and you get your products out. When the modulator involves a delay then you clearly have to wait for the result until all the products have caught up and that will affect the temporal response (pulse shape etc.). The growl will take time to build up. Our hearing system deals with low frequency sounds a bit differently and the 'tuning' of low strings and organ pipes is less critical. The growl is down there.

I think you may be referring to the way the attack of a note is affected by (or described by) the maximum length of the temporal response of the system.

One funny thing is that the subharmonic is not even necessary for you to hear the deep sound, because the nonlinearities in the ear/brain regenerate the subharmonic even if its not present. What matters is that the product frequencies will "mess up" the harmonics, so your brain can no longer clearly identify the clarinet harmonics

 

[Daniel Petka]

Edit: it totally works on a flute. Guess I didn't record a spectrogram when I made that claim

Ok I might have an explanation for all of this. This comes a bit late because I focused on Uni instead of this peculiar effect.

I tried another experiment: sending a single frequency into my mouth cavity while singing a note (Don't try this at home unless nobody is around lol)
And guess what, this still creates a subharmonic. Doing this with a flute doesn't.
Which leads me to believe that the reason why you hear a subharmonic when singing while playing a flute is that the sound from the flute gets into your vocal chords.
The flute is just a resonator so it's not surprising that a single frequency that's not resonant doesn't do much.
The vocal chords and the reed are different. It's awesome that this works just with vocal chords because we can ignore the whole resonator part. Yes it's there, but like Sophiecentaur pointed out, it doesn't matter if the Q factor is low.

Let me say this: I doubt that the nonlinearity or the switching of a diode is a good analogy, because the modulation works even when singing a quiet falsetto, which creates almost a pure sine wave. Because the frequency comes from an external speaker, the resonators are definetly not coupled, in other words, the voice doesn't affect the speaker.

The simplest explanation that I could come up with is that higher air velocity => higher amplitude. Hence, sinusoidally varying air velocity (sound) => sinusoidally varying amplitude, multiplication of the 2 sinusoids. I tried simulating this in Python by multiplying a recorded clarient tone by (1 + cos(wt)) -> constant DC air and AC perturbation. The DC part leaves some of the even harmonics and the AC multiplication adds odd harmonics. The resulting sound is extremely close to the real thing! This is no proof or anything, but I thought it was worth sharing

 

[Baluncore]

The simplest explanation that I could come up with is that higher air velocity => higher amplitude. Hence, sinusoidally varying air velocity (sound) => sinusoidally varying amplitude, multiplication of the 2 sinusoids.

Two sounds together, in the same linear space, does not explain the non-linearity required to mix or modulate the sine waves.

The vocal chords and the reed are different.

But they are both switching and are both non-linear.


I am confused because you are somehow combining two different sine waves, yet you are referring to harmonics and sub-harmonics, not to sum and difference frequencies.

Harmonics are exact integer multiples of one sine wave, they are not the sum-frequency resulting from multiplying two different sine waves. Harmonics are generated by a non-linear distortion. If the distortion is symmetric about zero, odd harmonics will be generated. If asymmetric about zero, even harmonics are generated.

Mixing two sine waves will produce a difference frequency, but a sub-harmonic must be an exact sub-multiple of one sine wave. Sub-harmonics are generated by information or energy storage.

You need to measure frequencies carefully to identify what is a phase locked harmonic, and what is a sum or difference cross modulation of two sine waves.

 

[Daniel Petka]

Edit: it totally works on a flute. Guess I didn't record a spectrogram when I made that claim

Ok I might have an explanation for all of this. This comes a bit late because I focused on Uni instead of this peculiar effect.

I tried another experiment: sending a single frequency into my mouth cavity while singing a note (Don't try this at home unless no-one is around lol)
And guess what, this still creates a subharmonic. Doing this with a flute doesn't.
Which leads me to believe that the reason why you hear a subharmonic when singing while playing a flute is that the sound from the flute gets into your vocal chords.
The flute is just a resonator so it's not surprising that a single frequency that's not resonant doesn't do much.
The reed is different.
The nonlinearity or the switching of a diode can be eliminated by singing quietly and putting the sine wave into the vocal chords.
The most simple explanation that I could come up with is that higher air velocity => higher amplitude. Sinusoidally varying air velocity (sound) => sinusoidally varying amplitude, multiplication of the 2 sinusoids.
I tried to simulate this in Python by multiplying a recorded clarinet .wav file by this DC air and AC perturbation, basically
Growl(t) = clarinet(t)*(1+cos(wt))
Where w is the angular frequency one fifth above the clarinet's fundamental. The resulting sound is very close to the real thing.

 

[NTL2009]

It seems simpler than all this to me (which might be my error, I'm a simple-minded person!). Amplitude modulation of a signal produces sum and difference frequencies (stop me here if this is not correct, but I'm sure that it is).

Singing is alternate compressions and rarefactions of the air. Wouldn't we expect that to amplitude modulate any musical instrument that is driven by our air supply? If I blow softer or harder (within limitations) on a wind instrument, the sound is louder/softer - isn't that amplitude modulation?

So if I play a 400Hz note on a wind instrument, and sing at 600 Hz, I would expect there to be a 200Hz difference signal, the original carrier of 400 Hz, and a sum of 1000Hz.

OP said he did not observe this difference frequency on a flute? Perhaps the flute is not so sensitive to the variations in air pressure that singing produces (and it's rather tough to sing loudly while maintaining the embouchure required to sound a note on a flute! - But we hopefully have some fans of the Jethro Tull band and Ian Anderson.). So not observing may not equate with "not existing"? And perhaps what I assume is the far greater non-linearity in a reed instrument, that the effect is far more pronounced with a reed? I no longer have a clarinet at my disposal, so I refer here:

https://www.britannica.com/art/reed-instrument

Single reeds may hit against a frame (beating reeds), as in a clarinet mouthpiece, or may vibrate freely through a closely fitting frame (free reeds), as in an accordion; the term single reed usually refers to a beating reed.

I believe this one-sided constricted motion of the reed is why @Baluncore earlier made the comparison of the reed to a diode.

So, isn't that all that is needed? I don't understand, from the definition of amplitude modulation, why any sort of 'storage' is required?

 

[Baluncore]

So, isn't that all that is needed? I don't understand, from the definition of amplitude modulation, why any sort of 'storage' is required?

AM can generate the difference frequency of two sine waves, fd = f1 - f2.
To generate a sub-harmonic from one sinewave, fs = f1 / n, requires 'storage'.

I suggest we are NOT looking here at a sub-harmonic, but at a difference frequency. The use of the correct term is important, as it indicates the mechanism that produces the sound.

We need to know if the lower frequency generated, is an exact sub-multiple of one input, or if it is a difference frequency of two inputs.

 

[NTL2009]

... I suggest we are NOT looking here at a sub-harmonic, but at a difference frequency. The use of the correct term is important, as it indicates the mechanism that produces the sound.

We need to know if the lower frequency generated, is an exact sub-multiple of one input, or if it is a difference frequency of two inputs.

Yes, the correct terminology is important. And a difference frequency could also be a sub-harmonic . As in my example of a 400 Hz note and 600 Hz singing/modulating note, the difference frequency is 200 Hz, and 200 Hz is an octave below the 400Hz note ( I think a musician would call the 200 Hz note a sub-harmonic, and it is also the difference frequency).

Some sort of controlled lab set up would help, the instrument (carrier) and the modulator (human voice) all have fairly complex harmonics, and specific conditions aren't so easy to replicate. But I expect that the OP could do enough examples to verify with some confidence whether or not the lower note is always a difference frequency or not?

 

[NTL2009]

Adding to the above, I see in the video in the OP, it states "By playing a note and simultaneously singing a quint to it, the tone will sound an octave deeper". A musical quint is 1.5x the frequency of the reference note. So my example of 400 Hz note, and singing 600 Hz, fits this scenario. And 200 Hz is an octave deeper, so it is a sub-harmonic, and a difference frequency as well.

I suspect it will always be a difference frequency (based on the amplitude modulation effect), it just happens to be a sub-harmonic as well in this case.

 

[Baluncore]

So my example of 400 Hz note, and singing 600 Hz, fits this scenario. And 200 Hz is an octave deeper, so it is a sub-harmonic, and a difference frequency as well.

I disagree, it is a difference frequency if it comes from two inputs. A sub-harmonic is phase locked to a single higher fundamental. The method of generation is more important than the idealised numerical frequency.

A 10 mHz frequency variation of one of the two inputs, would sound the same to a listener, but the difference frequency would not be phase-locked to either one of the higher frequency inputs, so it is NOT a sub-harmonic.

 

[Daniel Petka]

It seems simpler than all this to me (which might be my error, I'm a simple-minded person!). Amplitude modulation of a signal produces sum and difference frequencies (stop me here if this is not correct, but I'm sure that it is).

Singing is alternate compressions and rarefactions of the air. Wouldn't we expect that to amplitude modulate any musical instrument that is driven by our air supply? If I blow softer or harder (within limitations) on a wind instrument, the sound is louder/softer - isn't that amplitude modulation?

So if I play a 400Hz note on a wind instrument, and sing at 600 Hz, I would expect there to be a 200Hz difference signal, the original carrier of 400 Hz, and a sum of 1000Hz.

OP said he did not observe this difference frequency on a flute? Perhaps the flute is not so sensitive to the variations in air pressure that singing produces (and it's rather tough to sing loudly while maintaining the embouchure required to sound a note on a flute! - But we hopefully have some fans of the Jethro Tull band and Ian Anderson.). So not observing may not equate with "not existing"? And perhaps what I assume is the far greater non-linearity in a reed instrument, that the effect is far more pronounced with a reed? I no longer have a clarinet at my disposal, so I refer here:

https://www.britannica.com/art/reed-instrument


I believe this one-sided constricted motion of the reed is why @Baluncore earlier made the comparison of the reed to a diode.

So, isn't that all that is needed? I don't understand, from the definition of amplitude modulation, why any sort of 'storage' is required?

I hear what you are saying, you think that the amplitude modulation should work for any wind instrument, not just one that uses a vibrating membrane.
To find the subharmonic on a flute, I used an external sine generator, because singing might give you a false signal - the subharmonic may come from the vocal chords! I turned the volume way up and tried many instruments that are similar to a flute (glass bottle, wine glass, etc... Helmholz resonators essentially), but failed to find the subharmonic. Honestly, though, this is what I expected, because a linear sum of two sine waves obviously doesn't produce any new frequencies. The FFT itself is linear, all it tells you is the amplitude and frequency of sine waves that need to be summed to get the measured signal.

 

[Daniel Petka]

AM can generate the difference frequency of two sine waves, fd = f1 - f2.
To generate a sub-harmonic from one sinewave, fs = f1 / n, requires 'storage'.

I suggest we are NOT looking here at a sub-harmonic, but at a difference frequency. The use of the correct term is important, as it indicates the mechanism that produces the sound.

We need to know if the lower frequency generated, is an exact sub-multiple of one input, or if it is a difference frequency of two inputs.

This is a good point! It's actually a difference frequency generally speaking. It just happens to be a subharmonic whenever the frequencies form a nice ratio (like 3/2, 4/3, and so on)

 

[Daniel Petka]

Yes, the correct terminology is important. And a difference frequency could also be a sub-harmonic . As in my example of a 400 Hz note and 600 Hz singing/modulating note, the difference frequency is 200 Hz, and 200 Hz is an octave below the 400Hz note ( I think a musician would call the 200 Hz note a sub-harmonic, and it is also the difference frequency).

Some sort of controlled lab set up would help, the instrument (carrier) and the modulator (human voice) all have fairly complex harmonics, and specific conditions aren't so easy to replicate. But I expect that the OP could do enough examples to verify with some confidence whether or not the lower note is always a difference frequency or not?

Yes, I can say with certainty that it's a difference frequency. To address your worry about the voice having too many harmonics: It doesn't necessarily have to. Like written above, singing quietly and in falsetto leads to almost a sine wave.
I put my phone near my mouth and played a tone at, say 800 Hz, while singin. If you ask me, everyone has a phone and everyone has vocal chords, so this as easy to replicate as it gets and you can feel free to do it if you doubt the results (if no one is around xD). Note that the scale is logarithmic, so the harmonics are not nearly as strong as they appear to be, but yeah it's hard to get rid of the first harmonic in the falsetto.

from bottom up: sine at 800Hz; falsetto voice at 800Hz/1.5; both

To prove that this is indeed a difference frequency, I simply increased the pitch of my voice and, as you would expect, as the fundamental frequency of my voice got closer to 800Hz, the difference frequency dropped. And you see another difference frequency that increases, as the first harmonic of my voice gets further away from 800Hz.

 

[Daniel Petka]

I disagree, it is a difference frequency if it comes from two inputs. A sub-harmonic is phase locked to a single higher fundamental. The method of generation is more important than the idealised numerical frequency.

A 10 mHz frequency variation of one of the two inputs, would sound the same to a listener, but the difference frequency would not be phase-locked to either one of the higher frequency inputs, so it is NOT a sub-harmonic.

You are not wrong, but to a listener, the phase doesn't matter. So a musician would just call it a subharmonic. With that being said, I should have measured it..

 

[NTL2009]

I disagree, it is a difference frequency if it comes from two inputs.

But don't we have two inputs here? The clarinet (carrier) and the singing tone (modulator)?

Or maybe I'm misreading your comment - are you agreeing that it is a difference frequency, but objecting to calling it a sub-harmonic? I can see that distinction, because as soon as I shift the frequency of either by a bit, it would no longer appear (to a musician) to be a sub-harmonic (it would not be harmonically related).

I hear what you are saying, you think that the amplitude modulation should work for any wind instrument, not just one that uses a vibrating membrane.
To find the subharmonic on a flute, I used an external sine generator, because singing might give you a false signal - the subharmonic may come from the vocal chords! I turned the volume way up and tried many instruments that are similar to a flute (glass bottle, wine glass, etc... Helmholz resonators essentially), but failed to find the subharmonic. ...

We'd need more details of your set up, but I suspect that it was not replicating the effect of the human voice well enough. I don't really understand how the signal generator was being used - you "turned the volume up" - what was the generator connected to, and how?

I think you need the signal amplified, and run to a speaker that is modulating the flow of air to the wind instrument. Just pointing a speaker at it (if that is what you did), might not have the desired effect. I'm thinking something along the lines of a speaker mounted to a closed box, with a controlled source of air running in and out of the box through hoses, to the wind instrument. This, I think, would provide a stream of air that has those 600 Hz (using my earlier examples) variations in pressure, but always positive, so as to keep the air column oscillating.

I'd love to set this up and test it myself, but I have a few other things going on at the moment.

 

[Baluncore]

Or maybe I'm misreading your comment - are you agreeing that it is a difference frequency, but objecting to calling it a sub-harmonic?

I am insisting that it is a difference frequency, and that in a discussion of the physics, it should NOT be called a sub-harmonic.

One input, can generate harmonic or sub-harmonic frequencies.
Two inputs, can generate sum or difference frequencies.

You are not wrong, but to a listener, the phase doesn't matter. So a musician would just call it a subharmonic.

This is a physics discussion. The use of the correct physical term avoids confusion, and makes it possible to identify the mechanism of generation.

It just happens to be a subharmonic whenever the frequencies form a nice ratio (like 3/2, 4/3, and so on)

It is a difference frequency, that may have the same frequency as a sub-harmonic of one of the inputs.

Take a 1 kHz master oscillator and distort the output with a diode, then use two resonators to select the 2 kHz and 3 kHz harmonics generated. If you then multiply the 2 kHz and 3 kHz harmonics together, you will get a 1 kHz difference frequency that happens to be phase locked to the inputs. But that is not a sub-harmonic, it is a difference frequency, and it is exactly equal to the original master oscillator fundamental. To generate a sub-harmonic of that fundamental, takes more than amplitude distortion of the signal, it requires energy or information storage.

As an example, digital processors generate EMI across a wide frequency band. The frequencies below the processor clock are sub-harmonics, since any software loop, or digital frequency divider, is a sub-harmonic generator. Those sub-harmonics are rectangular digital waveforms, which have harmonics ranging upwards, well above the processor clock, where they fall in the spectrum between the integer harmonics of the processor clock.

 

[NTL2009]

I am insisting that it is a difference frequency, and that in a discussion of the physics, it should NOT be called a sub-harmonic. ...

OK, thanks, we are in agreement on that now. It is a difference frequency, that in some specific cases, can be harmonically related.

 

[Daniel Petka]

But don't we have two inputs here? The clarinet (carrier) and the singing tone (modulator)?

Or maybe I'm misreading your comment - are you agreeing that it is a difference frequency, but objecting to calling it a sub-harmonic? I can see that distinction, because as soon as I shift the frequency of either by a bit, it would no longer appear (to a musician) to be a sub-harmonic (it would not be harmonically related).



We'd need more details of your set up, but I suspect that it was not replicating the effect of the human voice well enough. I don't really understand how the signal generator was being used - you "turned the volume up" - what was the generator connected to, and how?

I think you need the signal amplified, and run to a speaker that is modulating the flow of air to the wind instrument. Just pointing a speaker at it (if that is what you did), might not have the desired effect. I'm thinking something along the lines of a speaker mounted to a closed box, with a controlled source of air running in and out of the box through hoses, to the wind instrument. This, I think, would provide a stream of air that has those 600 Hz (using my earlier examples) variations in pressure, but always positive, so as to keep the air column oscillating.

I'd love to set this up and test it myself, but I have a few other things going on at the moment.

It's not as complicated as you think, the signal generator is just my phone playing a sine wave from an online tone generator. That's the whole setup, my mouth and my phone near it.

Edit: once again, this setup doesn't include the clarinet. This is what makes it reproducible. While only a few people have a clarinet, all people have vocal chords and a phone

 

[NTL2009]

It's not as complicated as you think, the signal generator is just my phone playing a sine wave from an online tone generator. That's the whole setup, my mouth and my phone near it.

Edit: once again, this setup doesn't include the clarinet. This is what makes it reproducible. While only a few people have a clarinet, all people have vocal chords and a phone

OK, thanks - I got mixed up between the earlier statements about not being able to produce the difference frequency with a flute.

It's not clear to me how a voice near a speaker would end up amplitude modulating that speaker tone, wouldn't they simply add? It's different from the voice being a source of wind for an instrument, where it would affect the amplitude. It's also possible (likely?) that a significant non-linearity exists in the phone's microphone and/or signal chain?

 

[Daniel Petka]

No, the voice doesn't affect the speaker. That's the joke, the speaker affects the voice and not the other way round. You can call it "nonlinearity" but all it is, is multiplication of the 2 sound waves.
The vocal chords are like the reed.

 

[NTL2009]

No, the voice doesn't affect the speaker. That's the joke, the speaker affects the voice and not the other way round. ...

That seems like a theory. I'm not discounting it, it may be correct, but do we really know that's what's happening?

I hope I can get to setting up my own experiment as I described above. I have a soprano and a tenor recorder, an air compressor and amps and speakers, and a workshop to build a box as I described. I suspect modulating the air stream would be enough, that the vocal cords/reeds would not be needed, but I don't know. Or maybe they are acting the same, and the speaker is modulating our non-linear vocal cords?

... You can call it "nonlinearity" but all it is, is multiplication of the 2 sound waves.
The vocal chords are like the reed.

Well, my understanding is there must be a non-linearity (and they don't care what/if I call it anything! :) ). In a linear system, the signals add. With non-linearity, there are harmonics and/or sum/difference frequencies.

 

[Daniel Petka]

That seems like a theory. I'm not discounting it, it may be correct, but do we really know that's what's happening?

I hope I can get to setting up my own experiment as I described above. I have a soprano and a tenor recorder, an air compressor and amps and speakers, and a workshop to build a box as I described. I suspect modulating the air stream would be enough, that the vocal cords/reeds would not be needed, but I don't know. Or maybe they are acting the same, and the speaker is modulating our non-linear vocal cords?

Well, my understanding is there must be a non-linearity (and they don't care what/if I call it anything! :) ). In a linear system, the signals add. With non-linearity, there are harmonics and/or sum/difference frequencies.

Yes, you are right of course, I just like to use the more precise term whenever possible.
Sure, it is a theory. I'm not gonna repeat myself, this is why I think it happens:

Ok I might have an explanation for all of this. This comes a bit late because I focused on Uni instead of this peculiar effect.

I tried another experiment: sending a single frequency into my mouth cavity while singing a note (Don't try this at home unless no-one is around lol)
And guess what, this still creates a subharmonic. Doing this with a flute doesn't.
Which leads me to believe that the reason why you hear a subharmonic when singing while playing a flute is that the sound from the flute gets into your vocal chords.
The flute is just a resonator so it's not surprising that a single frequency that's not resonant doesn't do much.
The reed is different.
The nonlinearity or the switching of a diode can be eliminated by singing quietly and putting the sine wave into the vocal chords.
The most simple explanation that I could come up with is that higher air velocity => higher amplitude. Sinusoidally varying air velocity (sound) => sinusoidally varying amplitude, multiplication of the 2 sinusoids.
I tried to simulate this in Python by multiplying a recorded clarinet .wav file by this DC air and AC perturbation, basically
Growl(t) = clarinet(t)*(1+cos(wt))
Where w is the angular frequency one fifth above the clarinet's fundamental. The resulting sound is very close to the real thing.

Edit: I just did it with a subwoofer playing at 800Hz while not even singing directly at it. I doubt that the voice had any effect on the sub

 

[sophiecentaur]

Amplitude modulation

The appropriate term is not Amplitude Modulation; AM is a very specific no-linear process. Every instrument / player combination will have a unique non-linear characteristic with unique spectral products. If you don't have an accurate model then you can say very little about the resulting sounds. You can't even talk harmonics in describing instruments or vocal systems.

This is even harder in the case of an instrumentalist with an instrument. There is feedback with the person's perception which can easily 'pull' what they do with their vocal system and cause misconceptions of what they hear / feel. The experimenter becomes an unreliable witness. You'd have to invent some sort of double blind test to eliminate this problem.

This doesn't matter but I'm just pointing out that conversations about this sort of topic are unlikely to reach safe conclusions. You would need to build a hardware model - good luck with that.

 

[Daniel Petka]

The appropriate term is not Amplitude Modulation; AM is a very specific no-linear process. Every instrument / player combination will have a unique non-linear characteristic with unique spectral products. If you don't have an accurate model then you can say very little about the resulting sounds. You can't even talk harmonics in describing instruments or vocal systems.

This is even harder in the case of an instrumentalist with an instrument. There is feedback with the person's perception which can easily 'pull' what they do with their vocal system and cause misconceptions of what they hear / feel. The experimenter becomes an unreliable witness. You'd have to invent some sort of double blind test to eliminate this problem.

This doesn't matter but I'm just pointing out that conversations about this sort of topic are unlikely to reach safe conclusions. You would need to build a hardware model - good luck with that.

Very few things in science reach safe conclusions, but I would say it's safe to call this AM.
I don't quite see why perception matters here - the witness is my phone, not my ear.