Why singing a note while playing a note creates a difference frequency

[Daniel Petka]

TL;DR Summary

Playing a note (on e.g. a flute) and simultaneously singing into the flute creates a difference frequency. Also, the flute doesn't matter! It works with a tone generator as well, so the nonlinearity that is causing the modulation happens in the vocal chorda. What is causing the nonlinearity?

If you play a note of a certain frequency on a flute and simultaneously sing a note at a different frequency, then you create a third frequency that wouldn't be there if you play or sing in isolation - and the frequency of this subharmonic is the difference of the flute frequency and the voice frequency (I'm talking about the fundamental frequencies for simplicity). This is caused by modulation (product rule: cos(f1x)cos(f2x) = 0.5(cos(f1-f2)+cos(f1+f2)), which in turn suggests some nonlinearity in the system.

I made some observations about this effect:

1. It is not unique to flutes- works on any wind instrument (commonly known as Growling on saxophones) I asked this on a different post specifically about clarinets, but it seems to be very general
2. It is NOT the Tartini tone / missing fundamental / difference tone. I know it might be tempting to classify it as such, but keep in mind that the Tartini tone is just in your head (caused by some nonlinearity in the hearing apparatus). The subharmonic mentioned here is clearly measurable on a spectrogram.
3. Whistling into the flute does not work. Sure, it produces a beat frequency (linear superposition) and a Tartini tone (illusion in your head), but the subharmonic is not visible on the spectrogram. This suggests to me that the nonlinearity that causes the modulation is somehow in the vocal chords. The problem might be somehow linked to coupled oscillators, but beyond me right now.

Do you have an intuitive explanation for this? Thanks!

 

[Baluncore]

This suggests to me that the nonlinearity that causes the modulation is somehow in the vocal chords.

Strictly speaking, it is a difference frequency, not a sub-harmonic.

The vocal cords appear to be a non-liner mechanism, capable of transferring energy from the excitation frequencies, to the difference frequency.

The blown air, flowing across the hole, must also be switching path, as it reinforces the resonant note of the flute. That flow-path switching, could also mix the two signals to produce the difference frequency.

 

[Daniel Petka]

Strictly speaking, it is a difference frequency, not a sub-harmonic.

The vocal cords appear to be a non-liner mechanism, capable of transferring energy from the excitation frequencies, to the difference frequency.

The blown air, flowing across the hole, must also be switching path, as it reinforces the resonant note of the flute. That flow-path switching, could also mix the two signals to produce the difference frequency.

Yes, you are right of course, I changed the title entirely (also got rid of the flute) I automatically took that from the last post I made.

I am now pretty certain it happens within the vocal chords and the flute has nothing to do with it, as I played a tone on an online generator at a reasonable volume and sang a difference tone at the same time. The result is clear, it's the vocal chords, but I don't understand what's causing the nonlinearity.

 

[Baluncore]

The result is clear, it's the vocal chords, but I don't understand what's causing the nonlinearity.

The cords can open and close like a switch or a diode. That non-linearity generates the higher voice harmonics.

The cords can be forced open or closed, but they can also be biased close to the edge of flow. The changes in air pressure can then move the cords, to open or close the flow. The difference frequency of two tones can then be detected by the cords. That process can transfer energy to the frequency of the envelope.

Maybe this will help understand the non-linear switching.
https://en.wikipedia.org/wiki/Vocal_cords#Function

 

[Daniel Petka]

Hm I'm not sure if this it. If you sing loudly, then sure, the vocal cords clash. But you can also hum a quiet falsetto and that shows up almost like a sinusoid on the spectrogram. And yet, the difference tone is clearly there. I'm thinking that it could be somehow related to the Bernoulli equations. The sound wave is a change in pressure and if the pressure in the vocal cords drops momentarily, their vibration volume increases and vice versa

 

[Baluncore]

I'm thinking that it could be somehow related to the Bernoulli equations.

I think airflow has everything to do with the Bernoulli equation, which simply follows packets of air through the system, conserving energy, in its different forms. It is just a question of when, where and how the energy in the flow changes form, as it flows through the system of passages.

If the vocal cords are positioned and tensioned by the singer, to close as the airflow increases, then that is a non-linear system. High velocity air at low pressure, is slowed, to become low velocity air at higher pressure, which then in turn, opens the passage through the vocal cords, to repeat the process. That process is sensitive to the envelope of the two simultaneous tones.

The sound wave is a change in pressure and if the pressure in the vocal cords drops momentarily, their vibration volume increases and vice versa.

What is meant here by "vibration volume"? That sentence lacks the detail and definition necessary for analysis.

 

[Daniel Petka]

I think airflow has everything to do with the Bernoulli equation, which simply follows packets of air through the system, conserving energy, in its different forms. It is just a question of when, where and how the energy in the flow changes form, as it flows through the system of passages.

If the vocal cords are positioned and tensioned by the singer, to close as the airflow increases, then that is a non-linear system. High velocity air at low pressure, is slowed, to become low velocity air at higher pressure, which then in turn, opens the passage through the vocal cords, to repeat the process. That process is sensitive to the envelope of the two simultaneous tones. What is meant here by "vibration volume"? That sentence lacks the detail and definition necessary for analysis.

I meant loudness. If the loudness of the sound increases then so does the pressure difference. So the fact that air travels faster when the pressure is lower (and vice versa) is directly responsible for the modulation

 

[Swamp Thing]

But you can also hum a quiet falsetto and that shows up almost like a sinusoid on the spectrogram. And yet, the difference tone is clearly there.

Even if two pure tones reach the ear from different directions (i.e. without interacting with each other beforehand), the ear itself has nonlinear phenomena that can "rectify" the resultant envelope and produce the sensation of hearing a difference tone. For example, see http://szhorvat.net/pelican/combination-tones.html

 

[Daniel Petka]

Yeah, this is a very interesting phenomenon (Tartini notes) but see my second point, it's not relevant because the difference tone is visible on the spectrogram. Actually, I should be careful phrasing this, it's definitely relevant when it comes to the perception of the sound, but here I'm more interested in the generation of the subharmonic. People on pf here explained it with nonlinearity and compared it to a diode, but imo a better electronic analogy is a voltage controlled oscillator. The air speed through the opening controls the pressure difference and thus the force on the reed, so an oscillating air speed modulates the sound and the difference frequency comes from the product rule. That's my best theory yer.
Anyways, you might find it interesting that the Tartini tones are actually not produced in the ear, as you can send each frequency to a different ear and the combination tone is still there (binaural beats) The difference frequency happens somewhere in the brain (probably also as a result of multiplocation)

Edit: ah ok you mentioned from different directions. I think you're already referring to binaural beats. If the frequencies are in different ears, then it can't possibly be due to the ear's nonlinearity