# Is the human singing voice different from a linear, time-invariant (LTI) system?

Typically the human voice is modelled by a linear, time-invariant (LTI) system.

An LTI system cannot add any frequency components to an input signal. This is obvious from the input-output relation in the frequency domain: $$Y(ω)=X(ω)H(ω)$$ where $$Y(ω)$$ is the output spectrum, $$X(ω)$$ is the input spectrum, and $$H(ω)$$ is the filter's frequency response. https://dsp.stackexchange.com/a/38199

However, I have been looking at spectrograms of singing and it seems, that the better the singer the more able they are to remove frequency components from harmonics outside their formants and transfer them to the harmonics in their formants.

On the other hand, this could just be that in-order to get $$||H(w)||$$ to approach $$1$$ around the formants, $$||H(w)||$$ must approach $$0$$ elsewhere, due to the physics of the vocal track shape.

Is the resonance that vocal coaches talk about, merely the feeling getting $$||H(w)||$$ to approach $$1$$, or does the nasal cavity act akin to the sound box of a string instrument.

• I'm eager to exchange ideas with you. The stackexchange comment section is not intended for conversation. There is a link to the contact page of my website on my stackexchange profile page. If it's ok with you then do contact me. Commented Dec 5, 2021 at 10:31

I'm a singer. (Well, it's not that I earn my living with singing, but other than that I qualify myself as a singer.)

So, researchers have used various ways to investigate the process of singing.

For instance, there has been research where tiny microphones were lowered down the trachea, past the vocal folds. The wire was small enough to not interfere with voice production. That way the researchers could investigate what spectrum of sounds were present below the vocal folds.

My understanding is that there is significant acoustic interaction between the sound spectrum below the vocal folds and the sound spectrum above the vocal folds.

It seems highly plausible to me that there is a lot of feedback mechanism between the vibration of the vocal folds and the sound spectrum in the vocal tract.

(In my personal experience: when I am singing in the pitch range where my passagio is some vowels tend to interfere with the intended voice production. My hypothesis is: at that pitch it so happens that the formants for the particular vowel are at odds with getting the reinforcing feedback to the vocal folds. I'm still practicing at working around those oddities.)

So in the case of voice production I don't think there is a meaningful distinction to be made in terms of 'input frequency spectrum' and 'filter frequency response'.

It seems plausible to me that the resonances of the vocal tract feed back to the vocal folds, interacting with the very way the vocal folds are vibrating.

In the comment section a very fruitful exchange developed. I have had it happen in the past that a moderator would move a fruitful exchange to chat. In my opinion only bickering should be moved to chat, and fruitful exchange should be allowed. I now copy the existing comment exchange to this post, and I will update as necessary.

Tom Huntington:
"interacting with the very way the vocal folds are vibrating". Would have never though of that. Thanks! –

Cleonis:
@TomHuntington I just remembered that in december 2018 I had submitted an answer about the physics of note bending with a mouth harmonica. Researching that question I found out that a harmonica instructor (David Barrett) had managed to arrange for him to be MRI scanned while note bending, for the purpose of imaging his internal adjustments. (A harmonica builder was commisioned to construct two harmonicas using only materials that do not magnetize.)

Tom Huntington:
I've found that in-order to sing with volume in chest voice, I must widen my throat right down at the cords (probably this is what is meant by open throat). This would agree with your feedback/interaction theory.

Cleonis:
@TomHuntington I never do any intentional widening. Never any yawning. That said, the following is among my means for relaxation of the singing apparatus. Years ago I noticed something about the neck of a saxophone player. Everytime he started playing the girth of his neck increased by centimeters. His trachea must have been crazy elastic. I do that thing, but with minimal air pressure. I put my thumbs to my nostril holes, allowing just a smidgeon of air to escape, mild breath pressure, sometimes with humming. It feels like the (mild) internal pressure stretches the trachea/throat.

Cleonis:
@TomHuntington The whacky thing about singing is the total mismatch between the pitch (of the fundamental) that is produced by the vocal folds and the size of the resonant cavity. Resonance is your friend, but that resonance is (especially for male singers), in a range that is octaves higher than your fundamental. For a classically trained singer the fundamental is in effect irrelevant, the training is towards developing the singer's formant I think of singing in terms of tuning resonance, never in terms of "open throat"

Tom Huntington:
Yes, singer's formant is what we're working for. I've been thinking of singing in terms of source (phonation of vocal cords) and filter (shape of vocal tract). The best read on singing is Jack Livigni's article You have to get large harmonics in your singings formant, at the source, before the shape of the vocal track starts mattering. Laryngeal tilt with cord closure (adduction) is needed to find the the "ring" in your voice, and then bring that ring back to chest voice.

Cleonis:
I concur of course that in order to populate the singer's formant with high harmonics those harmonics must be produced in the first place. If you do fourier analysis of a sound: a sound that is a saw-tooth on the oscilloscope is a sound with a lot of energy in the higher harmonics. That is how I understand the emphasis that Jack LiVigni places on bringing the vocal folds close together. However, I believe there is a trade-off. Bringing the vocal folds closer together (making the opening and closing more explosive) will shift energy to generating higher harmonics, but at the expense of less interaction of sound spectrum below and above the vocal folds. I visualize there is a sweet spot, and I aim for that. I aim to have my vocal folds close just enough such that full sound is produced, while minimizing breath presssure.

• "interacting with the very way the vocal folds are vibrating". Would have never though of that. Thanks! Commented Dec 4, 2021 at 21:02
• @TomHuntington I just remembered that in december 2018 I had submitted an answer about the physics of note bending with a mouth harmonica. Researching that question I found out that a harmonica instructor (David Barrett) had managed to arrange for him to be MRI scanned while note bending, for the purpose of imaging his internal adjustments. (A harmonica builder was commisioned to construct two harmonicas using only materials that do not magnetize.) Commented Dec 4, 2021 at 21:57
• I've found that in-order to sing with volume in chest voice, I must widen my throat right down at the cords (probably this is what is meant by open throat). This would agree with your feedback/interaction theory. Commented Dec 5, 2021 at 1:49
• @TomHuntington I never do any intentional widening. Never any yawning. That said, the following is among my means for relaxation of the singing apparatus. Years ago I noticed something about the neck of a saxophone player. Everytime he started playing the girth of his neck increased by centimeters. His trachea must have been crazy elastic. I do that thing, but with minimal air pressure. I put my thumbs to my nostril holes, allowing just a smidgeon of air to escape, mild breath pressure, sometimes with humming. It feels like the (mild) internal pressure stretches the trachea/throat. Commented Dec 5, 2021 at 2:10
• @TomHuntington The whacky thing about singing is the total mismatch between the pitch (of the fundamental) that is produced by the vocal folds and the size of the resonant cavity. Resonance is your friend, but that resonance is (especially for male singers), in a range that is octaves higher than your fundamental. For a classically trained singer the fundamental is in effect irrelevant, the training is towards developing the singer's formant I think of singing in terms of tuning resonance, never in terms of "open throat". Commented Dec 5, 2021 at 2:26

The resonance of the vocal tract can't be modelled by the simple first-order linear theory of musical instruments, in which we would expect to see just one peek of frequencies at each formant. Instead we see what looks like the harmonics of the vocal cords after being passed through a filter (i.e. a LTI system).

However, my empirical observation indicated that the harmonics of the vocal cords were being amplified in the formants proportional to the strength they were being removed outside of the harmonics. This can be explained by mode locking which is necessary to correctly model musical instruments.

Technically, this is the process that physicists and engineers call mode locking, and is an effect characteristic of nonlinear oscillators. When oscillations at two frequencies $$f_1$$ and $$f_2$$ are input to an non-linear system, they produce what we call sum and difference terms: vibration components with a range of frequencies including $$f_1 + f_2$$ and $$f_1 − f_2$$. www.phys.unsw.edu.au/jw/brassacoustics.html

Thus two harmonics outside of the formants can contribute to the resonance so long that the difference or sum is inside a formant. This is why cord closure is so important because it creates harmonics above the singer's formant which then can add to the resonance of the singers formant.

It may make a good paper to numerically solve for this resonance using the equations (5) and (6) from here.