The ability of "hearing out" harmonic components depending on the fundamental frequency of complex tones

[BC32]

Dear forum,

I am looking into the following issue. On many musical instruments, it becomes easier to hear out individual partials (overtones) the lower the fundamental frequency of the complex tone is. This applies, for example, to string instruments and the piano, but also to the clarinet or organ pipes. I am searching for a psychoacoustic explanation for this phenomenon.

The topic is hinted at in many places, but I have not found a truly satisfactory answer to my question. In Helmholtz’s “On the Sensations of Tone as a Physiological Basis for the Theory of Music”, I do find an implicit observation of the same effect, but it is explained there in terms of the construction of the respective instruments. According to him, a piano is built in such a way that fewer loud overtones are produced at higher fundamental frequencies.

In my context, I am also trying to understand the concept of critical bands. In many places, it is stated that the boundary between resolved and unresolved harmonics lies somewhere between the 7th and 8th partial(?). However, I am not sure whether this approach can explain my problem, since the ability to hear an overtone separately should depend much more clearly on the fundamental frequency (and less on its position in the harmonic series).

Does the construction of instruments perhaps play a larger role after all, or have I overlooked another psychoacoustic principle?

I would be very grateful for any answers, and please excuse any mistakes—English is not my native language

 

[kemmler3D]

I have not looked deeply into this, but this makes total sense to me. Our hearing is most sensitive in the 400hz- 6khz range and especially around 2-4khz. Much less sensitive once you go past 8-10khz or so.

If you have a low note at say 60hz fundamental, (~B1) the even harmonics show at 120, 240, 480, 960, 1920hz, etc.

Some of the harmonics line up with our sensitive hearing band.

If the fundamental is high, say 2000hz, (~C7) then the harmonics go to 4khz, 8khz, 16khz... They quickly go past our sensitive hearing range.

So generally the lower the note, the more harmonics will be easier to hear.

The way the instrument works probably also plays a role but I don't know much about that part.

 

[Curvature]

Lower fundamentals are more likely to have the overtones farther apart and therefore in different critical bands (based on the Bark scale) or ERBs (based on the Cam scale, more accepted today), as well as less masked. As frequency increases, ERBs become wider.

Audibility also depends on the phase of the overtones, although I don't know exactly how this works apart from noting that artificial situations where the phase of the overtones is scrambled (e.g., in experimental settings) causes significant issues in auditory processing (clarity in particular).

In my context, I am also trying to understand the concept of critical bands. In many places, it is stated that the boundary between resolved and unresolved harmonics lies somewhere between the 7th and 8th partial(?).

The comment about partials seems like an overgeneralization. I wouldn't take it too seriously. Main way to think about overtones is in terms of their level and distance from the fundamental. IMO, too much emphasis in musical literature is given to the ratios between overtones.

ERBs are groups of frequencies where one tone is expected to interfere with another tone through masking or other phenomena. These are specifically related to 3500 inner hair cells along the length of the basilar membrane. Each inner hair cell is its own filter, so each ERB (the shape and bandwith depends on frequency) is effectively a bundle of hair cells working together and firing signals as part of the tonotopic (frequency-specific) organization of the auditory cortex in the brain.

In Helmholtz’s “On the Sensations of Tone as a Physiological Basis for the Theory of Music”

This is well over 100 years old A foundational text, very important for developing the general concepts, but specific mechanics are better addressed with more recent literature.

 

[kyuu]

I am looking into the following issue. On many musical instruments, it becomes easier to hear out individual partials (overtones) the lower the fundamental frequency of the complex tone is. This applies, for example, to string instruments and the piano, but also to the clarinet or organ pipes. I am searching for a psychoacoustic explanation for this phenomenon.

The topic is hinted at in many places, but I have not found a truly satisfactory answer to my question. In Helmholtz’s “On the Sensations of Tone as a Physiological Basis for the Theory of Music”, I do find an implicit observation of the same effect, but it is explained there in terms of the construction of the respective instruments. According to him, a piano is built in such a way that fewer loud overtones are produced at higher fundamental frequencies.

It's not that pianos (or other instruments with strings) are deliberately built in a way that results in more overtones at low frequencies, it's that it's a consequence of the fact that the wavelengths for the lowest notes are long. Therefore, in order to get a string that plays a clean fundamental in the lowest octave, you need very long strings. For the lowest notes, you need strings too long to fit even in a concert grand piano. Luckily, you can also tune the string to a lower frequency by adding mass instead (hence why the bass strings are wound with copper). The practical effect of using mass rather than length for the tuning is that you get a lot of overtones, sometimes to the point that the fundamental is drowned out.

This is why small uprights (particularly consoles) and baby grands generally sound rather "muddy" in the lowest octave: they have to use more mass on more strings and so they have a lot of overtones on those low notes.

 

[sam_adams]

Lower fundamentals . . .

It would be extremely helpful if you labelled your example diagrams as to their nature and source.

 

[Duke]

If you have a low note at say 60hz fundamental, (~B1) the even harmonics show at 120, 240, 480, 960, 1920hz, etc.

Maybe I am completely wrong, but I thought the overtone sequence of a 60 Hz fundamental would be 120, 180, 240, 300, 360 Hz, etc... ?

Lower fundamentals are more likely to have the overtones farther apart...

Again maybe I'm completely wrong, but wouldn't the overtone sequence be spaced more closely for a lower fundamental? Doesn't the fundamental frequency predict the spacing of the overtones?

On many musical instruments, it becomes easier to hear out individual partials (overtones) the lower the fundamental frequency of the complex tone is.

My understanding is that, on a grand piano, the fundamental of A0 is essentially inaudible because it is simply not very loud. Same thing for the first overtone of A0, though to a lesser extent. My understanding is that it is the second overtone of A0 that is finally loud enough to be clearly and solidly audible. The ear hears the 27-Hz interval between the overtones and infers the 27 Hz fundamental even though it's not audible.

My understanding is that the piano would need to be several times larger than it normally is in order for the fundamental of low-A to be as loud as the second overtone is on the pianos we have.

On many musical instruments, it becomes easier to hear out individual partials (overtones) the lower the fundamental frequency of the complex tone is.

Could the relatively low acoustic energy in the lowest fundamentals, and sometimes even in the first overtones, play a role in what you are observing?

 

[kyuu]

My understanding is that the piano would need to be several times larger than it normally is in order for the fundamental of low-A to be as loud as the second overtone is on the pianos we have.

I believe the piano is nearly 6 meters long in order to have a long enough string for A0.

 

[kemmler3D]

Maybe I am completely wrong, but I thought the overtone sequence of a 60 Hz fundamental would be 120, 180, 240, 300, 360 Hz, etc... ?

You know, I had the same thought when I was writing the comment and I hadn't finished my coffee yet either...

e: yep

 

[Curvature]

Again maybe I'm completely wrong, but wouldn't the overtone sequence be spaced more closely for a lower fundamental? Doesn't the fundamental frequency predict the spacing of the overtones?

I should have been more clear.

I meant the ERB bandwidths are tighter in LF than in HF, so from the perspective of filter spacing the overtones are farther apart.

 

[Curvature]

It would be extremely helpful if you labelled your example diagrams as to their nature and source.

They are very common diagrams often reproduced in studies.

First one is from: (2008) Fundamentals of Acoustics and Noise Control, Technical University of Denmark.
The second is from: (2012) Application of Perceptual Filtering Models to Noisy Speech Signals Enhancement, Journal of Electrical and Computer Engineering.

Did you mean you want the diagrams explained as well?