Missing Fundamental

[Florin Andrei]

Another try...?
You see him playing C0...?

It doesn't matter. If you can't see the fundamental in the spectrogram, then it was not actually recorded. You may hear artifacts, beats, etc, and you may end up convincing yourself you "hear" it - but the source of truth, ultimately, is the spectral analysis. If the measurement says it does not exist, then it does not exist.

It's quite possible the microphone they used for that recording was not capable of capturing that frequency. Or the acoustics of the room prevented it. We can speculate along these lines forever, the only fact that remains true is that there is no significant amount of energy in that sound below approx 60Hz or so.

What you think you "hear" is some kind of artifact.

 

[RayDunzl]

Spectrum of the part where he plays the last few keys starting at 1:50.

Lowest key is active - black

Lowest tone about 65Hz

 

[LeftCoastTim]

Haha, I was recently explaining "Missing Fundamental" in a different thread. These are perfect examples.

Edit: To summarize, pianos and strings do not produce very low frequency (<60Hz) sound, and thus your audio system doesn't need to reproduce them. Organs and drums on the other hand are not the same!

 

[F1308]

Haha, I was recently explaining "Missing Fundamental" in a different thread. These are perfect examples.

Edit: To summarize, pianos and strings do not produce very low frequency (<60Hz) sound, and thus your audio system doesn't need to reproduce them. Organs and drums on the other hand are not the same!

88 keys pianos go down to 27.5 (A0) should they be tuned to equal temperament A4@440 Hz.
Then add what is called STRECHED TUNING, meaning you really tune to a higher pitch the keys far on the right and to lower pitch the ones far on the left, out of equal temperament but more pleasant to the ear, and you go even lower than 27.5 Hz.
The octave D1D2 plays 36/72 Hz and is very much used as a foundation for the bass.
Should we use the Versailles tuning fork, at 392 Hz, then we start with A0 at 24.5 Hz and D1/D2 would play 32/64 Hz.
As for strings, most are mainly designed to play above 65 Hz, no surprises.

What I heard from several essays on the tone is that when you play 100,200,300,400 simultaneously, our ears will tell us we are hearing just 100.
That was the very reason for many well trained, outstanding composers to err BY A WHOLE OCTAVE the tones they were asked to identify. Tartini, a celebrated violinist and theoretical musician, and Henrici being among them.

 

[infinitesymphony]

88 keys pianos go down to 27.5 (A0) should they be tuned to equal temperament A4@440 Hz.
Then add what is called STRECHED TUNING, meaning you really tune to a higher pitch the keys far on the right and to lower pitch the ones far on the left, out of equal temperament but more pleasant to the ear, and you go even lower than 27.5 Hz.

Sure, but those frequencies were not present in the recording you posted. They are not available to hear.

To sum up, you hear bass fundamentals where none exist and elsewhere you claim frequency content above 12 KHz doesn't matter because the "musicality" is mostly in the fundamentals (even if you can't hear them). Okay.

 

[LeftCoastTim]

88 keys pianos go down to 27.5 (A0) should they be tuned to equal temperament A4@440 Hz.
Then add what is called STRECHED TUNING, meaning you really tune to a higher pitch the keys far on the right and to lower pitch the ones far on the left, out of equal temperament but more pleasant to the ear, and you go even lower than 27.5 Hz.
The octave D1D2 plays 36/72 Hz and is very much used as a foundation for the bass.
Should we use the Versailles tuning fork, at 392 Hz, then we start with A0 at 24.5 Hz and D1/D2 would play 32/64 Hz.
As for strings, most are mainly designed to play above 65 Hz, no surprises.

What I heard from several essays on the tone is that when you play 100,200,300,400 simultaneously, our ears will tell us we are hearing just 100.
That was the very reason for many well trained, outstanding composers to err BY A WHOLE OCTAVE the tones they were asked to identify. Tartini, a celebrated violinist and theoretical musician, and Henrici being among them.

Yes, I know about stretch tuning. But my point about "missing fundamental" still stands.

Try this out: Try playing the video segment of the lowest notes on your built-in LAPTOP speakers. You'll hear the 16Hz "buzz" from the Imperial's C0 note. Your laptop is not capable of producing 16Hz. And the piano is not producing it either. The lowest note sound it makes is its 3rd harmonic, which should be about 64Hz, perfectly agreeing with @RayDunzl 's spectrogram. But due to how the perception of sound works, your brain makes up the 16Hz. Heck, human ear can't hear 16Hz.

Human perception is pretty flawed. We see the color yellow when our TV is not capable of making it. Our brain just makes it up. Real yellow exists in nature, obviously, but our brain is doing a lot of signal processing, most of it unconscious.

edit: I think the phenomenon of the missing fundamental should be taught in audiophile school It's the perfect illustration of how flawed our hearing is, how our mind "just makes up" a sound, and only a partial understanding of physics gets one into justifying one's spending habits (e.g. My speakers need to reproduce 16Hz in order to play piano solos).

 

[LeftCoastTim]

Note: This was my first post of this thread, to move the off-topic discussion here. Since then, the mods have moved the off-topic discussion above this post.

I'm going to start with a myth:

The lowest note on a normal grand piano is A0, which has a fundamental frequency of 27.5Hz. The lowest note on a Bosendorfer Imperial (The Big One) is C0, which has a fundamental frequency of 16Hz. Therefore, my audio system must be able to reproduce these extreme low frequencies in order to reproduce piano solo music.

Busting the myth: Try playing this video on your LAPTOP speakers. You will hear the 16Hz buzz of the lowest note, even though 16Hz doesn't exist in the recording itself, and your laptop speakers is not able to reproduce them. What happened? Your brain made it up!

This phenomenon is called the Missing Fundamental, where our brain perceives the fundamental frequency but the sound only consists of the fundamental's harmonics. For the Bosendorfer Imperial and it's C0, one of the members looked at the spectrogram and verified that only 65Hz (3rd harmonic) and above is recorded.

So what does that mean? In order to "perfectly" reproduce a Bosendorfer Imperial, one only has to reproduce 60Hz and above. For a normal piano, one has to reproduce 100Hz (3rd harmonic of A0) and above.

Edit: Video demonstration 1. Video demonstration 2. Video demonstration 3.

Caveat: Church organs do produce sound at the fundamental. So do some drums.

I'm writing this out here so that I can refer to this post whenever I run into this myth again.

 

[pma]

This is trivial, you hear harmonic overtones and not the fundamental. Like with the small speakers. However, to get the full sound impression, you need to hear full range down to fundamental. This covers mechanical vibrations as well (church organ).

 

[LeftCoastTim]

For church organs, where there is sound energy at the fundamental, of course.

But for the piano, the fundamental is missing, like the title of this post says. It's not there.

 

[F1308]

Sure, but those frequencies were not present in the recording you posted. They are not available to hear.

To sum up, you hear bass fundamentals where none exist and elsewhere you claim frequency content above 12 KHz doesn't matter because the "musicality" is mostly in the fundamentals (even if you can't hear them). Okay.

Oh, yes.
Happiness is so easily achieved.
Cancelling that double-subwoofer order.
I hear 10 Hz without me spending a single penny.
And without me fighting any awful standing waves !!!!

 

[JIW]

I have made some spectrograms of the recordings I referred to earlier of the same piece. There seems to be some correlation between the range of fundamental tones in the lowest octave (16-31 Hz) and higher octaves where the harmonics are.

Here is the first recording. A 2 minute segment where the lowest keys are used.

Here is the second recording.

Here is the third recording.

The first two recordings were converted from the YouTube video to WAV-files via https://youtube-converter.online/. The third recording was made from Tidal HiFi using Audio Hijack.

 

[staticV3]

I have made some spectrograms of the recordings I referred to earlier of the same piece. There seems to be some correlation between the range of fundamental tones in the lowest octave (16-31 Hz) and higher octaves where the harmonics are.

Here is the first recording. A 2 minute segment where the lowest keys are used.
View attachment 102500
View attachment 102501


Here is the second recording.
View attachment 102502
View attachment 102503

Here is the third recording.
View attachment 102504
View attachment 102505

The first two recordings were converted from the YouTube video to WAV-files via https://youtube-converter.online/. The third recording was made from Tidal HiFi using Audio Hijack.

If you want the best possible quality from YouTube, then you're better off using the YouTube-dl CLI, since that will return the original file that YouTube is streaming to viewers, instead of reencoding the audio yet again, this time to WAV.
(Though that's just my perfectionism speaking. Not sure if there's anything to gain from that just for spectral analysis)

 

[Francis Vaughan]

We see the color yellow when our TV is not capable of making it. Our brain just makes it up. Real yellow exists in nature, obviously, but our brain is doing a lot of signal processing, most of it unconscious.

This is a complete left field hijack - but that is a slightly flawed idea of how colour perception works. Our eyes intrinsically cannot differentiate pure spectral yellow from a combined set of wavelengths. Hence tri-colour vision theory. Even in nature your eyes cannot differentiate a mixed yellow from a pure yellow. This isn't part of how the brain works. It is already welded into the signals the optic nerve sends back to the brain. The brain "sees" in Lab colour. There is pre-processing of the colour occurring in neurons in the back of the retina, these take the stimulus from the colour sensors and perform a simple convolution, yielding Lab. As a really cool bit of neurophysiology, some time back a group of researchers reverse engineered the retina's function from study of the actual interconnections, and indeed, it exactly matched the previous empirically derived function. Very very cool.

As to the missing fundamental versus real fundamental, this is indeed the ear-brain system processing. Just where it occurs is open to debate. The ear operates with real time feedback from the brain (unlike the eye.) So there are all sorts of places the missing fundamental could come from.

[ETA, below - this is perhaps subsumed by @JIW 's post above, but reaches the same conclusion.]

I found another recording of the Imperial Bösendorfer, and it is perhaps more revealing. The fragment explicitly tried to emphasise the low keys. The spectrogram clearly shows that the recording itself is capable of carrying the low energy, as there is a very deep bass at about 22Hz. Possibly due to general thumping of the instrument. Tellingly there is a clear peak at about 25Hz, which matches a dominant peak at 50Hz. But it is well down in amplitude. I would conclude that there is some real energy at the fundamental, but it is, as one would expect, well down in amplitude compared to the second harmonic. Even a 9 foot grand is going to have significant trouble reaching 25Hz. The sound must come from the sounding board, and the Q of that would need to be so low as to make it floppy, just to get it to get moving much at those frequencies. Something that would wreck it for the rest of the notes. Maybe if someone built an 18 foot grand.

 

[JIW]

This is a complete left field hijack - but that is a slightly flawed idea of how colour perception works. Our eyes intrinsically cannot differentiate pure spectral yellow from a combined set of wavelengths. Hence tri-colour vision theory. Even in nature your eyes cannot differentiate a mixed yellow from a pure yellow. This isn't part of how the brain works. It is already welded into the signals the optic nerve sends back to the brain. The brain "sees" in Lab colour. There is pre-processing of the colour occurring in neurons in the back of the retina, these take the stimulus from the colour sensors and perform a simple convolution, yielding Lab. As a really cool bit of neurophysiology, some time back a group of researchers reverse engineered the retina's function from study of the actual interconnections, and indeed, it exactly matched the previous empirically derived function. Very very cool.

As to the missing fundamental versus real fundamental, this is indeed the ear-brain system processing. Just where it occurs is open to debate. The ear operates with real time feedback from the brain (unlike the eye.) So there are all sorts of places the missing fundamental could come from.

[ETA, below - this is perhaps subsumed by @JIW 's post above, but reaches the same conclusion.]

I found another recording of the Imperial Bösendorfer, and it is perhaps more revealing. The fragment explicitly tried to emphasise the low keys. The spectrogram clearly shows that the recording itself is capable of carrying the low energy, as there is a very deep bass at about 22Hz. Possibly due to general thumping of the instrument. Tellingly there is a clear peak at about 25Hz, which matches a dominant peak at 50Hz. But it is well down in amplitude. I would conclude that there is some real energy at the fundamental, but it is, as one would expect, well down in amplitude compared to the second harmonic. Even a 9 foot grand is going to have significant trouble reaching 25Hz. The sound must come from the sounding board, and the Q of that would need to be so low as to make it floppy, just to get it to get moving much at those frequencies. Something that would wreck it for the rest of the notes. Maybe if someone built an 18 foot grand.

View attachment 102513

I also made some spectrograms of another demonstration where only the low keys are used.

From 3:00 to 3:13 in the video.

Also, the spectrogram you posted looks like the one you posted of the fragment earlier.

 

[Francis Vaughan]

I also made some spectrograms of another demonstration where only the low keys are used.

Yeah, seems pretty much in agreement. Something there, but hardly the dominant part of the sound.

Also, the spectrogram you posted looks like the one you posted of the fragment earlier.

Ugh, yes it is. At least I'm consistent.

 

[dasdoing]

it's not actualy missing, it's just nor audible because too quite (you probably wouldn't hear it even when isolated).
and those deep organ tones are manly overtones, too

 

[LeftCoastTim]

Don't forget that the Bosendorfer Imperial is a 9.5 ft piano. Its soundboard is enormous. A more "typical" piano's A0 would have zero energy at 28Hz.

Here is a video.

 

[MRC01]

This is psychoacoustic, the brain "perceiving" the tone that it has learned must be there. There are some individual differences in how people perceive this. Most do, some don't, and some can choose to hear it either way, similar to an optical illusion that you can make yourself "see" it both ways. Some people believe this makes it less important for speakers to reproduce frequencies below 30-40 Hz. You'll "hear" those low sounds if the speaker misses the fundamental but catches all the harmonics.

The Bosendorfer example from the other thread seems a bit suspicious to me. Just because the fundamental is missing from a recording, doesn't mean that fundamental didn't exist. It could mean it wasn't picked up by the microphone, which could be for any number of reasons. I have a few piano recordings that do show energy at 28 Hz, so I think that fundamental does exist, at least under ideal conditions (room, distance, mic, etc.).

PS: It would be really interesting if we could test newborn infants before they have a chance to "learn" the frequency patterns of natural sounds to detect missing fundamentals. But then, infants can hear while in the womb, and the brain is almost fully formed by the 3rd trimester, so maybe that would not be a definitive test since they could have already learned it by the time they're born.

 

[LeftCoastTim]

Red X is where the fundamental should be:

 

[dasdoing]

I can't hear it, and have a hard time believing that someone can.
do those 4 last pedals realy sound deeper then the above for someone?