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Thanks Prout, that clears up my cognitive dissonance!

Mea Culpa. I must sincerely apologize to PW for my contribution to mis-information. The addition of sine waves, as shown by the video posted by PianoMan51, is indeed accurate. My own tests using the technique shown in the video produce the same results. The octave below (missing fundamental) is clearly heard on my headphones.

My knowledge of the math has now been enhanced. That being said, the graphs and pictures I have posted above are accurate and do represent observed reality.

Here are graphs of the sum of 4 sine waves - 120Hz, 180Hz, 240Hz, and 300Hz, all 60 Hz apart and of decreasing amplitude. The top sine wave shown is 120Hz and the next one is the sum of the four sine waves. This produces a 60Hz 'fundamental', clearly seen and heard. However, if you look at the FFT analysis of the same .wav file, you will see that there is NO ENERGY at 60Hz, only at the 4 sine wave frequencies. You may hear it, but it isn't there.

I looked at some other analyses by the same guy. See the youtube below. I wasn't really interested in how we hear, but rather in the surprising lack of fundamental in the low end of acoustic pianos. When I read closer, I agree with Prout's statement that smaller pianos have little power below 100Hz, and large pianos have little power below 60Hz. This can be seen in the youtube below.

If you can accept that, then we can move into the ideal sound transducer for piano alone. This is where I look at the typical frequency range of decent studio monitors, not 'tiny', but not X-Large. They typically are reasonably flat down to 60Hz. This should be plenty for acoustic piano reproduction. And as Prout states, the emphasis should be on mid-range quality.

My own experience is with the Baldwin SF-10 (7' semi-concert grand), which I chose over the 9' because the added bass resonance interfered with crisp jazz bass lines. I bought this 30 years ago.

For Digital Piano players I can think of two reasons why they might want to buy a sub-woofer. The first is that their current transducers don't go to 60Hz. The second, and more unpleasant, is they don't want their DP to like a real piano, but rather want their DP to sound like some idealized piano in their dreams. Everyone has their own tastes.

For the first case, get a good set of studio monitors, and set them up as near-field, in the traditional triangle pattern. You will love this.

For the second case, go wild baby. It's your sound and you can do what you want. But as a pianist for the last 55 years, I can tell you that even a 2 or 3 dB of bass boost in a recorded piano sounds unnatural to me.

The problem of the missing fundamental is not related to the length of the string. Because of the copper wire, a short string does vibrate at a low frequency.

However, the mechanical impedance of the soundtable is high at a low frequency, then, the lower partials doesn't produce a lot of sound.

Bigger grands have a larger soundboard and are able to produce lower partials.

"Physics of the piano" is a interesting book to read.

The video PianoMan51 posts just above is interesting in that it shows how little energy is produced below about 100Hz. The presenter tries to argue that the idea of superior clarity in European pianos over other pianos is inconclusive. He does not mention, however, the differences in the use of muting felt in the area between the bridge and the hitch pins.

On many pianos the felt extends over every single string. This reduces non-tuned sound from competing with the tuned strings. On many other pianos the felt extends only over a portion of the lower keys and is left off in the treble. On these pianos the unmuted portion of the strings vibrate and produce noise. In the Steinway, for example, the noise is untuned and adds brilliance to the sound, not clarity. On other pianos, the unmuted portion is carefully tuned to either the fundamental, fifth above, octave above or 12th above. This adds clarity to the sound.

It is my opinion that both either muting all the non-speaking portions of the strings or tuning them adds to clarity. I think is actually seen in the video. The S&S is shown to be the noisiest.

I don't understand what part of your earlier statement you're taking back.If you linearly add pure sine waves at frequencies 2f, 3f, 4f, 5f, etc, the result contains NO energy at 1f. This is what I think you said earlier, and it remains true. So what exactly are you retracting?

If one does find some output energy at 1f, it's because either:
(a) the original sine wave sources were impure, containing some energy at 1f
or
(b) the mixing was non-linear, and the non-linear mixer products have produced harmonic and intermodulation distortion, some of which will be at 1f

But I don't think you intended to introduce these anomalies into the discussion. So wuzzup with the retraction?

@MacMacMac : we could add :

(c) we can ear missing frequency because of an ear illusion due to the way our brain process the decomposed sound.

We have an article about the missing frequency : https://en.wikipedia.org/wiki/Missing_fundamental

I see. No, first of all I don't accept that. Even though fundamental is significantly weaker (on large grands it is -25 db for A0 compared to strongest sound component according to book "Acoustics and the performance of music", p. 71), it is still there. And such low frequencies are very perceivable, even on tactile level.

Besides it is very arguable question if near-field monitors can match the sound of subwoofer (even if they are flat down to 60Hz). Many people will tell that it's a completely different experience, maybe because of different sound diffusion in space.

Yes, that's the whole point:This is missing from the charts given earlier. The charts can only show physical waveforms. They cannot show perceptual results.

I think you're onto the right point.My take is this: You don't lose much if your speaker response drops off below, say, 60 Hz.

But you do lose something. Not much, but something.

Still, I wouldn't add a subwoofer. Big bass doesn't move me.
These are pianos, not bass guitars.

The graphs in my first post showed sine waves at multiples of 130Hz, which explains why the resulting waveform had no peaks at 65Hz. My lastest graphs show sines at multiples of 60Hz starting at 120Hz, which obviously converge on a peak at 60Hz, which is what the video shows, even though there is still no energy at the fundamental in spite of there being peaks at the fundamental.

You, Frédéric L and I are all saying that there is no energy at the fundamental produced simply by the addition of sine waves that are multiples of the fundamental. What we hear is the result of a non-linear system, either in our ears or in the audio equipment or both.

That being said, my piano produces A0 24dB down from the peak partial, as can be seen on my posted image, so there is still audible, if weak, energy at the fundamental. I can sense it, but don't think a sub is necessary to reproduce it on a DP.

Here's where I'm losing you. I don't understand this "convergence" ...

I don't think the illusion is about a non-linearity of the ear. An inner hair cell of our ear has no reason to vibrate at 1f if we have 2f, 3f... frequencies. I think it is about the way our neurons are connected. It is expected that when we ear a 3f 4f 5f frequencies, the "1f" neuron is connected to the 3f 4f 5f neurons and the addition of all the signals stress the first neuron. I think it is expected that related frequencies neurons are connected in order to sum-up what we ear.

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When we read https://nanohub.org/resources/18884/download/2013.06.19-Giordano-REU.pdf page 17... we can't recognise the -24dB fundamental at 1f. We only found a -40dB noise at 1f. Here, the frequency is realy missing.

Poor choice of words on my part. The video shows, as each succeeding partial is added to the previous partials, the peak amplitude of the 'missing fundamental' increases as did my own tests.

My reading over the past 5 years on the acoustic processes of the ear shows that there are a number of theories regarding how we perceive pitch. The most prominant one is the one that applies here. It states that pitches (that is Cs, Ds, and so on, not frequency) are binned together in the the cochlea. Thus the same group of stereocilia perceive all Cs - C1, C2, C3 and so on. We use other cues to differentiate if the pitch seems higher or lower. This is a theory mind you, but, based on it, it seems resonable that non-linearities in the cochlea give rise to the missing fundamental.

A simple test of this is to produce a 500Hz sine wave in your left ear and a 400Hz sine wave in your right ear using audacity and headphones. You will clearly hear a 500Hz sound in the left ear and a 400Hz sound in your right ear, and nothing else, since the pitches are binned separately. Now mix and render the two tracks to a new combined track and listen again. You will now clearly hear both pitches in both ears PLUS a new third pitch at 300Hz. You will not hear any other pitches. We should hear 100Hz but we don't. However, if the pitches are binned, then 300Hz is simply binned along with 100, 200, 300, 400, and 500 Hz. Perhaps it is its proximity to the other pitches that makes it audible.

A stereocillia is just a plain mechanical oscillator. It should act as narrow bandpass filter. I don't think it can vibrate at different frequencies. However, a C3 stereocillia will be vibrate when you play a C2 note since this one has a C3 partial. Is this what you are talking about ?

I have read that stereocillia are ordered by frequencies, never read something about group of related stereocillia. Have you got some pointers.

Here is one that will start you off in your research.

The traditional theory is called 'Place code theory' formulated by Helmholtz. It posits that the frequency is mapped in the cochlea by position. The Temporal Code theory is more modern and refutes the Place code theory. It posits that frequency is not mapped to cochlear position but to neuronal firing rates. This allows for mode locking and non-linear effects at the level of the neuron.

I havn't read that stereocillias are binned together : we have here a distribution with ordered frequencies. We have a plot with 3 neurons : but the text doesn't tell the relationship between these frequencies.

Perhaps we should understand by binned : some neuron associated to different frequencies are connected to a focal neuron which detect a given pitch.

Here is an article on the psychophysical study of pitch perception. It describes in detail the various models, including the spectral and temporal theories and an in depth discussion of the 'missing fundamental' phenomenon, including non-linear processing in the auditory complex. Scan down to 'A Historical Perspective' to read about it.

It is interesting that all the theories can explain portions of pitch perception, but none capture our complete ability to discrimate pitch, especially when the detectable pitch change is less than the neural tuning rate. Example given - at 250Hz we easily hear a 1Hz change in pitch, but the tuning of neural fibres is not narrow enough to account for the small difference. All this implies that we cannot think of the stereocilia as simple mechanical vibrators that respond to a given frequency which is the frequency we perceive. If that were the case, there would have to be an essentially unlimited number of fibres in the cochlea, which defies the basic laws of physics regarding volume.

The previous article shows that the band pass filter is not so narrow. Then with a given frequency, multiple stereocillias will vibrate. I suppose that the brain would have to compute a "maximum" in order to ear the right pitch. A "simple mechanical vibrator" has a given bandpass, then we mustn't have an infinity of them to hear all frequencies.

This would explain some masking used in MP3 compression : if a given frequency makes vibrate neighbors stereocillias, a sound at a near frequency won't produce much effect (the corresponding stereocillias already vibrate).