However it happens, the acoustic seems to have an almost infinite amount of tonal breadth - I believe the pianist has so much more control over the process in an acoustic - I have also heard the only thing we can control is the velocity of the hammer but there is so much more that goes into the tone production as Ithaca has mentioned, the quality of the hammer, the type of felt used, how the hammer is shaped, the strings etc. - I can get so much more variety of tone from an acoustic - and I have much better control of say, playing the left hand part much quieter than the right hand part (if needed). I have limited control of this on my P515. I always wondered about which DP or midi controller would have a better action so I can see if it's just limited to my P515. I am curious to try Iternabe's test on my P515 when I get home tonight.
Truth and myth on tone quality?
This has been debated endlessly at PW and other places. Many pianists (even very good ones) are adamant that there is a difference in tone quality independently of the velocity of the key but physics says that is simply not the case. While you can use body movements to help you make the sound you want the explanations given by many pianists amount to basically voodoo.
iternabe the “harshness” is not from the note itself, but its relations with the notes that’s surrounding it.
Yes, it's all contextual. For another example, "legato" is not only about continuity of sound but about continuity of the musical line. You can play completely detached and without pedal and it can still sound more legato than if you play with the pedal down but without good phrasing.
Ithaca I think (totally not sure) that you control the hammer velocity and the damper behavior via the impulse that you apply at the keys.
I agree. This is also something I have been pondering over. I think I can reconcile the impulse and the velocity.
When I said velocity, I am think the velocity of the hammer at the moment it leaves the escapement. From then on, it is out of the pianist's control. This velocity is the result of the impulse from the finger, or more accurately from the whole playing apparatus (finger, knuckles, wrist, arm, shoulder, back).
To understand the impulse, I'd like to think its polar opposite. Think an example of an extremely heavy moving object collide with an extremely very light stationary object - the light object will accelerate almost instantly to the heavy object's velocity. Make that analogy with a heavy arm, falling at a controlled constant velocity from high above the keys, and striking with steely fingers - that will move the key instantly to match the fingers velocity. That would be super accurate control of the velocity at the with the hammer strikes. Only that we humans, or most of us, cannot do it that way. The key need to gradually accelerate from stationary to a certain velocity at the let-off point. This is the impulse. And this gradual acceleration can also vary a lot based on how we perform the impulse, and even where we start the impulse from. Although the goal is to control the velocity at let-off, I doubt any human is capable of sensing speed at that precise moment. We have to control the velocity at let-off indirectly by controlling how we execute the impulse. So the real question, again, is what kind of motion gives us the best control of the impulse and the resulting hammer velocity.
Just to clarify, there are many things that go into tone production and a good pianist can certainly control it to an extent using the pedals. I often use partial pedaling or add a touch of una corda as additional effects that I control mostly by ear but that is not the main point of contention. The controversial point is only whether there exists a separate dimension of "harsh" independent of volume for a single isolated note and physics says there isn't. Harshness is an illusion of playing something out of place in the musical context.
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Ithaca
I don't get your point about impulse. Impulse is just the time rate of change of momentum and since the system is contant mass it means it's the rate of change of the acceleration. In the end it all translates to velocity.
Ithaca For this kind of problem I would want good empirical data.
You would have to do it with a machine of some kind because there is no way a human can control precisely how much acceleration a key has.
I am not denying that different gestures help you control the sound. There are many subtleties of your body movements that can help shape the resulting sound in relation to everything else you hear. What I'm saying is just that all of that sound shaping is a contextual illusion not the result of some undefined tonal quality of single notes.
BartK While you can use body movements to help you make the sound you want the explanations given by many pianists amount to basically voodoo.
If the method works, who cares if the explanation is voodoo, right? 
Well, I was thinking what if a method work on many but fails on a few. It is then that having voodoo explanation can be of little help. Because if the reason of why something works is incorrect, it may lead one to keep trying ineffective solution. Having the explanation align with the underlying physics will have a much better chance of finding the right solution.
Ithaca I think people who say “physics says” aren’t modeling the mechanics accurately enough. An idealized model will likely fail to explain/predict some real-world behaviors.
I am in agreement with you here. Almost all models are approximation. We like to start simple, and think that it's not necessary to include things that don't affect the result meaningfully or perceptibly. But a critical part of scientific training is as soon as models (theory) and reality (experimental results) deviate, more rigorous investigation is warranted, and minds/models/theory should be ready to be changed.
Good luck obtaining grants for this kind of research on piano sound, though! 
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iternabe I am in agreement with you here. Almost all models are approximation. We like to start simple, and think that it's not necessary to include things that don't affect the result meaningfully or perceptibly. But a critical part of scientific training is as soon as models (theory) and reality (experimental results) deviate, more rigorous investigation is warranted, and minds/models/theory should be ready to be changed.
Yeah, OK, but if you want to propose a different model you have to at least present an argument of why the simple model fails and a tentative explanation. I'm not getting what exactly it is about impulse that changes the way the hammer hits the string. Some people have proposed that the elasticity of the hammer stores some angular momentum but I'm very sceptical that this effect is non-negligible compared to the total momentum of the hammer. I'm too lazy to do actual calculations but I'm open to arguments (with math) if you think you can convince me otherwise.
iternabe Good luck obtaining grants for this kind of research on piano sound, though!
That might be a good challenge for the science Youtubers. I don't know if you watched it but a few years ago there was a wager between Steve Mould and ElectroBOOM where each attempted to give an explanation of the Mould effect when dropping a chain of beads from a height. There were a lot of experiments and back and forth arguments and other Youtubers like Veritassium also put in their 2 cents. It was very entertaining to watch. Maybe someone is up for a piano tone challenge. 
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A lot of comments about the beautiful, rich and expansive tonal color palette of acoustic piano are making me longing for one! Weather, tuning and regulation, and just the analog and infinite grades of motion of the piano action is just fascinating. Though, some of these factors are affecting the tonal quality over time, or after mechanical adjustment.
I was contemplating more along the line of what a pianist can do with their interaction with the keys at a particular moment to affect the tonal quality.
In my opinion it’s more about controlling the hammer better up to the point of no return. Since that’s all we can really do. I found an interesting university study that seems to touch on this discussion. It’s lengthy and I just started reading it:
https://public.websites.umich.edu/~brentg/Publications/Thesis/Chapter2.pdf
danno858 that's interesting. It seems to explain @iternabe's observation with human perception:
By carefully governing the timing overlap of notes as they follow one another, the musician can evoke a certain percept in listeners which will not be labeled in terms of timing at all; it will be labeled by the ear and auditory perceptual centers in terms of the frequency domain, that is, timbre. Although it is not, in physical terms, independent control of timbre and intensity, it is perceived (and labeled) as such.
I was going to suggest it might be an auditory illusion. That article seems to go to similar direction. But it also suggest that the pianist have some control over the perception. I will read the rest of it now 
Human brain is complicated. I suspect brains of different individuals process the same auditory input differently. Not only that, the input data is different too. Because our "hardware" is different. We all have different physical ears. Some people even have different left & right ear canals/structures.
danno858 I found an interesting university study that seems to touch on this discussion. It’s lengthy and I just started reading it:
https://public.websites.umich.edu/brentg/Publications/Thesis/Chapter2.pdf
@danno858 I cannot thank you enough for find this chapter, as it is truly enlightening. I have yet to read the second half of the chapter on synthesizer. This paragraph has already made me want to get the entire book.
More thoughts of the moment as I am reading the chapter posted by @danno858:
The author pointed out a major difference between synthesizer/digital piano’s method of determining key velocity vs acoustic piano. In acoustic piano, the velocity that matters is at the instant the hammer escape from the action. In digital piano, the velocity is the average velocity of the key passing through the distance between two sensors. Needless to day, the same gesture (i.e. accelerating the key from stationary to motion at the let-off point) can produce very different velocity value between acoustic action and digital action. Moreover, effect of gestures that produces non-linear acceleration of the key/action can get lost by DP actions simplified averaging algorithm.
This also makes wonder about the effectiveness of the best hybrid actions, like Kawai’s NV10S. If its sensors is designed to detected the velocity of the traveling hammer before it strike the string, then it is a much better approximation of acoustic action in that the velocity detected is almost the instant velocity of the hammer escaping the action. I still say almost because if the velocity sensing mechanism employs a pair of sensors placed apart, then it is still an average speed over distance, which might still be a bit off from the final string striking velocity if the hammer does not maintain constant speed due to friction or gravity.
danno858 I found an interesting university study that seems to touch on this discussion. It’s lengthy and I just started reading it:
https://public.websites.umich.edu/brentg/Publications/Thesis/Chapter2.pdf
Here is the link to the entire thesis by R. Brent Gillespie: Haptic Display of Systems with Changing Kinematic Constraints: The Virtual Piano Action
I found another published study on this topic:
The study concludes: Our results suggest that only some musicians are able to distinguish between a struck and a pressed touch using the touch noises as cue, especially the finger–key noise that characterises a struck attack, whereas others could not tell any difference. Without those touch noises none of them could tell a difference anymore. When they could not hear the touch differences, they tend to rate louder tones as being struck, and soft tones as being pressed.
Ithaca Randomly pulling stuff out of my hat (sorry, I'm tired - 3 nights of very short sleep) - think about two mechanisms which can make contact at a single point, each of which has a squishy balloon-like or somewhat elastic section somewhere inside. (Say, the finger pad and compressible joints in one mechanism, and a soft hammer + leather accoutrements in the other.) Those soft/elastic sections absorb and then dissipate energy over time, and probably in a non-linear way. So if you whack the first mechanism at the far end in one way, the way that force over time gets transmitted through the first mechanism and then through the second mechanism to produce a hammer taking vertical flight isn't so obvious. If you've modeled systems with multiple springs with different K constants, that might give you a better sense of what you're looking at. But even then, typically when people look at these systems, they're not looking at the time factor, unless they're studying something with a constant-period oscillation, so again, in steady state.
It's certainly interesting to look at the mechanics of your hand and finger but that's not what the contention is about.
There is a cut off point where the escapement mechanism throws the hammer free like a projectile. It doesn't matter how the key action was moved before the escapement point. After this point the hammer is in free fall not touching anything until it hits the string. The only thing that can physically affect the sound at this point is the momentum of the hammer. Unless you can show that it's possible for the action (and all that comes before the escapement) to impart some angular momentum such that the hammer hits the string at a different angle that is controllable by the player then there isn't any physical way to produce a sound that has a different quality at the same volume. If the piano had a mechanism where there is a physical connection between the player and the strings at the point of sound production, like in the harpsichord or clavichord, then I would believe you. But the piano has this air gap which disconects the cause from the effect.
Ithaca I propose that what the musician hears as control over timbre is effected by careful control over
timing. For a particular tone, the pianist is indeed able to select only two parameters beyond pitch:
the hammer strike speed and the hammer strike time (although the time of tone damping is also
important). I propose that by selecting these two parameters, a pianist can produce a percept in
his or her listener of timbre being controlled independently of intensity
That doesn't make sense. How does the player control the strike time? The travel distance of the hammer is constant so the time it takes for the hammer to travel depends on the initial speed. How do you get two independent variables?
