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Recording is used to capture samples of the audio at regular intervals. Once captured, the set of them is a digital form of the waveform and they form inputs to a mathematical model to recreate the waveform. That’s why a DAC (whether in the keyboard, in a computer soundcard, or standalone) is needed to produce the analog waveforms that form the sound.

In time, everything will be modeled.

Peace
Bruce in Philly

Yes -- I should have said:

. . . A sample-based VST uses data that's as close to a "real recording" as it can be, while using digital storage.

That’s not actually true. The tone is not sampled/recorded until it does out, nor at all possible dynamic levels, not with all possible pedal positions. Both sampled pianos and modeled pianos are mathematical models used to approximate the actual waveform. That the parameters of the model are determined empirically by sampling some of but not all of the possible waveforms created by an actual is incidental to the mathematical nature of the model. Physical modeling likely also uses sampling in the process of determining its model parameters, but the model is further derived using other techniques. Both are mathematical models to approximate actual waveforms.

We must not confuse between "sound modeling" and "sound synthesis".
Sampling and modeling are both two forms of sound synthesis, but they are not the same thing.

In sample-based engine, as the word says, the main data for the sound generation comes from the "sample" itself. For "sample" we don't refer to a single waveform of a periodic signal (that would be Wavetable Synthesis or Digital Waveguide Synthesis), but to a long sequence of waveforms (usually, 1-2 seconds or more) recorded from a real sound. Then you can do some "processing" to the sample to add resonances and other effects, or to get an arbitrarily longer duration (engines with looped samples) if the hardware has no much memory to store the full original sample. Looped samples make the notes more artificial sounding, because to make the loop smooth sounding you have to give up to all those little details that were present in the original full length sample. When on your DP the sound enters in its looped part, you feel it become boring and repetitive. That's the loop...

In "pure" modeling engines the stored data are parameters that should relate to some physical properties from the real world. Then some equations should get these parameters together and generate a sequence of amplitude values (at a certain playback rate) as the output. Unfortunately the equations to simulate the generation of a piano sound in an accurate way are too demanding for current hardware, so actual modeling engines do many approximations. But the distinction between sampling engines and modeling engines should be clear: in the first case, you start from the playback of a sample at a certain frequency and then make some post-processing to add effects (and some manufacturers call 'modeling' even this). In the second case you start from a set of parameters to generate (from scratch) and modify the waveform in real-time from the start to the end of the note. This means no looped boring sounds, more timbral variations based on velocity, and you can change the character of the instrument in a much deeper way too, so that you can emulate the sound from different acoustic piano brands and types (or even create an instrument that doesn't exist!). Of course, all of that requires a very good piano modeling engine.

Once the recorded sound is sampled, you have a parametrized representation of the waveform. In pure sampling, that is a time domain model. If you transform it with a discrete Fourier transform, you will get a parametrized model in the frequency domain which takes up much less space to store, and may even be stored as constant parameters in the rendering software. As a result, the label physical modeling is more easily applied to the latter.

Besides, it may turn out that sympathetic resonance is just affecting some of the sine waves coefficients (matching partials), so it’s trivial to implement resonances and also call this modeling. And I’m just hypothesizing.

1) If you do a DFT (Discrete Fourier Transform) to transform a digital signal from the time domain to the frequency domain, the output array size will be exactly the same. To store it in less space you have to give up some frequencies, maybe because they have a so low amplitude that the human ear should not able to perceive the difference... But that's a (lossy) compression algorithm, unrelated to the DFT itself.

2) A sampled piano note is just a "digital" (discrete) representation of the original analogic sound.
As I said before, when I talk about parameters for a piano model, I'm not talking about pure mathematical data like amplitude, frequency or phase, but some values directly related to the physical thing, like string length, string thickness, strike position of the hammer on the strings, number of strings for each note, hammer hardness, soundboard size, etc... You cannot easily relate these physic parameters to the amplitude values (or amplitude+phase values in frequency domain) of a sampled piano note... Of course, you can make some little variations in post-processing, but there are limits on what you could achieve just with samples playback+post-processing.

That would be a way of modeling sympathetic resonances. But you have to consider that from a computational point of view, changing continuously the amplitudes of more than 2-3 overtones in a played sample is more costly than just playing another overlapped note with a different envelope. And consider that the lower notes often have more than 70 overtones (for a single note)...

So, I think that sympathetic resonances in most (if not all) digital pianos with sample-based engines are emulated by playing just the looped part of a sympathetic note at the lowest velocity timbre, but with a different volume envelope.

But we’re discussing modeling and the way of modeling with summing sine waves. So what I’m saying is the resonances can be implemented by modifying the overtones of an already sounding tone which would be the honest of way of doing this and calling it “modeling”. Otherwise there’s no difference between modeling and sampling.

Maybe someday Roland will spill the beans and tell us exactly what is inside my V-Grand. According to Roland, nothing in my V-Grand was sampled. Until one of these companies (Roland, Pianoteq, ???) ponies up with the goods, we can only guess.

Tony

You are correct. I was thinking it would be half the size since it only needs to represent frequencies up to the Nyquist frequency (half the sample rate) but you are correct in stating that it will be the same size.

From what you say I think you are assuming that the only way to generate a sound from scratch (when you don't play it from a recorded sample) is summing sine waves. That is the old and classic "Additive Synthesis" approach. But in the last decades, researchers developed many other approaches to the sound synthesis, like:

- subtractive synthesis
- frequency modulation (FM) synthesis
- wavetable synthesis
- digital waveguide synthesis

and many others...

Pure additive synthesis is not much attractive to developers, because it's very demanding. In the past it was used for the very first synthesizers, with something like 2 or 3 oscillators that you could combine. The sound resulted in a "videogame"-like timbre because real sounds have many many overtones. For example, if a real piano note in the lower register has ~70 overtones, with additive synthesis you should generate 70 sine waves and then sum them. This is overkill, considering that with the same computational power required to generate just a single overtone (sine wave) you can play a full-detailed sample recorded from a real piano. Of course with additive synthesis is very simple to change the way an overtone affects the sound (you have just to change the parameters of the oscillator that generates that particular sine wave), but even with current hardware you would have a max polyphony of just a few notes. And I insist that this would not be physical piano modeling, because you are not altering parameters directly related to physical properties.

Of course, if a digital piano manufacturer would implement a sample-based engine but with options to change "all" those physical variables that you can change, for example, in the Pro version of Pianoteq, then in a sense this would be a form of physical piano modeling too. For now, in DP sample-based engines, we can change just some little things, typically the incidence of some added resonances on the output waveform, which are very approximative solutions to a piano modeling problem. I don't think you can go much further than that with sample-based piano engines without using some real physical modeling... What I mean is that you can have a beautiful and rich sampled piano sound, but if you want to have the option to deeply change that piano sound timbre (in a realistic way) as you wish, physical piano modeling is the way to go. With sample-based engines I cannot make a Yamaha piano sound timbre sound like a Steinway timbre (or the other way around). With physical piano modeling, theoretically, I can generate the timbres of all the brands of pianos, because I'm not strongly tied to the samples of some recorded piano notes. And, of course, another advantage of physical piano modeling is that it doesn't require tons of gigabytes of samples and fast storage devices like an SSD.

No, that's not what I am assuming. It's you who are assuming I'm assuming It's either my bad English or you not reading me well but it's getting regular now that I need to reply to you to clarify what I wanted to say...

I know VERY well all types of synthesis and we've discussed that in other threads, with your participation as well. No need to reiterate something again from scratch. We've commented on how the first (and only) Pianoteq patent is about summing sine waves. And we've commented that in HUGE details and you also participated in that thread. My comment about summing sine waves is related to that notion and how resonances fit nicely in that approach. I'm NOT assuming that's the only way of recreating piano sound.

P.S. It may sound like a bitter comment, but sometimes it won't hurt to remember forum aliases and try to remember what has been discussed with whom. Otherwise it's getting tedious to always start from 101 without any particular need for that.

Gee, all I do is press my fingers onto the keys ... and sound comes out! I'm glad I don't have to sweat the tech.

Sorry if I misunderstood (and you are right I can be a little forgetful of previous threads), but I think I read somewhere on this forum that the Pianoteq patent is old and the developers now do things in a different way (and of course they don't tell us "how"). Anyway I was talking about physical piano modeling more in a general way, not necessarily related to exactly what Pianoteq does (that of course we don't know!).

What are you doing here in this utter boring and useless thread!? Sit in front of the piano and make some good practice, instead!

Jokes aside, I think that when there is much passion about something (i.e. a musical instrument, a car, a motorbike, a computer, etc.) and everything that revolves around it, we would like to learn everything we can on that subject.

@CyberGene:
I tried with Pianoteq connected to my DP. If I play just a silent note, the polyphony counter will increase of 1 unit even if you don't hear nothing. Now if you keep that key pressed and play, for example, 3 other notes, you will notice that the polyphony will increase to 4 slots and the silent note begins to resonate (sympathetic resonance). Now, if you release just the other 3 notes, the polyphony will back to 1 and you will hear just the sympathetic resonance.
So, I think that Pianoteq reserves 1 polyphony slot for each note that could "eventually" play in sympathetic resonance. For example, if you play 3 silent notes, the polyphony slots will increase of 3 units. This suggest me that for sympathetic resonances (with right pedal "not" depressed), Pianoteq doesn't modify the waveform of the played (not silent) notes, but it generates new notes by using the reserved polyphony slots related to the keys you keep pressed.

^ That’s an interesting discovery. It means only the resonances can be used separately as we discussed in another thread, say use the samples from a sample-based VST and the resonances from Pianoteq. But they need to expose such a functionality I guess.

One thing I have been doing as of late is in mainstage. I layered Pianoteq Steinway D with the Grand Piano sample from Logic/Mainstage. I tweaked a few of the Pianoteq settings and they are at a slightly lower level than the Apple sample. It sounds great it really does sound more real than either plugin alone. I think that is what I'm hearing more of the sample but the resonances from Pianoteq are somehow definately helping here.