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The sound of the piano is something very complex. There are various elements to consider such as action, hammer, soundboard, sympathetic resonance, tuning age and even room reverb.
Some companies (such as Yamaha, Roland, Viscount and Modartt) have succeeded in realizing a realistic piano tone using physical modeling synthesis.

How did they achieve that amazing result? Where can I explore or even just get started with physical modeling? Searching online, I could only find incomplete scientific papers. Thanks in advance.

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    $\begingroup$ Unless you're able to dig up any relevant patents, you're probably out of luck. All semi-convincing piano models are proprietary. Maybe just do your own research and start from scratch with a simple model that you then make more realistic step by step? A simple model would be one that just simulates the two strings per hammer, maybe using commutative excitation synthesis. Then add stiffness to the strings and coupling to the soundboard. And then try to understand what's still missing and model that. $\endgroup$
    – Jazzmaniac
    Nov 30, 2022 at 19:36
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    $\begingroup$ It's not physical modeling, but if memory is cheap, you can sample all 88 notes of a piano with various key velocities, from $pp$ to $fff$. You can even have multiple samples of the same note, same key velocity, and randomize which one is selected to playback. Of course the PCM sample playback has to be truncated gracefully when the key is released (MIDI Note-Off). These guys have an extensive library of sampled piano notes for PCM sample playback. I have used them in my research to extract wavetables out of. $\endgroup$
    – robert bristow-johnson
    Nov 30, 2022 at 22:11
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    $\begingroup$ @robertbristow-johnson unfortunately, the ringing of not struck but open strings is important for realistic piano sound. $\endgroup$
    – Dmitri Urbanowicz
    Dec 1, 2022 at 9:26
  • $\begingroup$ @robertbristow-johnson Unfortunately, sampling 22 keys with 5 or 6 velocity layers only one time may take up even 10 GB, imagine sampling 88 keys with tens of velocity layers and more than one time... It would take up Terabytes of memory for just one instrument. And you can't custom parameters like string lenght, soundboard impendance, string types, hammer hardness... it would be impossible to achieve effects such as sympathetic resonance or damper pedal harmonics and you can't get extended keyboard (for instance, instead of A0 to C8, C-1 to G9 like MIDI). Physical modeling can do all of this. $\endgroup$
    – Alessandro Costantino
    Dec 1, 2022 at 17:36

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Physical Audio Signal Processing by Julius Smith is probably something you'd be interested in! The full book is available online, and there's a whole chapter devoted to piano synthesis.

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  • $\begingroup$ Latest approaches likely mix in machine learning, but one has enough on the plate with piano physics... $\endgroup$
    – OverLordGoldDragon
    Nov 30, 2022 at 20:14
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    $\begingroup$ @OverLordGoldDragon: No. Latest approach is sampling. It works well and is the same for almost all instruments. $\endgroup$
    – Hilmar
    Dec 1, 2022 at 4:02
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    $\begingroup$ @Hilmar Sampling doesn't cut it for a piano where the vibration patterns are different depending on which combinations of keys have the hammer resting on the string or not. One key will have a different sound depending on if the adjacent keys are held or not. $\endgroup$
    – pipe
    Dec 1, 2022 at 8:08
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    $\begingroup$ @Hilmar, as someone who has been working in the audio industry for nearly three decades I can tell you that sampling has never been the latest approach but rather the final resort. There are very few instruments that really work well with sampling. As soon as you have interactions between the notes or an instrument that can be played with more expression than an initial velocity, sampling stops working well. Sampled pianos have always been a compromise and for a long time the only feasible approach. This only changed with the advent of convincing sound models (both physical and data modelled). $\endgroup$
    – Jazzmaniac
    Dec 2, 2022 at 18:46

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