Isn’t it just a synthesiser that emulates a piano?

Well, you finally have an answer to your question.

I think a good way to appreciate the distinction might be to consider a good old analog synthesiser, then compare a traditional emulation of that synth in software, and an emulation created using physical modeling.

The analog synth creates waveforms by generating specific voltages in appropriate physical electronic circuitry, so if a sine wave is desired, an oscillator using transistors, or maybe an Integrated Circuit, passive components like resistors and capacitors etc creates that waveform.

A traditional emulation of this in software to create a software synth would create a sine wave using the mathematical formula for a sine wave, inserting values to derive the necessary frequency and amplitude. Or an alternative would be a wavetable synthesiser, which draws the basic waveform from a digitally recorded sample of that waveshape.

An emulation using physical modeling would use equations that would define the operation of the components in the circuit that generates the sine wave in the analog synth. So all the calculations would be done to represent the interaction of active components like transistors, the frequency and phase relationships in resistor/capacitor (RC) networks, the changing resistance of potentiometers to set frequency and amplitude. The aim of this would be to achieve a better emulation by hopefully capturing the uniqueness of the sound associated with the way the specific circuitry in the analog synth generates that sound.

So both emulations would be a software synthesiser, but in order to know what means they're using to achieve that, and thus to have an idea what differences are likely in the results, we need to know one is a wavetable synth or whatever, and the other is Physically Modeled.

So I'll call it a Piano Emulator then. Nice and simple.

When I was just a teenager, music students from Watts and Venice in California boarded buses which were bound to arrive at the University of the Pacific in Stockton, California and performed together there.

I got a recording of a local product that was just a local girl:

Portugese Love, Teena Marie

And don't explode many floors of a building...
Despite covid19 be killing nearly twice of that per day.

Gaston wrote:

The big difference with a flight simulator is that even  if you make a huge crap with a real piano you don't risk your life

Just as at a digital piano any pianist reasonably can assume per desktop PIANOTEQ probably involves and takes into some account many more computations comprising algebraic formulas and mathematical equations (more than widely marketed sample based softwares), probably it better exploits computer resources meaning today computer technology and more efficiency also.

However from a computer scientist software conceivably is a lot simpler to me to explain scientifically than the events of 911!

Certainly, a day might eventually come when you’ll have modelled pianos at your desktop and with them you’re finally able to discern virtual software modelling and what it’s like, easily, from just any old shoddy appearing computer 3D graphics video as now a poor illustration taken from yesterday.  In which and in hindsight the poorly attempted model depiction of the once somewhat tall structure only depicted it collapsing alarmingly fast.  Sure, as one such production is unpleasant it equally is unimpressive only by today standards, since in reality I myself can now take any freebie such as DazStudio and convincingly replicate truly similar effects, actually in 3D with more photorealistic detail given to shadows.  But, mine compared to it will look a whole lot less shady, after applied Adobe After Effects...

Sure too, a lot of correlations one vividly may draw from the virus COVID-19 and the viral videos of 911 events, remarkably those of 2001 oddly enough.  To me they’re as plain as the nose on one’s face!  Perplexing perhaps is suddenly guys who deal with ones (1s) and zeros (0s) day to day refuse to arrive at a sum of two (2) plus two (2) in 2021!

The analog synth creates waveforms by generating specific voltages in appropriate physical electronic circuitry, so if a sine wave is desired, an oscillator using transistors, or maybe an Integrated Circuit, passive components like resistors and capacitors etc creates that waveform.

A traditional emulation of this in software to create a software synth would create a sine wave using the mathematical formula for a sine wave, inserting values to derive the necessary frequency and amplitude. Or an alternative would be a wavetable synthesiser, which draws the basic waveform from a digitally recorded sample of that waveshape.

An emulation using physical modeling would use equations that would define the operation of the components in the circuit that generates the sine wave in the analog synth. So all the calculations would be done to represent the interaction of active components like transistors, the frequency and phase relationships in resistor/capacitor (RC) networks, the changing resistance of potentiometers to set frequency and amplitude. The aim of this would be to achieve a better emulation by hopefully capturing the uniqueness of the sound associated with the way the specific circuitry in the analog synth generates that sound.

So both emulations would be a software synthesiser, but in order to know what means they're using to achieve that, and thus to have an idea what differences are likely in the results, we need to know one is a wavetable synth or whatever, and the other is Physically Modeled.

The big difference with a flight simulator is that even  if you make a huge crap with a real piano you don't risk your life

Not wishing to derail the thread, but this reminds me of one of the page of jokes at a small airfield nearby:

A good landing is one where you can walk away.  A great landing is when the 'plane can be used again.

Now, jokes aside, what about the first sampler, the Mellotron ?

https://www.youtube.com/watch?v=N07-YAKtRAw

https://www.youtube.com/watch?v=tTKPW92ndUA

https://www.youtube.com/watch?v=HdkixaxjZCM

The differences are quite comparable. There are two major ways to fly a virtual plane in a virtual world, both feel very similiar when you first try it out. But only at the beginning:

1) Technical in the background are TABLES. Do this and that with your plane - the plane behaves so and so. There are of course a limmited number of variations.

2) Real modelled air, airflow, winds, wings, rotors... the plane behaves the way, the physically modelled air, airflow, shape of your entire airplane, air density, humidity, ..., is the case in a specific situation. There are de facto an unlimmited number of variations and situations possible.

Whats the difference when fiddling with it? Number 2 is much more realistic, esp. in extrem situations, more possibility to 'feel' the air, guiding the plane. Like with PIANOTEQ.
Pianoteq is virtualizing more of the physical factors, much more, than say a SAMPLED piano does (that would compare to number 1), which have an very limmited number of varions - and therefor an limmited number of possibilities of expressing yourself. 

In short: pianoteq and a real air and surface model of a flight simulator gives you:

- more reality based feeling (because it IS more on the sinde of the reality)
- more, much more, freedom for you (be it simulating extreme situations in flight or expressing yourself with a piano)
- more finetuning based on real physics

Pianoteq is much more a 'real' piano than say a sampled one. The underlaying 'engine' is very, very different.

And what about OTHER SYTHESISERS? Well, they could be very limmited or even have a lot of possible variations. In later case pianoteq is such a 'sythesiser'. But a very good one! Because it handles physical factors of the reality. Which in my opinion is much, much harder to code than just playing and fiddeling with 'some' factors oft audio which sounds funny, even great, but have little to do with physical factors of a real instrument.
What we want to have is a 'real' physical instrument - and this is PINANOTEQ. I guess very hard to code because you have to deal with a very specific kind of physics - the physical factors of a real instrument.

And so - in my opinion - it deserves another name than just 'sythesiser'. Because that task ist very hard and tries to simulate the REALTIY not just 'something'. And yes, you could name it 'sythesiser' - as both kinds of flight simulators are called 'flight simulator'.

That's in my understanding the short storry uppon this question.

With either PIANOTEQ or them, I make a mix so that a listener hears a number of the often under-appreciated  intricacies that could go otherwise unnoticed into the instrument recording, including accentuated and rapidly changing motifs, unanticipated but coherent phrasings and various textures which move from one abruptly sudden extreme onto another and without any of them requiring any automation.  Indeed you imagine you hear the person playing at the acoustic instrument.  Specifically v.7 realistically is doable I attest, and completely believable without a hint it’s really only software.  But, you have to mix it just as you do the acoustic pianos such as any can come with a lot of unique complexities, obviously, among the most experienced of audio engineers and mastering engineers alike.

You change your attitude!

Because I openly acknowledge I’m myself just an amateur, I feel I am even more so in a very easy position to humble myself quite more than enough for me to learn adequately from many of my artificial intelligence packages as each of which can ultimately culminate in my further acquiring a potentially much broadened understanding, resultantly  ––about the general and specific goals of audio mixing and recently its mastering...

So throughout my learning from indeed some new artificial intelligence (A.I.), I will occasionally take off on my own as though I were flying 'solo,' and get the end results one usually expects from only the old acoustic instruments.  The software is now available as advertised.  You just need a methodology!

Thank you, for your topic!

Why do we we (at leat I do) use language like algorithmic modelling and mathematical simulation of a real piano sound when describing Pianoteq? Isn’t it just a synthesiser that emulates a piano? I’ve saved you a lot of words. So much less typing, so much less talking : ) (But the marketing does get a bit boring).

Because language conveys impressions beyond the strict meaning of the words. "Synthesizer" suggests a certain scope of sounds to most people, musicians and non-musicians alike. "Physically-modeled virtual piano" is four words; it would take many times that many words to explain to someone that "synthesizer" in this context does not mean what they think it means.