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Any piano can produce the main frequency of the bass notes. Apart from length, the tension and density of the string play a role in the main frequency as well.

The problem with shorter strings is rather simple. Shorter strings are stiffer than their longer equivalents. These strings produce sharp overtones, since there is less length of the string to vibrate on higher frequencies. Perfect overtones need perfect flexibility to be an exact multiple of the main frequency, which is physically impossible.

Good to know, can't remember where I read that.

I've first read about it on a more mathematically and scientifically inclined tuners page. It also described the effect that slightly untuned strings on the same pitch create a semi-chaotic effect which allows us to hear the tone for a longer amount of time, while raising the perceived loudness (caused by a very strong initial decay and a long linear decay).

There is a good explanation on Piano Acoustics linked from the wikipedia page you posted [1]

[1] https://en.wikipedia.org/wiki/Piano_acoustics

>For any piano and especially the upright, the bass strings are actually too short to produce any vibration of the main frequency,

Not sure what you're talking about here. For example, the lowest guitar string is an E2 which is 82.4Hz. If you calculate[1] the wavelength, it is 13.7 ft. The string lengths from bridge to nut[2] are only ~25 inches long and yet it's reproducing the fundamental of 82.4Hz without requiring the brain to fill in the gap.

I think that misunderstanding is similar to believing that a speaker can't reproduce 82.4Hz because the cone is not 13.7 feet in diameter, or you can't hear an E2 in a small room because the width and height of the walls is less than 13 feet. If this were true, when you listen on earphones, all 88 keys of a piano and entire range of guitar and vocals would be "audio illusions" of the fundamental frequencies since the physical sizes of the transducers and ear cavity are all less than 1 inch. The highest 88th key on piano is a C8 with a wavlength of 3.2 inches.

[1] http://www.mcsquared.com/wavelength.htm

[2] https://www.guitarlessonworld.com/lessons/parts-guitar-learn...

> In fact, this is part of why a piano sounds like a piano and guitar sounds like a guitar.

My understanding is that it is a bit more complex than that, literally!

The the final waveform is a not just

W = sum(a_i * f_i) = Psi

where a_i is the amplitude and f_i are the fundamental frequencies.

It is actually

W = sum(a_i * f_i + sqrt(-1) * (b_i * f_i))

  = Psi + i * Phi

Loosely, the imaginary part plays a significant role in making an instrument sound like it.

Of course, the brain fills up a lot of stuff that is still a mystery but the elec keyboards can set the a_i, b_i to change from "guitar" to "reed organ".

That is way too simplistic as well, as this model is time invariant. Most instruments, due to physical nature of excitation, are variable in both frequencies and phases over time.

It is the main reason why modelling physically is the best way for realistic results right now - lossy lumped finite element models typically - digital waveguides are one of such models.

In such a model you can incorporate nonlinear damping and resonance functions over time at desired accuracy.

I have a terrible maths background, so that's -way- above me.

However, I have a good grasp of why instruments sound like they do, so I'm hoping that your statement is a complex way (no pun intended) of showing that the waveform has many harmonics, and that those harmonics vary over time? Not looking for any kind of argument, just hoping for a bit of explanation of the above; From what I've learned over the years it's the balance of harmonics and the way that they change over time that gives an instrument its timbre and explains the difference in tone between instruments despite them playing nominally the same note (i.e. fundamental at the same frequency).

This is Stretch Tuning, right?

https://en.wikipedia.org/wiki/Stretched_tuning

To my understanding it's more like Equal Temperament vs Just Intonation..?

No, those work even with sine waves. What the piano tuner is talking about is timbre: overtones can be not-perfectly-in-tune with the fundamental.

Has to do with the thickness of the core windings, I think? Something to do with the string acting partly as a rigid rod, not just an idealized wire. I hear this also in electric bass strings with super-heavy core wires.

Right, you're absolutely correct. I actually "get" inharmonicity, I think lack of sleep and the fact that tuning and the imperfect interval of an octave comes into it (I didn't know that tunings were actually shifted around inharmonicity) made me start thinking about JI/ET.. how embarrassing.

As a piano player you might find this guy interesting, https://en.wikipedia.org/wiki/Lubomyr_Melnyk His playing style focuses nearly exclusively in using harmonics and resonance to create his music.