During sympathetic resonance in a piano, are new frequencies generated?

Question

Sympathetic resonance in a piano is the phenomenon of one string being excited, transmitting its sound to other strings that will then start vibrating if they have common frequencies.

For example C2 strings will have a fundamental frequency F and harmonics n times F, n being an integer. And C3 strings will also resonate after C2 is pressed, when sustain pedal is down (i.e. when the strings are free to ring) because C3 fundamental is C2 first harmonic.

My question is this. The strings that will resonate in sympathy have common frequencies with the strings that were hammered. But they also have frequencies that are not harmonics of the initial strings excited.

So will these other frequencies also happen or will the strings only resonate in a limited number of modes? Is that even possible? Can modes of a string be excited separately?

Beware. I am not asking about a simplistic mathematical model of the strings. I am asking about the real piano.

Not a physicist, but familiar with acoustics.

Answer 1

For sympathetic vibrations, the main frequency will be the driving frequency, but non-linearities will also couple the sympathetic vibrations into vibrational modes with other frequencies as well... but these other modes will in general have frequencies that are not multiples of the driving frequency so they will not resonate and will be minimal (ie, much more minimal than in the normal hammer driven modes where there is not a "driving frequency"). Below I describe this in more detail:

For the case where everything is linear, as you know, only the driving frequency will be present (even in a perfect string that has many perfect harmonics, there's just nothing to get any power into those modes).

For the non-linear case, I'll start with an important distinction between continuously driven and impulse driven instruments:
It's common for people to refer to "the fundamental and its harmonics", but as generally used this is a misnomer. The key point in physics that makes this important is that for continuously driven oscillations (like a violin, flute, oboe, etc), the overtones are harmonic, that is exact multiples of the fundamental (because anything that's not a harmonic changes phase relative to the driver and is built up when in phase and diminished when out of phase, ie, not resonant); but in impulse driven instruments (like a piano, guitar, etc) there is nothing forcing the overtones to be harmonic, and they generally aren't.

For a typical hammer-driven piano sound, there are a few non-linear things going on. The most important is the stiffness of the strings, which generally stretches out the overtones (and this is what makes pianos difficult to tune). There are other things going on as well. For example, the typical string vibrations we think of are transverse, but there are also longitudinal modes as well, and the transverse modes couple to and drive longitudinal modes. (Basically, I think the longitudal modes are excited by the stretching of the wire during large transverse deflections.) Because transverse and longitudinal modes have different speeds, there's no intrinsic relationship between their frequencies, and they are inharmonic. These longitudinal modes are perceptible, and sometimes included in synthesis.

So why am I saying that these non-linearities are unimportant for sympathetic resonance? The key point is that sympathetic resonance is being driven by the other vibration at a single frequency, and as with other instruments with driven oscillations, the only vibrations that persist to any degree are those that are continually supported by the driving frequency. (Also, eg, the transverse vibration never become large enough to stretch the string for longitudinal vibration, but anyways, these wouldn't be resonant.) That is, when struct with a hammer, the non-linearities of the string play a critical role in the sound of the piano, whereas the same modes will not be supported in sustained sympathetic vibration because they are not resonant with the driving frequency.

Should you believe this?:
Overall, I've studies this topic (physics of music) but this is predictive based on what I know and you shouldn't believe it 100% (like all predictions). I'm quite confident in all of the supporting info I'm basing this on, but the exact conclusion that other modes aren't relevant, well, I'd like to see and hear some data myself.

Answer 2

To your first question the answer is yes, you can excite individual modes of a string. It is easy to experience this by plucking a guitar string at different locations while enforcing a node by touching the string lightly at a point where a node is expected for a mode that you want to excite. With a little practice, you can excite at least the first 5 or 6 modes this way, one at a time.

About sympathetic vibrations: When a hammer hits a string it will excite many modes of the string. The frequencies of the lowest modes are close to harmonics of the fundamental, but high harmonics become increasingly sharp relative to the harmonic series (as explained by tom10) All these modes can sympathetically excite modes of other strings. All modes decay, so they have a width in frequency that you can estimate as 1/(decay time). For one mode to excite another it is only necessary that their resonant widths overlap sufficiently. It is also true that non-linearities could excite modes with frequencies that are not close to any mode frequency of the struck string. I suspect this is not important in the situation you are considering, but I’m not sure. The most likely new frequencies f_new would be low-order mixing products of some strongly excited modes f1 and f2. In other words, f_new = n f1 + m f2 where n and m are small integers.