While experimenting with piano acoustics I came across this weird phenomenon. Having an open B string and then hitting the C above will cause the B to resonate! This also works with open C and playing B below or B above! I cannot explain this with overtones since there isn't any close harmonic relationship between m2 or M7. Also how can any note produce frequencies below the fundamental?

  • If that's the case, it should happen for every two notes which are a semitone apart. Does it? How are you 'having an open G string? Holding the damper pedal down? Is the piano properly in tune? – Tim Jan 23 '19 at 11:08
  • Does it happen when you play another note besides C (with open B)? I'm thinking if it happens with C, it should happen with every other note too. – coconochao Jan 23 '19 at 12:00
  • In order to open a string I lightly press the corresponding note without hitting it until the damper is up. About my piano I had it properly tuned a few weeks ago so I guess this isn't the reason. Also this phenomenon occurs in almost every m2 relationship but in some cases (especially in the lowest area) the following happens: "while G1 is open and I hit A1b then the A1b resonates inside G1". I tried also hard hitting other intervals Tritone m6 m7 and it seems something similar happens but with lower intensity . It might be the case that something else inside piano transfers the vibrations. – Musical junkyart Jan 23 '19 at 12:19
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    Yeah, it's not a harmonic relationship just the piano resonating. Imagine how difficult it would be to stop the B resonating when you play the C. The dampers are there because they're necessary. . – PeterJ Jan 23 '19 at 12:33
  • Yea but still what is transferring the vibrations of ex. D4b lowered in order to match the fundamental frequency of C4? And also how is it possible for a ex. A1 to resonate in the frequency of B1b? – Musical junkyart Jan 23 '19 at 12:42

The answer lies in a mixture of engineering and "pure" physics. And a lot of Fourier Transforms :-) .
As we all know, the strongest resonances happen when the second string's pitch is an integral multiple of the first string. To see why, let's start with just the sound wave in air reaching the second string. The first crest excites the string, and just as the string's natural resonance pulls it back, the first valley helps "pull" the string in that direction as well. Thus the second string absorbs lots of energy from the sound wave. Next, look at an emitted frequency of F and a string resonating at 2F. Here the energy transfer matches the resonance only half the time (roughly speaking).
Now, if the second string's resonance is something like 12/11 *F, the sound wave is almost aligned with the resonant frequency, so the string will absorb energy for a few cycles -- and then it will dump energy as it goes out of phase! This leads to the "beat" phenomenon we all hear when two instruments are close but not exactly in tune.

Now to the engineering part: a physical piano does not have perfect acoustic reflection of the first string's travelling wave at the two ends of the string. Some energy, both acoustic and mechanical, gets into the sound board and the piano body. This energy starts dispersing (i.e. into random frequencies) and feeding back into the posts of all the other strings, thus exciting every string to some extent. How much energy makes it into a given note depends on the resonant profile of the entire system, not just the acoustic wave generated directly from the first string.

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  • Thanks for the answer! Two more question tho, is it possible to hear the beat phenomenon in piano? And also can any string produce lower frequencies than fundamental due to some nonlinear effect combined with strong hitting? I was hopping this to be the case in order to explore it some more :D – Musical junkyart Jan 24 '19 at 22:03

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