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When two tuning forks stand near one another and one is excited, the other rings as well. When high notes are struck on a piano, lower notes are also heard. If I understand correctly, this is called sympathetic resonance.

What is the principle behind this effect, and how can it be described mathematically? I've seen it used in analogy with quantum entanglement--is such an analogy accurate?

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The vibrations of a tuning fork cause vibrations in the air with the same frequency. This process is symmetric in time: if you happened to have vibrations in the air which matched the frequency of the tuning fork, the tuning fork could pick them up and start to vibrate. A second, identical tuning fork is a good way to produce vibrations in air with the correct frequency.

You get the same thing with the strings in well-tuned piano, but there is a difference. The strings associated with each key on a piano have a different fundamental frequency. If you play the mid-range keys on a piano with the middle pedal pressed (so that the low-range strings aren't damped), you don't hear the fundamental frequencies of the low strings; you hear their harmonics, which are the same frequencies as the fundamentals of the higher strings. (The harmonics of the higher strings also excite the higher harmonics of the lower strings, but that's a smaller effect.)

There are so many garbage analogies floating around in the popular literature on quantum entanglement that I'm going to leave the last part of your question unaddressed.

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    $\begingroup$ (Years later) The example of a "well-tuned piano" has a couple of pitfalls: tempered intonation, and "stretched" piano tuning. Other than octaves, in tempered tuning (TT) the harmonics of a note are never exactly equal to any other note of the scale. The harmonics are rational multiples of the fundamental, but the tempered higher notes are irrational multiples. E.g. tempered 5ths are ~2 cents off from just intonation (JI) 5ths, Major 3rds about 14 cents sharper: ... (1 of 2) $\endgroup$ – BrianO Nov 2 '18 at 22:18
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    $\begingroup$ Concretely: for A4 = 440 Hz, its 3rd harmonic (2nd overtone) = 660 Hz, the JI E above it; but in TT, E5 =~ 659.25 Hz; the 5th harmonic of A4 is 550 Hz, the JI Maj3rd above it, but in TT, C#5 =~ 554.4 Hz. In "stretched" piano tuning, higher notes are tuned sharper, and lower notes lower, than their theoretical frequencies; octaves are bigger than an octave. It's my understanding that all pro piano tuners do this, resulting in ~35 cents discrepancy across a small piano [the wiki article linked above relates this]. "well-tuned piano" is a surprisingly wiggly term! ... (2 of 3 ; ) $\endgroup$ – BrianO Nov 2 '18 at 22:44
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    $\begingroup$ In the case of a piano with middle pedal down, these discrepancies must result in the lower strings' harmonics sounding more quietly (having less amplitude) than they would if the higher struck string had been from the JI perspective, "in tune". (3 of 3) $\endgroup$ – BrianO Nov 2 '18 at 23:00
  • $\begingroup$ @BrianO Thanks for the observations. There are lots of subtleties in professional piano tuning that I consider beyond the scope of this question; for my purposes the piano at your grandma's that isn't too honky-tonk but has gone a half-step flat still has a functioning middle pedal. I agree with all of your statements. $\endgroup$ – rob Nov 3 '18 at 13:16

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