What is sympathetic resonance? 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?
 A: 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.
