Piano tuners tune pianos by listening to the beats between the harmonics of two different strings. when properly tuned, the note a should have the frequency 440 hz and the note e should be at 659 hz . the tuner can determine this by listening to the beats between the third harmonic of the a and the second harmonic of the e.
a tuner first tunes the a string very precisely by matching it to a 440 hz tuning fork. she then strikes the a and e strings simultaneously and listens for beats between the harmonics. the beat frequency that indicates that the e string is properly tuned is 2.0 hz
the tuner starts with the tension in the e string a little low, then tightens it. what is the frequency of the e string when she hears four beats per second?