Q. Piano tuners tune pianos by listening to the beats between the harmonics of two different strings. While properly tuned, the note A must have the frequency 440 Hz} and the note E must be at 659 Hz}. The tuner can find out this by listening to the beats between the third harmonic of the A and the second harmonic of the E.

Part A: A tuner first tunes the A string very precisely by matching it to a 440 Hz tuning fork. She after that strikes the A and E strings simultaneously and listens for beats between harmonics. What beat frequency indicates that the E string is properly tuned?

Part B: The tuner starts with the tension in the E string a little low, after that tightens it. What is the frequency of the E string while she hears four beats per second?