Piano tuners tune pianos by listening to the beats between the harmonics of two different strings. When properly tuned, the note A should have a frequency of 440 Hz and the note E should be at 659 Hz.
a) What is the frequency difference between the third harmonic of the A and the second harmonic of the E?
b) 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. What beat frequency between higher harmonics indicates that the E string is properly tuned?
c) 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?