Harmonics can be used by piano tuners by listening to the be

Harmonics can be used by piano tuners 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. What beat frequency indicates that the E string is properly tuned? answer= 2 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?

Solution

beat frequency = (440 x 3) - (659 x 2)

= 2 hz

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subtract freq E by one harmonic

= 659 - 1 = 658 Hz