Piano tuners tune pianos by listening to the beats between the harmonics of two

Question

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 440Hz and the note E should be at 659Hz. The tuner can determine this by listening to the beats between the third harmonic of the A and the second harmonic of the E.

Part 1- A tuner first tunes the A string very precisely by matching it to a 440Hz 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?

Part 2- 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?

Explanation / Answer

The beats between the two different strings then the note A should have the frequency 440 Hz

and note E should have 659 Hz

a )   The tuner can determine third harmonic of A is 3 * 440 Hz

                                                                          = 1320 Hz

        second harmonic of E is   2 * 659 Hz

                                              = 1318 Hz

The beat frequency is 1320 - 1318

                                    = 2.0 Hz

b ) when she hears beat frequency 4.0 Hz

   the frequency of E string is   1320 Hz - 4.0 Hz

                                              = 1316 Hz

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