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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. What beat frequency indicates that the E string is properly tuned

User Astrofrog
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1 Answer

1 vote

Answer:

The beat frequency = 2.0 Hz

Step-by-step explanation:

From the given information

The 3rd harmonic frequency of the note is equivalent to thrice note A fundamental frequency.

i.e.


f_(3A) = 3f_A

where;


f_A = note A fundamental freq.

replacing 440 Hz for
f_A


f_(#a) = 3 (440 \ Hz)


f_(3A) = 1320 Hz

However, the 2nd harmonic of the E is equivalent to two times the fundamental frequency of the note E.

i.e.


f_(2E) = 2f_E


f_E = note E fundamental freq.

replacing 659 Hz for
f_E


f_(2E) = 2(659)


f_(2E) = 1318 \ Hz

Finally, the beat frequency when the E string is properly tuned is:


\Delta f = f_(3A)-f_(2E)


\Delta f = 1320 \ Hz - 1318 \ Hz


\Delta f = 2.0 \ Hz

Thus, the beat frequency = 2.0 Hz

User Al Polden
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6.8k points