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

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ajeigbeibraheem ajeigbeibraheem · Answer 1 · 2021-04-18T04:16:37+02:00

Answer:

The beat frequency = 2.0 Hz

Explanation:

From the given information

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

i.e.

[tex]f_{3A} = 3f_A[/tex]

where;

[tex]f_A =[/tex] note A fundamental freq.

replacing 440 Hz for [tex]f_A[/tex]

[tex]f_{#a} = 3 (440 \ Hz)[/tex]

[tex]f_{3A} = 1320 Hz[/tex]

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

i.e.

[tex]f_{2E} = 2f_E[/tex]

[tex]f_E =[/tex] note E fundamental freq.

replacing 659 Hz for [tex]f_E[/tex]

[tex]f_{2E} = 2(659)[/tex]

[tex]f_{2E} = 1318 \ Hz[/tex]

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

[tex]\Delta f = f_{3A}-f_{2E}[/tex]

[tex]\Delta f = 1320 \ Hz - 1318 \ Hz[/tex]

[tex]\Delta f = 2.0 \ Hz[/tex]

Thus, the beat frequency = 2.0 Hz