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 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?

Step by step solution

01

Discussion on the path difference

The path difference also deals with the phase differences between them and the third speaker.

02

Explanation on phase difference

Provided sloution

$\mathcal{l}=1320-1318=2Hz$

03

Step 3:

The most clearly audible beat will be just the one found. Therefore, the beat frequency will be 2Hz. The frequency given by$$\begin{gathered} f_b=f_{3a}=f_{2e} \\ {f}_{3a}=1320 \end{gathered}$$

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