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 between higher harmonics indicates that the E string is properly tuned

Suppose, 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.

We need to calculate the third harmonic of A

Using formula of harmonic

[tex]f_{3}=n\times f[/tex]

Put the value into the formula

[tex]f_{3} =3\times440[/tex]

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

We need to calculate the second harmonic of E

Using formula of harmonic

[tex]f_{2}=n\times f[/tex]

Put the value into the formula

[tex]f_{2} =2\times659[/tex]

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

We need to calculate the beat frequency

Using formula of beat frequency

[tex]f_{b}=f_{3}-f_{2}[/tex]

Put the value into the formula

[tex]f_{b}=1320-1318[/tex]

[tex]f_{b}=2\ Hz[/tex]

Hence, The beat frequency is 2 Hz.