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 between higher harmonics 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?

(a) 2 Hz

The frequency of the nth-harmonic is given by

where

is the fundamental frequency

Therefore, the frequency of the third harmonic of the A () is

while the frequency of the second harmonic of the E () is

So the frequency difference is

(b) 2 Hz

The beat frequency between two harmonics of different frequencies f, f' is given by

In this case, when the strings are properly tuned, we have

- Frequency of the 3rd harmonic of A-note: 1320 Hz

- Frequency of the 2nd harmonic of E-note: 1318 Hz

So, the beat frequency should be (if the strings are properly tuned)

(c) 1324 Hz

The fundamental frequency on a string is proportional to the square root of the tension in the string:

this means that by tightening the string (increasing the tension), will increase the fundamental frequency also*, and therefore will increase also the frequency of the 2nd harmonic.

In this situation, the beat frequency is 4 Hz (four beats per second):

And since the beat frequency is equal to the absolute value of the difference between the 3rd harmonic of the A-note and the 2nd harmonic of the E-note,

and , we have two possible solutions for :

However, we said that increasing the tension will increase also the frequency of the harmonics (*), therefore the correct frequency in this case will be

1324 Hz

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