Tuning a Piano

1. Nov 23, 2005

quicknote

Hi,
I was just hoping that some one could check my answers...particularily b and c.
Thanks!

The note A should have a frequency of 440Hz and the note E should be at 659Hz.

a) What is the frequency difference between the third harmonic of the A and the second harmonic of the E?

$$f_{beat} = 3f_{A} - 2f_{E} = 2Hz$$

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 simultanesouly and listens for beats between the harmonics. What beat frequency indicates that the E string is properly tuned?

$$f_{beat} = f_{1} - f_{2} = 659-440 = 219 Hz$$

but I don't think it's possible to hear 219 Hz is it? Would the answer be the same as part a, since you are trying to tune E?

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?

$$f = 659 \pm 4$$
therefore f = 655 or 663 Hz but what's throwing me of is that it says "it starts with the tension in the E string A LITTLE LOW" which would me that the frequency is not equal to 659...

2. Nov 24, 2005

Andrew Mason

The beat is between the 2nd harmonic of E and 3rd of A. That is 2 Hz.
Assume you start lower, say around 650 hz. The first frequency at which one hears 4 beats/sec is 658 Hz.

AM

3. Nov 24, 2005

quicknote

Thanks for checking it over...but can you explain this a bit more? I don't understand how you would get 658Hz

4. Nov 24, 2005

Andrew Mason

The second harmonic of 658 hz is 1316. The third harmonic of 440 hz is 1320. The difference is the beat frequency: 4 beats/sec.

AM