Question

One of the strings of a guitar has a fundamental frequency of 440 Hz and another has a fundamental frequency of 660 Hz. Which of the following set of frequencies could be produced on both of these strings?

Choices:
a. 440 Hz, 880 Hz, and 1320 Hz
b. 660 Hz, 1320 Hz and 1760 Hz
c. 880 Hz, 1320 Hz, and 1760 Hz
d. 1320 Hz, 2640 Hz, and 3960 Hz
e. None of the above.

The answer is D and I don't know why it is the answer.

Answer 1

The frequencies of subsequent harmonics of a string are integer multiples of the fundamental frequency (`f_1`).

• `f_n = n f_1`

On that basis:

• string 1 can produce: `440, 880, bb1320,...`

• string 2 can produce: `660, bb1320, 1980,...`

`lcm(440, 660) = 1320`, so that is the first common frequency, and points to option d)

Next check with your calculator that 2640 and 3960 are also integer multiples of both 440 and 660.

They are, and so it is option d)

Answer 2

We know that harmonics of any fundamental `f` are its integral multiples.

`f_n = (n+1) f`
where `n` is an integer `=1, 2,3,4....`

Therefore

First string can produce : frequencies `440, 880,1ul320,1760...` and
Second string can produce : frequencies `660, ul1320, 1980,...`

From above we see that the first common frequency is `1320\ Hz`.

It would always be possible to produce integral multiples of this frequency by both strings.

On inspection we see that Options (a) to (c) contain frequencies lower than lowest common frequency.

Option (d) contains first common frequency and its integral multiples. Hence (d).