Question

In a music situation, two notes are said to be an octave apart when one note is exactly twice the frequency of the other. Suppose you have a guitar string playing frequency f0. To increase the frequency upward one octave to 2f0, by what factor should you decrease the length?

Answer 1

Answer:

$\frac{1}{2}$

Explanation:

Let's see how we can approach the problem:

Using a simple simulation:

One octave separation = 2 x the length = 1/2 length

Two octaves separation = 4 x the length = 1/4 length

Three octaves separation =  8 x the length = 1/8 length

Four octaves separation = 16 x the length  = 1/16

and so on...

As we can see, the length is decreasing by a factor of $\frac{1}{2}$