Question

In music, 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 by an octave, to 2f0, by what factor would you have to (a) increase the tension or (b) decrease the length?

Short answer

a) 4 times

b) 2 times

Step-by-step solution

Given information

Frequency of guitar string is being doubled by keeping everything constant except for a) Tension, b) length.

Relation of frequency of a string to tension and length.

$f=\frac{1}{2L}\sqrt{\frac{T_s}{\mu }}$

Part a): Dependency on Tension

We can see frequency is directly proportional to square root of tension. So in order to increase frequency by 2 times we have to increase the tension in the string by 4 times.

Part b): Dependency on Length

We can see frequency is inversely proportional to the length of string, so to increase the frequency by 2 times, length has to be decrease by 2 times (or halved).