Question

A musical interval of an octave corresponds to a factor of 2 in frequency. By what factor must the tension in a guitar or violin string be increased to raise its pitch one octave? To raise it two octaves? Explain your reasoning. Is there any danger in attempting these changes in pitch?

Short answer

To raise its pitch one octave the tension in a guitar or violin string be increased by factor 4 and to increase by two octaves then increase the tension in the string by factor 16.

Step-by-step solution

  Relation between frequency and tension.

The frequency of a string is directly proportional to the square root of its tension.

The relation between frequency and the tension is

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

Here, L is the length of the string, T is the tension, $\mathrm{\mu }$is linear mass density.

The tension in the string for two-octave waves.

The octave corresponds to a factor 2 in frequency. So, raise its pitch to one octave which means factor of frequency, then the tension must be of factor .

2f = 2T

= 4 T

Therefore, the tension in a guitar or violin string be increased by factor 4.

To raise its pitch two octaves which means factor 4 of frequency, then the tension must be of factor .

4f = 4T

= 16 T

Therefore, the tension increases by sixteen times as the tension before.
If the is increased then tension in the string increases large which causes strings to break so there is danger in incrementing the pitches of the string.