Question:
The strings of a cello have a length L.
By what length 
must they be shortened by fingering to change the
pitch by a frequency ratio r.
Find , if 
and 
.
Solution:
The condition for resonance on a string of length L with two fixed ends is
, where k is the wave number and 
is an integer.
Using 
and 
, we can show that
is the frequency corresponding to the integer
n.
Now shortening the string by an amount , means that each
resonance frequency is increased to
.
Now 
, and hence
.
So for
r=6/5 gives 
,
r=5/4 gives 
,
r=4/3 gives 
,
r=3/2 gives 
.