Actually, the opposite question is more relevant

• why doesn't a string vibrate at all the frequencies? When you hit it, you aren't providing just one vibration node, but putting in a chunk of energy.

It comes down to where the nodes, or zeros are (have a look at @MatthewRead's answer over here) as with the ends of the string fixed, any wave which has an element of movement at those points will not propagate (the end point just won't move)

So what that does is limit the possible frequencies to those which have a wavelength which fits into the string a whole number of times.

Actually, the opposite question is more relevant

• why doesn't a string vibrate at all the frequencies? When you hit it, you aren't providing just one vibration node, but putting in a chunk of energy.

It comes down to where the nodes, or zeros are (have a look at @MatthewRead's answer over here) as with the ends of the string fixed, any wave which has an element of movement at those points will not propagate (the end point just won't move)

So what that does is limit the possible frequencies to those which have a wavelength which fits into the string a whole number of times.