The Length of a string

Some unusual use of terms here.

For a guitar String, [or an organ pipe] we have a Fundamental mode of vibration [Fundamental frequency] and a whole bunch of overtones - the other frequencies that can be forced to occur.

The term Harmonics is a pure arithmetic. It is a series based on the fundamental.

First harmonic = 1 * Fundamental
Second Harmonic = 2 * Fundamental
Third Harmonic = 3 * Fundamental etc.

I am not certain that when you described the first harmonic, you weren't in fact referring to the first over-tone? The first harmonic is the fundamental

[Note: for a guitar string, the first overtone is the second harmonic - but that's another story]

plucking a guitar string at different points along it will not change the fundamental frequency that sounds - though it can change the relative intensity of the various overtones.

I wasn't sure what you meant by "beating" the string, but to get a frequency higher than the fundamental of the open string, you have to press the string to the fret, effectively shortening it.

If the frequency you are after happens to be one of the harmonics, you could touch the string at a point, thus preventing the fundamental, and any other harmonics of lower frequency that the one you wanted.

For the 2nd harmonic - touch the string 1/2 way along [in the middle]
For the 3rd harmonic - touch the string 1/3 the way along
etc.

 

I am in Eastern Australia - so probably a different time zone to you.

When you pluck a guitar string, you always get a standing wave - actually a whole lot of then.

The lowest frequency - the fundamental, or first harmonic as we call it - is the pitch our brain interprets the sound to be. [actually is the relative mix of the other harmonics that enable a practiced listener - eg a guitarist - to establish whether the guitar had a steel string or a nylon string]

Still not sure what you mean/is meant by "beat"

But if you take the lowest string on a standard guitar and pluck it / hit it you will hear the pitch musicians call E
Press at the 3rd fret and repeat you get G
Press an the 5th fret and repeat, you get an A

Touch the string in the middle while you pluck and you still hear a note called E by musicians, but a higher pitch to the first.
It is in fact the 2nd harmonic for the open string.

If you press and hold the middle fret, you will hear that same tone - and it will be the fundamental / first harmonic for that shorter length of string.

 

The problem needs to worded a lot better - or is perhaps written by someone who doesn't understand how strings work.
To change the fundamental frequency of a string you can change either the length, the tension or the material the string is made from - or an appropriate combination of them.