Jtdylktuy

A guitar is a plucked instrument and it is played by plucking a string at an off-centre point fixed at two ends. In general, Fourier analysis tells that all harmonics (the resonant frequencies of the string) will be excited and the string will vibrate in a superposition of different harmonics. It is true that for $n^{th}$ harmonic the amplitude goes like $1/n$ suggesting that the fundamental contributes the most. In this situation, I am confused. How does a guitar string produce a pure tone/pure frequency sound instead of a noise?

Human perception is involved here because when you humans talk about noise this generally means a sound that is aperiodic. However the tone produced by a guitar will be something like:

$$ A(t,x) = sum_{i=0}^infty A_i sin(nomega_i t - k_i x) $$

i.e. a superposition of the frequencies $f$, $2f$, $3f$, etc. The function $A(t,x)$ is periodic in time with frequency $f = 2piomega_0$ so the ear/brain team perceives it as a tone not a noise.

Constructing a noise is actually quite complicated as we need to include all frequencies, not just integer multiples of a fundamental, and there will be a phase term in the equation that is not constant i.e. the sine waves making up the noise are not coherent.

Usually a guitar does not produce a pure tone/frequency. If so, its sound would be very close to a diapason. The difference between noise and a musical tone is not that a tone is made by a unique frequency, but there is a continuum between a pure tone (one frequency) and noise (all frequencies, not only multiple of a fundamental, without any regular pattern among their weights), where many non-pure tones are still recognized as dominated by a fundamental frequency. The additional frequencies add what we call the tone color or timbre of the sound.

In general, the exact weight of each harmonics can be somewhat varid according to how and where the chord is plucked. You might find interesting this study on the subject.

For an ideal string, the key point is that all the harmonics are "harmonic" : their frequency is a integer multiple of the frequency of the fundamental. So the movement of the string is periodic and has a well definite frequency.

For an ideal string, the harmonics have frequency ${{f}_{1}}$ , $2{{f}_{1}}$, $3{{f}_{1}}$.....and ${{A}_{1}}cos (2pi {{f}_{1}}t)+{{A}_{2}}cos (2pi 2{{f}_{1}}t)+{{A}_{3}}cos (2pi 3{{f}_{1}}t)....$ is a periodic function of frequency ${{f}_{1}}$

To complete, the dependence in $n$ is rather $approx 1/{{n}^{2}}$ for a plucked string.

Sorry for my english !

You do not hear a pure note. What is presented to your ear is a whole mixture of frequencies, which are approximately integer multiples of each other. It is your brain that melds these sounds together to give you wnat sounds like a single pitch (with a variety of "tone colours" depending on the mixture of frequencies. To see the effect, take the top string of the guitar and wrape a small piece of masking tape around it right in the middle of the length of the string. (small means less that a square 1/2 inch on a side). You will now hear two notes, one approximately an octave higher than the other. Your brain now does not meld the even harmonics (frequencies which are even whole number multiples of the fundamental frequency of the string) and the odd, and instead hears them as two separate notes. The frequecies that go into the ear are almost the same here as they were when you plucked the string without the tape, so the processing to produce on or two notes out of those frequencies takes place in the brain.
Some argue that it is the periodicity of the tone, but it is clear that you hear one note even if someone in the room is talking while you play the note. Their talking ruins any periodicity in the sound coming into your ear. But still you hear only one note (or two if you put the tape on).
Another instance in which the ability of the brain to meld the harmonics into one note is with Tuvan Singit throat singing. It sounds like a drone and a high pitch flute over the drone. But the high pitch is an exact multiple of the drone, and your brain one would think would meld the two into one note (as it does in normal singing).

How or why the brain does this is still, AFAIK, poorly understood.