How harmonics are produced in a guitar string?

[Gabriel Maia]

My question is simply 'are all notes produced in a guitar produced by first harmonics?', but I will clarify what made me ask this question.

Now, if you have a wave driver you can make several harmonics in a string by increasing the frequency of the machine. In a guitar string, however, it does not matter how fast you pluck it, it always sounds the same. I understand that you would have to pluck them faster than it is humanly feasible for you to notice any change in the sound so if you want a different note from the same string you have to shorten its length.

Considering the relation

$\lambda = \frac{2\,L}{n}$

between the length of the string, the wavelenght of the note it is producing and the harmonic it is in, and the relation $v=\lambda\,f$ between the propagation speed, the wavelength and the frequency I have that

$f = \frac{n\,v}{2\,L}$

So, if I want to double my frequency, I can halve the string's length (L'=L/2) or I can move to the second harmonic (n=2). I know how shorten my string, but how can I move to the next harmonic keeping the original length? As I said, it would require more energy than a human being could afford to paly the string faster enough for it to move to the second harmonic, so this made me wonder if all notes played on a guitar are the first harmonic. Is there a note in a given length that you play and it is naturally on other harmonic than the first one?

What about instruments like flutes? You always see drawings of harmonics in instruments like that, and you have the impression that changing the tube's length changes the harmonic you're in, but just as was the case for the string, I don't understand how this works.

Thank you very much.

 

[CWatters]

Perhaps have a look at..
https://en.wikipedia.org/wiki/Guitar_harmonics

The string isn't the only source of sound on a guitar. The sound box has quite an effect by amplifying harmonics. I'm not a player but I recently spent time helping my son choose a new guitar. We looked at several very similar Yamaha guitars that differed mainly in the thickness of the guitar body. The thicker guitars had a noticeably richer and warmer sound than the thinner body guitars.

 

[Orodruin]

The speed at which you pluck the guitar has very little to do with what harmonics you excite. What is important is what shape and speed the string has when it is released. When you are using a tone generator to generate the harmonics, you are generating driven oscillations. Unless you strike the string in a very peculiar manner, you are going to excite all of the overtones as well as the fundamental tone. The higher overtones will be damped out relatively fast. This is very different from a string that is plucked and then allowed to vibrate freely.

 

[Gabriel Maia]

Perhaps have a look at..
https://en.wikipedia.org/wiki/Guitar_harmonics

The string isn't the only source of sound on a guitar. The sound box has quite an effect by amplifying harmonics. I'm not a player but I recently spent time helping my son choose a new guitar. We looked at several very similar Yamaha guitars that differed mainly in the thickness of the guitar body. The thicker guitars had a noticeably richer and warmer sound than the thinner body guitars.

I'm not interested in the sound per se, only in how the string is vibrating. when I pluck a string only once is it always in the first harmonic? If not, what determines the harmonic the string is in? If I want to double the frequency of my note I arrive at the following equation

$\frac{L^{\prime}}{L} = \frac{n}{2}$

Where L' is new length of the string (when I pressure it at some point). So, I can produce the same note by changing the length of the string or the harmonic. What I don't know is how the harmonic is changed?

When you play a string is it naturally in a given harmonic? How can you tell?

 

[Gabriel Maia]

The speed at which you pluck the guitar has very little to do with what harmonics you excite. What is important is what shape and speed the string has when it is released. When you are using a tone generator to generate the harmonics, you are generating driven oscillations. Unless you strike the string in a very peculiar manner, you are going to excite all of the overtones as well as the fundamental tone. The higher overtones will be damped out relatively fast. This is very different from a string that is plucked and then allowed to vibrate freely.

Perhaps using the guitar example was a bad idea. I'm not interested in the particularities of the sound itself. I want to consider an ideal vibrating string that is not dumped in any way. You can see videos on youtube of a string attached to a wave driver. The length of the string never changes, so when the frequency of the machine is increased the string moves from one harmonic to the next (at specific frequencies). Now, think about the player's finger as the wave driver. If he never pressures the string, the frequency of his tapping the string will be the frequency of the oscillations (ideal, no dumped oscillations, remember). So if he wants to move to higher harmonics he needs to tap faster. If, however, he is allowed to change the string's length, he has another way of obtaining different frequencies. So when you change the string's length how do you know if you are on the first harmonic still? Does changing the length equals changing harmonics?

 

[Orodruin]

I'm not interested in the particularities of the sound itself. I want to consider an ideal vibrating string that is not dumped in any way.

What I said was not at all specific to a guitar string. It holds for any plucked string. If you do not have any damping (not dumping), you will generally excite all eigenfrequencies and the oscillations will never die out.

Now, think about the player's finger as the wave driver.

The finger is not a wave driver. The wave driver drives the string with a continuous driving frequency. The finger just plucks the string.

If he never pressures the string, the frequency of his tapping the string will be the frequency of the oscillations

No it won't. The finger is not an oscillator.

So if he wants to move to higher harmonics he needs to tap faster.

No, this is an incorrect assumption. The general plucked string will have all harmonics excited.

 

[rcgldr]

The harmonic is setup depending on where you pluck the string. Wiki article:

https://en.wikipedia.org/wiki/Guitar_harmonics#Overtones

Overtones are not quickly dampened. Frequencies that are not natural harmonics for a string result in moving nodes and those do get dampened fairly quickly.

 

[Orodruin]

Frequencies that are not natural harmonics for a string result in moving nodes and those do get dampened fairly quickly.

Frequencies that are not natural harmonics are not present in the ideal string at all. It is perfectly well modeled by an expansion in the harmonics. You are perhaps thinking of the driven string where a driving frequency different from the natural harmonics never gives any form of resonance.

In a damped string, overtones do get damped out relatively quickly, i.e., faster than the fundamental frequency. After plucking the string what will remain after some time is mainly the fundamental frequency. The very high overtones in a damped string will not even oscillate, but be overdamped and just go exponentially to zero without oscillating.

 

[rcgldr]

Frequencies that are not natural harmonics are not present in the ideal string at all. It is perfectly well modeled by an expansion in the harmonics. You are perhaps thinking of the driven string where a driving frequency different from the natural harmonics never gives any form of resonance.

A plucked string is a driven string. Take a look at 0:25 into this video:

In a damped string, overtones do get damped out relatively quickly, i.e., faster than the fundamental frequency. After plucking the string what will remain after some time is mainly the fundamental frequency. The very high overtones in a damped string will not even oscillate, but be overdamped and just go exponentially to zero without oscillating.

In the case of a guitar string, an overtone can be forced by placing a finger tip at the 1/2 or 1/3 point of the string. Just after plucking the string, the finger tip can be quickly removed and the overtone will be sustained.

 

[Orodruin]

A plucked string is a driven string.

No it is not and nothing in that video suggests it. There is no external driving force once the string is released. The string is vibrating freely and the shape that travels back and forth is a superposition of eigenmodes. He even says so in the video.

In the case of a guitar string, an overtone can be forced by placing a finger tip at the 1/2 or 1/3 point of the string. Just after plucking the string, the finger tip can be quickly removed and the overtone will be sustained.

The overtones are always there. The difference is that you are essentially removing the fundamental frequency. If you look at the spectrogram of a plucked string, you will see that the lower frequencies are sustained for much longer, with the fundamental frequency displaying the slowest decay. Again, you can even see it in the video, as the higher frequencies die out, the string takes the shape of the fundamental frequency.

That the string in the beginning has a pulse traveling back and forth and regains essentially its original shape is a result of the overtone frequencies being integer multiples of the fundamental frequency. After one period of the fundamental frequency, all modes will have acquired an integer number of periods and therefore the string returns to the same linear combination of Fourier modes it started with.

Everything that is said and seen in the video corroborates what I have said here.

 

[rcgldr]

The overtones are always there. The difference is that you are essentially removing the fundamental frequency. If you look at the spectrogram of a plucked string, you will see that the lower frequencies are sustained for much longer, with the fundamental frequency displaying the slowest decay.

When using finger tip to remove the fundamental frequency, my impression is that the duration of the sustained overtone is only somewhat less than the duration of a string with fundamental frequency, depending on the overtone. Higher overtones do decay a bit quicker, as quadruple frequency decays quicker than double frequency. Example starting at 3:15 into this video with a spectrum analyzer. The first snippet is the fundamental frequency, the next snippet is double frequency, and the next is triple frequency.

 

[Orodruin]

Here is the spectrogram of an acoustic guitar:

It should be rather clear that the higher overtones decay quicker than the lower and the fundamental frequency. One should also remember that the actual guitar string (or the string of any actual instrument) is really not an ideal damped string.