Sound/accoustics - guitar string question

[Carnivroar]

(I hope this is the right section). For those who play guitar, you might have noticed that plucking the open 1st string (E) causes the 5th (A) string to vibrate. Why is that? It doesn't happen with any other string/note that I'm aware of. I'm sure there's a physical explanation for it.

 

[DragonPetter]

(I hope this is the right section). For those who play guitar, you might have noticed that plucking the open 1st string (E) causes the 5th (A) string to vibrate. Why is that? It doesn't happen with any other string/note that I'm aware of. I'm sure there's a physical explanation for it.

The string vibrations are not pure sine waves, and so they have harmonics at higher frequencies. These harmonic pressure waves are picked up onto the other strings that resonate at common frequencies (like antennas for sound waves).

There are some instruments that are designed specifically to take advantage of this effect:
http://en.wikipedia.org/wiki/Hardanger_fiddle

 

[Rap]

(I hope this is the right section). For those who play guitar, you might have noticed that plucking the open 1st string (E) causes the 5th (A) string to vibrate. Why is that? It doesn't happen with any other string/note that I'm aware of. I'm sure there's a physical explanation for it.

Every string puts out its fundamental frequency and its harmonics. The harmonics are frequencies that are multiples of the fundamental frequency. For example the first string is E1 at 82.407 cycles per second (cps). Its harmonics are 164.814, 247.221, 329.628 etc cps, each harmonic is weaker than the previous one. The fifth string is B3 at 246.942 cps, almost exactly three times the E1 frequency. So the third harmonic of the first string will "ring" the fifth string.

But it rings more too - damp the fifth string and pluck the first string, you will hear a high note coming from the second string. (If you don't hear it, pluck the first string near the bridge - than gives you a twangy sound with strong harmonics). What happens here is that the fourth harmonic of the first string (329.628 cps) is ringing the third harmonic of the second string (330 cps). This note is E3 and it matches the sixth string (check it out). If you damp the second and fifth string, you can hear E3 again as the fourth harmonic of the first string rings the sixth string.

Having the third harmonic ring another string is not as strong as having a second harmonic ring another string. Play an E chord, damping the 5th string, and you will hear the second harmonic of the first string ringing the third string pretty loud.

 

[sophiecentaur]

They aren't "Harmonics" aamof. They are Overtones, which are not usually exact harmonics (multiples of the fundamental frequency) because of the uncertainty of the effective length of the string at different frequencies (end effect). This gives added 'colour' to the note and can even be affected by the way the string is plucked.

 

[Rap]

They aren't "Harmonics" aamof. They are Overtones, which are not usually exact harmonics (multiples of the fundamental frequency) because of the uncertainty of the effective length of the string at different frequencies (end effect). This gives added 'colour' to the note and can even be affected by the way the string is plucked.

But the overtones of a plucked string are, for all practical purposes, the harmonics, right? Sure, its not perfection, but its really, really close. If you pluck a string in the middle, you minimize the harmonics. If you pluck it at 1/4 or 3/4 of its length, you maximize the second harmonic, etc. If I take my guitar and damp the first string at the middle (where the double dots are), then pluck it, then let go of the damping, the fundamental is completely absent, and it sounds one octave higher than the fundamental. This harmonic is surely there when the string is simply plucked, right? I mean, what is an example of an overtone of a plucked string that is not nearly a harmonic?

 

[sophiecentaur]

Very close and "nearly an harmonic" but this is a Physics Forum so why not use the right terms? If you look at the overtones of many (brass, in particular) instruments they are very discernibly different from mathematical harmonics and it is this that gives them their distinctive sound (and strange natural scale).
Guitarists call them harmonics but there are many Science terms that are mis-used by artists and the general public which we wouldn't use when talking Physics. One of the reasons that the original synthesisers sounded 'wrong' or idiosynchratic was the difference between harmonics and overtones.

 

[someGorilla]

For those who play guitar, you might have noticed that plucking the open 1st string (E) causes the 5th (A) string to vibrate. Why is that? It doesn't happen with any other string/note that I'm aware of. I'm sure there's a physical explanation for it.

According to one of the many ways of numbering octaves, the sixth string is E2 and the fifth string is A2. The third harmonic (or second overtone, or call it what you like) of the 5th string is at about 330 Hz, or the note E4, which is also the fourth harmonic (third overtone) of the 6th string. So the A string is excited by the 330 Hz component of the low E string vibration. If you pluck the E string and then stop it you will hear the overtone from the A string.
It's the same principle you exploit when you tune your guitar using harmonics (fourth harmonic of E to third harmonic of A).
By the way, E4 is also the open high E string frequency, so that string also vibrates, you just don't see it because the amplitude is much smaller.
And it's not true that it doesn't happen with any other string. The same happens, to a certain amount, with all strings, but some of them at so high a frequency that 1) the overtone is already very weak and 2) the stiffness of the string immediately damps the vibration. The E-A combo is the most visible to the eye. But you should be able to see at least also A-D.
(On a classical guitar. On an electric guitar every effect is much smaller.)

The fifth string is B3 at 246.942 cps, almost exactly three times the E1 frequency. So the third harmonic of the first string will "ring" the fifth string.

Strings are numbered from treble to low on string instruments, so the first string is high E and the sixth string is low E.

what is an example of an overtone of a plucked string that is not nearly a harmonic?

Overtones on double bass played pizzicato, for example, can be quite different from pure harmonics.
High overtones in stiff strings, especially short stiff strings, can be quite out of range (though not very hearable).
Piano tuners normally "stretch out" the octaves a bit to compensate for the inharmonicity of the strings.
Tension, stiffness and diameter of the string all play a role in determining the inharmonicity.

 

[sophiecentaur]

I would suggest that the term 'overtone' would have been in common use before the term 'harmonic', when instruments were made and tuned by ear and when people didn't use Maths quite so fluently.

 

[someGorilla]

I would suggest that the term 'overtone' would have been in common use before the term 'harmonic', when instruments were made and tuned by ear and when people didn't use Maths quite so fluently.

In musical practice – who knows. Probably a whole lot of different terms in different places and different traditions.
In musical theory – maths has always been quite important, it was for the ancient Greeks, it was for the ancient Chinese, it was for virtually every civilization that knew maths.
I'm pretty sure that 'harmonic' was used long before 'overtone'. I don't know about English, but in many other languages the word for 'overtone' is a recent coinage and hardly ever used outside of technical jargon (say, live computer-aided sound filtering).

 

[DragonPetter]

Any mode of vibration is either a fundamental or a harmonic. If overtones are fundamental, they either vibrate with another fundamental of the same frequency or they vibrate with a harmonic of another fundamental. Is this not correct? Any fundamental is a first harmonic, mathematically n=1, so in that context, all modes of vibrations are harmonics of some kind. I think in both a physics, a mathematical, and a musical sense, harmonics is a valid term rather than overtone, which is a more general term that can encompass many harmonics or partial modes of vibration and is a less descriptive and more broad term to describe a sound. Overtone gets further away from the original question of how one string can cause another to vibrate, while a harmonic specifically addresses how 2 strings can share a resonance - they share at least one frequency mode of vibration.

 

[someGorilla]

What sophiecentaur pointed out is that 'harmonic' properly refers to an exact, integer multiple frequency. 3rd harmonic = exactly triple frequency
Physical strings being different from the idealized unidimensional perfectly elastic string exhibit different properties, in particular their overtones are not exact harmonics, though they can be very near. If you use 'harmonic' in this precise sense, then no, overtones are not (necessarily) harmonics.
Listen to a bell and tell me if its overtones are also harmonics.

 

[Rap]

Any mode of vibration is either a fundamental or a harmonic. If overtones are fundamental, they either vibrate with another fundamental of the same frequency or they vibrate with a harmonic of another fundamental. Is this not correct? Any fundamental is a first harmonic, mathematically, so in that context, all modes of vibrations are harmonics of some kind.

I think the correct teminology is that a vibrating system has a fundamental, or lowest possible, frequency. I guess any higher frequency which the system can sustain is an overtone, and if the overtones are integer multiples of the fundamental frequency, they are harmonics. You can describe a guitar string to good accuracy by a simple wave equation, and when you solve that equation, the only overtones you get are harmonics. (See http://en.wikipedia.org/wiki/Vibrating_string). A real string obeys the simple wave equation quite well, but not perfectly. I am assuming that the deviations from "perfection" explain the special cases mentioned above. Note that, even if you assume that the string is a Euler-Bernoulli beam, only harmonics are generated (See http://en.wikipedia.org/wiki/Euler–Bernoulli_beam_theory#Dynamic_beam_equation).

 

[DragonPetter]

What sophiecentaur pointed out is that 'harmonic' properly refers to an exact, integer multiple frequency. 3rd harmonic = exactly triple frequency
Physical strings being different from the idealized unidimensional perfectly elastic string exhibit different properties, in particular their overtones are not exact harmonics, though they can be very near. If you use 'harmonic' in this precise sense, then no, overtones are not (necessarily) harmonics.
Listen to a bell and tell me if its overtones are also harmonics.

If it is not a harmonic, it is another fundamental mode of vibration, right?

If you look at a damped resonator frequency response, there are a continuum of frequencies around the center oscillation frequency. They all will give a response, but some give a stronger response than others. Whichever ones are transferred to the oscillator will vibrate til they die down, and so they can be considered overtones, but they are also fundamental modes of oscillation.

 

[someGorilla]

If it is not a harmonic, it is another fundamental mode of vibration, right?
[...] but they are also fundamental modes of oscillation.

You seem to be using the term 'fundamental' in a strange way, or at least one I'm not aware of. Can you point to a reference for this meaning of 'fundamental'?

 

[DragonPetter]

You seem to be using the term 'fundamental' in a strange way, or at least one I'm not aware of. Can you point to a reference for this meaning of 'fundamental'?

Ok, maybe this will help clear things up, or make it more confusing . . I'm not sure.

http://en.wikipedia.org/wiki/Harmon..._fundamental.2C_inharmonicity.2C_and_overtoneI think, when I say fundamental I have been meaning partial with no lower multiple of itself (which is implied as a fundamental by the wikipedia article), as in a mode of vibration that has no multiple lower than it. If you see the way things are defined, overtone is a broad and general definition of all partials.

Every time I have said "mode of vibration" I have been referring to a partial. One string can only transfer energy to another string if they share a common frequency to transfer the energy. A partial of a lower string can have a higher multiple partial, and if another string shares that same higher multiple partial, it will vibrate. They are called harmonic partials. A harmonic partial is any collection of partials that share a common frequency - a fundamental.

An overtone is ANY partial, and so an overtone may or may not cause another string to vibrate. A harmonic partial will always cause the other partials of the common fundamental to vibrate. This is the distinction I've tried to make of why overtone is not the correct vocabulary to describe the answer to OP's question.

As an example, say a person plucks the lowest E string and it vibrates at the E tone, but also with its overtones - partials around that E - and then also the harmonics of the E and all those partials, which means you will have a harmonic of E and harmonics of all its partials that have harmonics. If another, higher pitch string, has a partial equal to the frequency of one of the E string's partial's harmonics, it will vibrate.

 

[maimonides]

sophiecentaur,
your statements about the (extent of) anharmonicity of overtones are a bit surprising to me. I´d like to learn more about it; so could you please point out your sources?

Edit: I know about the anharmonicity of piano strings, so no need to bother with that.

 

[someGorilla]

I think, when I say fundamental I have been meaning to say partial, as in a mode of vibration that has no multiple lower than it.

Hmm. "Partial" does NOT mean a mode of vibration that has no submultiple lower than it. It just means any mode of vibration. (I added "sub" for - I hope - obvious reasons.)

If you see the way things are defined, overtone is a broad and general definition of partials.

No, overtone is any partial which is not the fundamental, by the very wikipedia article you quoted: "An overtone is any partial except the lowest."

Every time I have said "mode of vibration" I have been referring to a partial.

This is ok.


Partials are all the modes of vibration: all frequencies in the compound sound, or I should say all more-or-less standing wave frequencies, since there's a lot more frequencies in the attack phase for example, producing more of a noise than a harmonic series.
Fundamental is the lowest mode of vibration (lowest frequency partial). Being the lowest, there is only one.
Overtones are all other modes of vibration (so, fundamental excluded).
Harmonics are partials with a frequency multiple of the fundamental frequency (the fundamental itself included).

I think you are using "harmonics" in the way musicians (me included eh, I'm being fastidious just because I'm on pf :) normally use it, that is, to mean overtones. That's why you say that every frequency is either a fundamental or a harmonic. Strictly speaking though it's wrong, you should say it's either a fundamental or an overtone.
And if it's not clearly definable as a fundamental or an overtone, you're probably more in the realm of engine roars than in the one of violin tones. The boundary is blurred of course, so this description only applies, well, as long as it applies.

 

[DragonPetter]

Hmm. "Partial" does NOT mean a mode of vibration that has no submultiple lower than it. It just means any mode of vibration. (I added "sub" for - I hope - obvious reasons.)

A partial without any sub multiple is also a fundamental.

Here is a drawing to describe what I am saying:

 

[someGorilla]

Fine, according to your personal definition. Just don't expect others to understand you.

 

[DragonPetter]

Fine, according to your personal definition. Just don't expect others to understand you.

Not my personal definition- my interpretation of the article.

Just let me wrap this up: Two strings will only vibrate if they share a common frequency. For those frequencies to be common, they must be multiples of each other (n = '1') and so are harmonics. If a higher pitch string has a common frequency to a partial of a lower pitch string, then that partial of the lower string is either a fundamental or a harmonic of a fundamental, and so the frequency is a harmonic.

The only way for a frequency to not transfer to another string is if that frequency is exclusive to that string, and so it IS NOT necessarily a harmonic, it IS still a partial, and it IS still an overtone. Please don't be so condescending when it is obvious that only frequencies shared by both strings, harmonics, are the only ones that transfer energy, while an overtone can refer to any frequency generated that isn't necessarily common between 2 strings.

 

[someGorilla]

Ok, I didn't see the drawing at first. I thought you meant something else, like considering a frequency 5.1 times the fundamental as a new fundamental.
Can you quantify how wide that frequency range is? I mean the "fundamental" frequency range. There's a lot of causes for having more than one single lowest frequency (string tension changing while the vibration is damped, attack noise, unavoidable indeterminacy for very short notes, etc.), but I think none of these allows to define more than one fundamental. So, how wide is it? Does it change over time? How wide is it during the "decay" phase of the note?

 

[someGorilla]

And by the way, the overtones which are not harmonics that we were talking about have nothing to do with your drawing!

 

[DragonPetter]

Ok, I didn't see the drawing at first. I thought you meant something else, like considering a frequency 5.1 times the fundamental as a new fundamental.
Can you quantify how wide that frequency range is? I mean the "fundamental" frequency range. There's a lot of causes for having more than one single lowest frequency (string tension changing while the vibration is damped, attack noise, unavoidable indeterminacy for very short notes, etc.), but I think none of these allows to define more than one fundamental. So, how wide is it? Does it change over time? How wide is it during the "decay" phase of the note?

You can't talk about LTI techniques and analysis if it is time variant. If you are changing the system as you play, then its response changes.

If you have a set of frequencies that are multiples of each other, then the lowest frequency is the fundamental in that set. It is that simple of what a fundamental frequency means.

 

[someGorilla]

Hey, when sophiecentaur started distinguishing between harmonics and overtones no one was talking of TWO strings anymore, just of the modes of vibration of ONE single string.

I realize now that maybe you don't know what a harmonic is... a SINGLE guitar string vibrating ALREADY contains a series of harmonics. Are you thinking of a harmonic as only the note that comes from the second string when it's excited? Then you're mistaken on terminology.

 

[DragonPetter]

Hey, when sophiecentaur started distinguishing between harmonics and overtones no one was talking of TWO strings anymore, just of the modes of vibration of ONE single string.

I realize now that maybe you don't know what a harmonic is... a SINGLE guitar string vibrating ALREADY contains a series of harmonics. Are you thinking of a harmonic as only the note that comes from the second string when it's excited? Then you're mistaken on terminology.

I'm talking about the response of the guitar's strings together. If you take a set of frequencies, the collection of all frequency response of all the strings, then the multiples of certain frequencies between strings are still considered harmonics.

The original question wasn't about a single string, it was about 2 strings sharing a harmonic frequency.

 

[sophiecentaur]

sophiecentaur,
your statements about the (extent of) anharmonicity of overtones are a bit surprising to me. I´d like to learn more about it; so could you please point out your sources?

I looked on the web to find a definitive answer / reference for you but it seems such a basic idea that you tend to find, on the one hand, many sites that do not distinguish between the terms (unaware of the difference) of other sites that do not even to draw the (obvious?) distinction.

All oscillators that are not reduceable to a single mass on a single spring (SHM) will have a number of natural modes of vibration. If you take an 'ideal' string then the natural modes of vibration are harmonically related. If you take any other oscillator, the modes may or may not lie on a harmonic series, based on a single fundamental. Once the speed of sound on the oscillator or the boundary conditions (effective dimensions) are different for the different frequencies then the modes depart from a harmonic series. Take a circular drum, for instance.
http://en.wikipedia.org/wiki/Overtone says that, for most musical instruments, the overtones are "close to" harmonics but that
"An example of an exception is a circular drum, whose first overtone is about 1.6 times its fundamental resonance frequency."

Clearly, the design of most musical instruments is aimed at a pleasing sound and so the overtones are 'controlled' better than that by tapering the bore, adding a bell etc etc. and (good) musicians 'pull' and 'shape' notes using embouchure for further polishing of the sound. Our brains tend to like 'harmony' but if you take just any old object and try to turn it into a musical instrument (teapot trumpet, for instance) the lunatic modes of vibration make it sound all wrong.

For another example, look at the designs for quartz crystal oscillators and at the specifications for such crystals. They often have fundamental and overtone frequencies marked (some are specifically made for overtone mode) and it is a common(?) pitfall for someone to make an oscillator that is going off at a harmonic frequency and they are surprised that this is not what they expected! They have to re-do the design to get the desired frequency and to suppress the harmonic.

 

[someGorilla]

I don't exactly understand what you mean, and I still think you're somewhat confused.

For example you wrote:
Overtone gets further away from the original question of how one string can cause another to vibrate, while a harmonic specifically addresses how 2 strings can share a resonance - they share at least one frequency mode of vibration.


This is not true. If you have string #1 made of a peculiar material, so that its fundamental is 100 Hz and its first overtones 230 Hz, 370 Hz... and then you have string #2 with its fundamental at 370 Hz, the first string will cause the second to vibrate, through an overtone which is not a harmonic.

 

[DragonPetter]

I don't exactly understand what you mean, and I still think you're somewhat confused.

For example you wrote:
Overtone gets further away from the original question of how one string can cause another to vibrate, while a harmonic specifically addresses how 2 strings can share a resonance - they share at least one frequency mode of vibration.


This is not true. If you have string #1 made of a peculiar material, so that its fundamental is 100 Hz and its first overtones 230 Hz, 370 Hz... and then you have string #2 with its fundamental at 370 Hz, the first string will cause the second to vibrate, through an overtone which is not a harmonic.

If they are the same frequency, then they are a multiple of 1 of a fundamental and are therefore harmonic frequencies.

 

[someGorilla]

Ok, I give up.

 

[DragonPetter]

Ok, I give up.

I'm not trying to argue, its simply a mathematical definition that no one can argue with:
http://en.wikipedia.org/wiki/Harmonic

 

[sophiecentaur]

I don't exactly understand what you mean, and I still think you're somewhat confused.

For example you wrote:
Overtone gets further away from the original question of how one string can cause another to vibrate, while a harmonic specifically addresses how 2 strings can share a resonance - they share at least one frequency mode of vibration.


This is not true. If you have string #1 made of a peculiar material, so that its fundamental is 100 Hz and its first overtones 230 Hz, 370 Hz... and then you have string #2 with its fundamental at 370 Hz, the first string will cause the second to vibrate, through an overtone which is not a harmonic.

There is no reason why the 370Hz resonance of either string should, in fact, be a harmonic of either 'fundamental'. All that is necessary is for the two systems to share (fairly closely) a natural mode with frequency of 370Hz for one to resonate with the other.

Ref to my earlier mention of circular drums, the first overtone of one membrane and the second overtone of the other membrane would definitely not have a common sub-harmonic in sight.

 

[someGorilla]

I'm not trying to argue, its simply a mathematical definition that no one can argue with:
http://en.wikipedia.org/wiki/Harmonic

I mean the 370Hz is not a harmonic of the first fundamental.
If you mean every note is a harmonic of itself, oh well guess what, I agree.

 

[DragonPetter]

I mean the 370Hz is not a harmonic of the first fundamental.
If you mean every note is a harmonic of itself, oh well guess what, I agree.

Yes, it does seem silly, but my point the entire time has been that harmonic is more accurate than overtone to describe why one string will vibrate from another. Why? Because a non-fundamental harmonic OR an overtone (still a harmonic technically) can BOTH cause a second string to vibrate. Perhaps overtones are the main contribution in guitars, I don't know, but saying harmonic is not as accurate was my contention.

 

[Rap]

I think, getting back to the OP, all of this is off-topic. The answer to the OP is that a guitar string can be quite accurately thought of as having a single lowest possible frequency. Let's call it the main frequency temporarily, to avoid further semantic thrashing. The string also contains integer multiples of that main frequency at various, usually lower intensities than the main frequency. These integers are 2,3,4,... Let's call them "non-main" frequencies. If any frequency (main or not) of a string is at or very near any frequency (main or not) of another string, then either string, when plucked, will "ring" the other - it will cause it to vibrate.

Then you can get nitpicky and say "what if a string is plucked in such a way that the frequency that rings the other string is absent?". Well, ok, then in that case it won't ring the other string.

 

[someGorilla]

Yes, apologies to the thread poster, we got carried away... Rap summed it up pretty well.