Are harmonics "real" in a vibrating string?

[Herman Trivilino]

What's the problem ?

You're reaching a conclusion that the harmonic is real when in fact the conclusion is that the (mathematical) model used to predict the behavior of the string works in this case.

Did you know, by the way, that if you leave the fundamental frequency out of a harmonic series, your brain puts it back in, so to speak? So, for example, when you strike the A 440 key on a piano you recognize the note as the A 440, but in fact there are overtones of various amplitudes, frequencies, and phase. If you have a machine that produces all of those overtones but not the fundamental frequency of 440 Hz, you will still report that you hear an A 440 note even though it sounds different from the A 440 note you hear from the piano.

 

[Herman Trivilino]

How can a physical object (my finger) interact with a mathematical/abstract object (the harmonics)?

There are two solutions:
1) the harmonics are real objects too and so interact with the finger in the physical world
2) the finger is "not real" and thus it interacts with the harmonics in the mathematical/abstract world.

3) Your finger interacts with the string. The mathematical model is used to predict the effect of that interaction.

 

[dRic2]

You're reaching a conclusion that the harmonic is real when in fact the conclusion is that the (mathematical) model used to predict the behavior of the string works in this case.

I'm starting to feel like you all are starting to make fun of me. I explicitly said the opposite in multiple posts. And in fact the part that you quoted is precisely it. The explanation 2) is what I said is correct and it is exactly what you are saying.

I think you are misunderstanding both my question, my reasoning and my conclusions. I expressed a perplexity in post #1 and solved it quickly after the enlightening post of @sophiecentaur who, I think, understood that I solved my problem (and, in fact, stopped posting).

Even then part that you quoted... I don't see how it can be misunderstood. I really don't.

Since you all seem to agree that I didn't understand it, it must be me.

I'm sorry if I'm getting worked up, but this is starting to annoy me. I honestly think the writing of mine is clear and if you read carefully you'll understand. Otherwise I apologize and it may be due to the fact that English is not my main language. Although something like this has never occurred to me before of PF.

I got the fact the it is a mathematical model and the question is pointless. I thank you all for the help and I'm happy that we all agree and that you care about my understanding. I don't want this tread to turn into a place full of hate so I think we should close this xonvers

 

[cosmik debris]

"Placing your finger on the metal strip kills some of the harmonics while leaving all the others"

What metal strip are you talking about here? Do you mean the fret? When I generate a harmonic on a string I rest my finger on the string. Alternatively there are pinch harmonics which requires a particular plucking technique.

Cheers

 

[dRic2]

@cosmik debris hey, you can rest your finger on the string but only at a very particular point. That point is the metal strip of the 5th, 7th or 12th fret. That if you want the harmonic of an open string. I don't get what kind of harmonic you are referring to. Pinch harmonics are totally different, I agree.

 

[Herman Trivilino]

The explanation 2) is what I said is correct and it is exactly what you are saying.

Here is your explanation 2)

2) the finger is "not real" and thus it interacts with the harmonics in the mathematical/abstract world.

I never said that the finger is not real. I never said that the finger interacts with the harmonics.

What I said is that the finger interacts with the string. This interaction can be explained using a mathematical model that involves harmonics.

You are confusing the phenomenon (finger interacts with string) with the explanation of the phenomenon. The phenomenon is real in the sense that it can be observed, the explanation of the phenomenon is real in the sense that it was created by humans.

What is real and what is not real is a philosophical issue that cannot be resolved using physics.

 

[BWV]

@cosmik debris hey, you can rest your finger on the string but only at a very particular point. That point is the metal strip of the 5th, 7th or 12th fret. That if you want the harmonic of an open string. I don't get what kind of harmonic you are referring to. Pinch harmonics are totally different, I agree.

Natural string harmonics are created by placing your finger lightly on the string on a node of vibration. For the 1st, 2nd, or 3rd partial (the 12th, 7th (or 19th) and 5th (or 17th) frets, respectively) the fret below is a good approximation for the correct position. This is not the case for higher partials, the 4th and 5ths partials are behind the 4th and 3rd frets

 

[rcgldr]

If you have a machine that produces all of those overtones but not the fundamental frequency of 440 Hz, you will still report that you hear an A 440 note even though it sounds different from the A 440 note you hear from the piano.

Resting a finger on the 12th fret of the A string results in a 220hz (plus some small multiples of 220hz) wave, with no noticeable 110hz component.

I haven't read any posts noting that a plucked string's initial state is triangular. A strummed string would be similar (momentum during the strum would make it less triangular).

 

[sophiecentaur]

I haven't read any posts noting that a plucked string's initial state is triangular. A strummed string would be similar (momentum during the strum would make it less triangular).

A point worth making. Very important that there is no longitudinal displacement if you want the string to behave 'normally' (no pun intended.) Some strumming styles could launch longitudinal waves, I suspect but musical instruments are complicated things and the'Physics' can only be described in the simplest situations.

 

[dRic2]

Natural string harmonics are created by placing your finger lightly on the string on a node of vibration. For the 1st, 2nd, or 3rd partial (the 12th, 7th (or 19th) and 5th (or 17th) frets, respectively) the fret below is a good approximation for the correct position. This is not the case for higher partials, the 4th and 5ths partials are behind the 4th and 3rd frets

They are usually played on the 5th, 7th or 12th fret. I've sometimes used and heard them played on the 4th fret with some good results. In my experience other frets do not sound that great or loud enough. I never hear and harmonics played on the 1st, 2nd or 3rd fret.

 

[BWV]

They are usually played on the 5th, 7th or 12th fret. I've sometimes used and heard them played on the 4th fret with some good results. In my experience other frets do not sound that great or loud enough. I never hear and harmonics played on the 1st, 2nd or 3rd fret.

You can find 4th and 5th partials in Fernando Sor’s music (early 19th century) they are not that uncommon.

 

[dRic2]

Fernando Sor

Thanks. I barely know his work.

BTW I had in mind electric guitar when I started the post and I've never heard those harmonics played on an electric guitar

 

[BWV]

Thanks. I barely know his work.

BTW I had in mind electric guitar when I started the post and I've never heard those harmonics played on an electric guitar

Even easier on an electric with some distortion. The 3-4-5 partials are a major triad.

 

[sophiecentaur]

I never hear and harmonics played on the 1st, 2nd or 3rd fret.

Listen to Joe Satriani and Steve Vai; power guitarists. They use many clever techniques with 'fretted' harmonics using the left hand both to set the note and a light fourth finger to make the harmonic.

 

[BWV]

Listen to Joe Satriani and Steve Vai; power guitarists. They use many clever techniques with 'fretted' harmonics using the left hand both to set the note and a light fourth finger to make the harmonic.

Yes, and that also is the technique for harmonics on bowed string instruments- touch the string a fourth above the fingered note to get the 3rd partial - a tone 2 octaves higher

 

[dRic2]

Listen to Joe Satriani and Steve Vai; power guitarists. They use many clever techniques with 'fretted' harmonics using the left hand both to set the note and a light fourth finger to make the harmonic.

Huge fan of both. I was talking about open string harmonics. Of course I know I can use my left hand but that's not what I was talking about. If you use your left hand or even the pick you can get all the harmonics you want.

 

[dRic2]

To be clear: this is what I was talking about. Place your finger on the metal strip and then struck the string, you'll get a good harmonic only at the 5th fret. 4th sounds ok too.

 

[Swamp Thing]

Sorry if this question has been addressed somewhere in the thread...

When we think about inharmonic partials, is that just the way that Fourier analysis would account for a gradual drift in the spectral content? For example, an MP3 encoder would take the spectrum of the input over some short time duration (maybe 10 to 100 cycles?). If, during that 10 cycle window, the timbre is changing appreciably, then that change would be represented as a slow beat effect between non-harmonic components. Is this a correct picture?

To put it more technically , if something goes boi-oi-oi-oi-oi-oinnggg, then each "oi-oi" would be a beat cycle between inharmonic partials, yes?

 

[BWV]

To be clear: this is what I was talking about. Place your finger on the metal strip and then struck the string, you'll get a good harmonic only at the 5th fret. 4th sounds ok too.

Here is a chart with the partials and the fingerboard locations

https://www.guitarlessonworld.com/content/lessons/images/harmonics-fretboard.gif

 

[sophiecentaur]

Fourier analysis

What is used in audio coding is a Discrete Transform (A 'Fast' implementation possibly. I seem to remember it's a Cosine transform but perhaps that's just for the picture processing)) The MP3 algorithm will make it sound ok - usually but not for people with Golden Ears. It's always a risk to assume that the results of this fit exactly with the basic Fourier Transform. You have to take finite time slots in which to do any transform and that assumes infinite repetition of the sequence. Any coding can be thrown out by subtle audio content.
The subtle details of musical sounds were beyond the early music synthesisers which were based on harmonics (dear old Hammond and its draw bars, for instance) I think it has to be done with sampling these days. (Memory is cheap so why not?)

 

[fizzle]

Then my question is: if you "mute" all the harmonics except for those having a node at that particular point, it means that they really do exist and that they're not just a mathematical trick.

It's a mathematical trick. The guitar string is not a sum or superposition of "substrings" vibrating at different frequencies with different amplitudes. You can *approximate* the string's behavior with Fourier or other mathematical analysis techniques ... but the string does what it does independent of your analysis!

Many moons ago, when I was a young student filled with mathematical techniques, I built a circuit that produced a square wave. When I slapped a scope on it, I somewhat expected to see harmonics in the output ... but the output simply went up and down, with some ringing on the transitions. The ringing was intrinsic to the circuit itself and not actually in the input. That really got me thinking about a how a narrow bandpass filter really works. You think of it "extracting" a signal from the input but it's actually a pseudo-oscillator responding to the input, i.e. it doesn't "extract" anything.

The next step is to wonder whether superposition is ever "real"...

 

[sophiecentaur]

When I slapped a scope on it, I somewhat expected to see harmonics in the output

Without a fair amount of dispersion and a non flat frequency response, it is not surprising that the scope produced a 'good' picture of the variation with time of the waveform. It was doing its job!

To display the frequency spectrum of a waveform requires (in effect) a swept filter with a display of what the filter is passing. Alternatively (but totally equivalent) you could do a Discrete Fourier Transform of a long sequence of samples of the time variation of the
waveform. The FT is effectively the result of finding the correlation between the signal and each of the harmonics of 1/(the time period of the sample).

 

[Ralph Dratman]

I agree with Ibix that the answer depends on what you take as "real" or "not real". Nevertheless I think I understand the motivation of the question, as I have often wondered whether, for example, a sharp click sound is "really" a combination of many different high frequency "tones" -- as the click might be portrayed using Fourier analysis. There is no single correct answer to that and similar questions, but it would be reasonable to study whether a few of the tiny hairs in our ears that sense pitch are in fact activated by a particular click sound.

 

[sophiecentaur]

It is not true to say that what we are aware of , as a ‘sound’, can be fully represented by a simple sum of single tones. Our ears work on how those hairs react to the vibrations. The response of a hair is certainly not ‘ideal’ but it’s perhaps the nearest thing to ‘real’ that you can get. Which makes a bit of a nonsense of the idea of ‘reality’ being applied to Science.

 

[Swamp Thing]

it would be reasonable to study whether a few of the tiny hairs in our ears that sense pitch are in fact activated by a particular click sound.

IIRC, the frequency analysis mechanism in the ear is the cochlea, which is a nonlinear(?) tapered transmission line. Different frequencies create different standing wave patterns along the length of the line, and each hair is just sensing the intensity of the standing wave at its own location in the cochlea. The brain then takes the entire standing wave's amplitude profile and works out its frequency spectrum. The brain probably also looks at the transient changes in the wave profile over tens to hundreds of milliseconds.

In the case of a sharp click, I would imagine a transient bouncing back and forth in the cochlea a few times, exciting all the hairs with pretty much the same amplitude. So if the brain got an input saying that all the hairs quivered equally during a span of a few tens of milliseconds, then it would conclude, "ah, yes, that must have been a click".

 

[sophiecentaur]

So if the brain got an input saying that all the hairs quivered equally during a span of a few tens of milliseconds, then it would conclude, "ah, yes, that must have been a click".

This would depend on a good timing mechanism in the brain itself and the ability to hear binaurally and image locate, supports this idea. It's clear that we use both frequency and time domain analysis. Breathtaking.

 

[Nik_2213]

Late to the party, but answer is 'YES'.

I could see the standing waves on my wife's viola's strings.
This took me back to fun college physics 'practicals', with paper 'riders' etc etc.

I'd been totally hopeless at music but, when my wife needed a 'Second Viola' and 'Pianist' for concert practice, I bought a PC, taught myself enough theory to run 'Cakewalk Home Studio', drove her Casio piano and a good 'Strings' sound-font via MIDI...

 

[tech99]

Then my question is: if you "mute" all the harmonics except for those having a node at that particular point, it means that they really do exist and that they're not just a mathematical trick. On the other hand this seems very strange to me because the string oscillates according to the function $u(x, t)$ which is something $per sé$, something real and physically observable: it is the "true" motion of the string.

An analog of the string is perhaps a metal box used as a cavity resonator for EM waves.
We can drive the cavity with a sine wave and many modes will spring up. If we insert a probe and connect it to a spectrum analyser we do not see any harmonics as a result of the various modes.
On the other hand, if we drive the resonator with a harmonic rich waveform, such as a square wave, our probe shows the harmonics, but altered in relative amplitude by the various resonances of the cavity.

 

[dRic2]

* To all the new posters please see post #4 by @sophiecentaur *
It's a very very good explanation of what I was trying to understand