Is there a context behind the overtone series ?

[m1469]

I have been meagerly beginning a study on the overtone series and in the process I am realizing that it is a bit of a missing link for me.  It seems to be the backbone or context of tone (of course) and harmonic progression.  But, I am wondering if there is something "beyond" the overtone series ?  Something that gives the overtone series a context, similar to how the overtone series seems to give a chord a context ?

Thanks .

[thalberg]

The answer is no.   There is nothing beyond the overtone series, nothing that gives it a context.  That is just how a string vibrates. 

Good for you for studying it, though.  It's good info to have.

[Bob]

If it helps...

The context being the halfs, thirds, fourths of the whole?  When something vibrates it can also vibrate in half and thirds, etc. 

So you end up with ratios.  1:1, 1:2, 1:3, 1:4...

In each ascending octave, the number of overtones doubles.  That adds up pretty quick and explains why dissonance isn't so bad if the tones are far apart.

Another interesting idea I heard about overtone, is that there is also undertones.  If the right overtones are emphasized, then you can hear a pitch below what is actually sounding.

[steve jones]

I thought the 'fundamental' was the lowest possible partial?

SJ

[timothy42b]

The answer is no.   There is nothing beyond the overtone series, nothing that gives it a context.  That is just how a string vibrates. 

Good for you for studying it, though.  It's good info to have.

I would cautiously say you are correct.

Cautiously, because we are at risk of veering off music and science and into philosophical discussions of naturalism.

However for those of us realists, there is nothing more in the nature of meaning.  There is a seemingly unlimited expansion of complexity if you need to understand the details.

Basically the Fourier theorem describes most of it.

From the physiological side, you would probably want to understand the five brain centers that respond to the five components of music.

From an engineering perspective (mine) you describe any vibrating system in terms of the governing equations (and probably transfer functions). 

Overtones in strings are different depending on whether freely vibrating, as in a hammered piano string, or driven, as in a bowed violin string.

To a tuner it is important to know that a piano string is just a narrow steel bar, and responds to being hammered exactly like a piece of railway track would, because it is stiff.  Of course we put enormous tension on it, causing it to combine the characteristics of a string and a steel bar, and making the overtone series very much unlike a wind column or any of the driven instruments. 

But despite the Lucy tuning adherents, there is no cosmic order behind it all, no perfect tuning that man lost to Adam's fall. 

If you are not fluent in temperament I would think that is one of the first places to start your study. 

[m19834]

Something I am wondering along these lines is if overtones have overtones (and perhaps those have overtones, too, etc).  Any note can act as a fundamental, and then within each fundamental can be found overtones, which are essentially tones that can be individually found on the piano.  So, do those tones within tones have their own recipe ?

For example, a "C" will have an overtone of a "G" within it.  If I were to play a "G" on the piano as a fundamental, I would find within it overtones.  Well, that means to me that each tone has a recipe of sorts.  Would a "G" which sounds as an overtone within a "C" have the same overtones that would be found if I just played a "G" as the fundamental ?

[Bob]

Do overtones have overtones?  Yes.  They're there, but not quite an overtone since the base note isn't that strong.

It's just math.

If 2 is the fundamental... 2, 4, 6, 8, 10, 12, 14, 16, 18, 20...
If you took 4....                    4,     8,       12,       16,       20....

Anything that the second partial (or whatever it's called exactly) has, the first one would have to have too.  Common multiples.

If that's what you meant.  Otherwise, they're will always be some common notes among overtones.  And once you get up a few octaves, it all blurs together anyway, so there are tons of close pithes then.

[timothy42b]

Do overtones have overtones?  Yes.  They're there, but not quite an overtone since the base note isn't that strong.

It's just math.

If 2 is the fundamental... 2, 4, 6, 8, 10, 12, 14, 16, 18, 20...
If you took 4....                    4,     8,       12,       16,       20....

No, the overtones don't have overtones of their own.  You have to go back to the physics of how overtones are produced in the first place.  Yes, there can be a very large series of overtones over the fundamental, and some of them might look like they are overtones to an overtone, but they are not.  See Benade.  Please.

Secondly, overtones are not x2.  Yes it is just math, but not a simple x2.  For certain ideal cases, like a massless limp string (limp meaning no stiffness at all) it might be.  Piano strings are made of steel, and a piano string is just as stiff as a wrench, therefore the timbre has some of the characteristics of the sound of a dropped wrench.  So the overtones will never be x2, x4.  They will be weird things like x2.0317896, etc.  That is called inharmonicity, and it is the reason pianos can't be tuned to equal temperament but must be stretched. 

[Bob]

I meant the overtones have their multiples in the multiplies of the original, fundamental pitch.  Not a true overtone, but if you x2 x3 x4, etc.  them, those numbers have to be there.  Theoretically.

[timothy42b]

I meant the overtones have their multiples in the multiplies of the original, fundamental pitch.  Not a true overtone, but if you x2 x3 x4, etc.  them, those numbers have to be there.  Theoretically.

That's what I'm trying to explain.  Mathematically, they do NOT have to be there.

They are there for driven instruments - things like wind instruments, bowed strings, etc. 

They are definitely NOT there for struck instruments like cymbals, piano, plucked violins, etc.  The math says they should not be there, the ear says they are not there, and if you put them on a scope (RTA with Fourier Analsysis) you can see they are not there. 

[keypeg]

They are there for driven instruments - things like wind instruments, bowed strings, etc. 

They are definitely NOT there for struck instruments like cymbals, piano, plucked violins, etc.  The math says they should not be there, the ear says they are not there, and if you put them on a scope (RTA with Fourier Analsysis) you can see they are not there. 

I believe that you can hear that if you listen carefully to a piano note with the damper down, and listen to what happens to the tone as it decays.  The sound changes in its quality or colour.

[Bob]

Yes, I'm getting confused.  Piano pitches have overtones.  I can hear them.  If they weren't there, it wouldn't sound like a piano, just a sine wave.

All I meant was that if 2 is the base number number and you take multiples of it, any of the multiplies of any of those multiples are also a multiple of 2.  If that's making sense.  If it's the product of 2 and something, whatever multiples it produes x2, x3, x4, etc. are also multiples of 2.  The the half, third, fourth vibrations of overtones are there.


I thought piano sounds had all the overtones.  I hear something, although it could just be in my mind I suppose.  Clarinets are just odd overtones I think -- just odd or just even, one of those.

[Bob]

Yeah, pianos have overtones.  Not mathematically perfect ones.  It's the real world.

https://sound-ideas.blogspot.com/2006/09/from-scratch-part-3.html
- That shows the fourier graph.  There's even an audio example to compare a piano sound a a pure sine wave.

I'm not going to hunt for more decent sites to post.  Piano strings produce overtones, although not perfect overtones of course.

[timothy42b]

Maybe I went too far into the details.  In engineering school we did a lot of calculation of system response higher modes, what you would call overtones on the piano.  Almost never did they line up as integral multiples (X2, X3, X4, etc.) 

To explain why, think of a couple of very simple systems.  Try a pendulum.  Move it off center, and it will return.  Only the force of gravity returns it.  That force acts down, but you can resolve it into two components, one at a right angle to the motion.  If you do the math for that very simple setting, you get an easy to solve first order equation.  But if there were a magnet nearby, then not only gravity would be acting on it.  You would have to add some additional terms to your equation, and now the answer would not be so simple.

Now move to a perfectly limp piano string, made out of nylon maybe.  Tug it off center and see what happens.  It will snap back, go a little past, come back, etc., vibrating until air resistance damps it out and it stops vibrating.  What pulled it back?  Only one force, tension on the string.  When you pull it down, it makes a triangle shape, with tension pulling on each end;  when you let it go tension will pull it back up.  Again that tension is at an angle but you can resolve it into two components, one horizontal and one vertical.  If you solve that equation, you get a fairly simple answer and the results will be in exact multiples just like you said. 

But a real piano string doesn't just have tension.  It has stiffness too.  It is made of steel, and the steel in a piano string is just as stiff as that of a wrench.  So when you pull the center of the string down, you get tension pulling it back, and you also get stiffness, just as if you bent a steel bar.  A thin one, true, but the properties don't change.  Now when you write your equations of motion you have two terms, and the stiffness term is fairly complicated.  Now your results are not in exact multiples.  They may not even be close especially for the bass strings. 

Does it matter?  Oh, yes, that's what makes piano tuning an art.  If you tune a piano to equal temperament based on the fundamental of each string, it will sound terrible because the overtones are now far away from equal temperament.  So you must compromise by shifting the fundamentals away from ET to get the best lineup of overtones.  This is called stretch, and it varies for every piano, because no two pianos have exactly the same amount of stiffness. 

[keypeg]

Timothy, that was very enlightening. 

I think the 2:1 3:2 etc. ratios are derived from the Greek time when the length of a string would be stopped half ways, two thirds, and thus an octave, a fifth, etc. were produced.  But it would not have been a precise measurement to the decimal.

I once heard that Galileo's father made strings for instruments and was considered approaching heresy because he wanted to consider pitch rather than proportion.  Has anyone ever heard of this?  It was in the course of a dinner conversation and I wondered about it ever since.

[timothy42b]

I think the 2:1 3:2 etc. ratios are derived from the Greek time when the length of a string would be stopped half ways, two thirds, and thus an octave, a fifth, etc. were produced.  But it would not have been a precise measurement to the decimal.

Strings or tubes like flutes, as long as they are continuously bowed or blown, will have those simple ratios. 

But now you're coming dangerously near talking about temperament!  which is entirely different from inharmonicity and stretch. 

I hadn't heard that about Galileo but it is intriguing. 

[Bob]

So, you're just saying the overtones are there for piano, but they're not perfect mathematically?

[timothy42b]

So, you're just saying the overtones are there for piano, but they're not perfect mathematically?

Oh, yes, they're there, for sure. 

Perfect mathematically?  Umm, that's kind of a loaded philosophical question.  To me they are perfect.  But after years of engineering education, most of us come to an acceptance that there is a one-to-one correspondence between math and reality.  Math IS reality. 

I think that to most of the rest of you math is kind of an interesting but arcane and irrelevant subject, best forgotten beyond the minimum to balance a check book.  This is a philosophical divide not often talked about explicitly. 

But for a more direct answer, yes the overtones are there, yes they obey mathematical laws, no they aren't simple multiples.