## Guitar Waking Up?

Discussions relating to the classical guitar which don't fit elsewhere.
prawnheed

### Re: Guitar Waking Up?

Alan Carruth wrote:
Thu Feb 15, 2018 3:09 pm
All the best evidence I'm aware of says that the guitar is vanishingly close to linear, but it's still complex!

A few years ago I set up a simple rig to look at the forces a vibrating string puts on the saddle top. I wanted to verify the math relating the transverse force with the twice-per-cycle tension change. In the process of making the measurements I found a longitudinal compression wave in the string, which seems to be primarily driven by the off-center-of-length pluck. Naturally verifying and tightening up the quantities on this is requiring a major upgrade in my apparatus, and a bunch of time, which has to be stolen from my other, paying activities . All of the equations that describe this are linear, but when you stack them up things get 'interesting'. And the string is the simple part of the system....
There is a whole branch of mathematics called chaos theory which deals with the fact that even simple, linear systems behave unpredictably based on very small changes to initial conditions.

chiral3
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### Re: Guitar Waking Up?

prawnheed wrote:
Mon Mar 26, 2018 5:40 pm
Alan Carruth wrote:
Thu Feb 15, 2018 3:09 pm
All the best evidence I'm aware of says that the guitar is vanishingly close to linear, but it's still complex!

A few years ago I set up a simple rig to look at the forces a vibrating string puts on the saddle top. I wanted to verify the math relating the transverse force with the twice-per-cycle tension change. In the process of making the measurements I found a longitudinal compression wave in the string, which seems to be primarily driven by the off-center-of-length pluck. Naturally verifying and tightening up the quantities on this is requiring a major upgrade in my apparatus, and a bunch of time, which has to be stolen from my other, paying activities . All of the equations that describe this are linear, but when you stack them up things get 'interesting'. And the string is the simple part of the system....
There is a whole branch of mathematics called chaos theory which deals with the fact that even simple, linear systems behave unpredictably based on very small changes to initial conditions.
There’s been hints of chaotic behavior in loudspeakers for some time. It’s easy to write down the equations and figure out things like Lyaponov exponents and such to figure out if there’s chaotic solutions. The problem that has arisen in loudspeaker design is that while there’s evidence of things like period doubling and whatnot in the time series the empirical conclusion have been shaky and there hasn’t been that much research. A designer of planar electrostatic transducers once suggested that near-field tests are not really feasible because of local (chaotic) breakup of the membrane. I would see an analogy with a guitar, but it wouldn’t amount to much. I agree with Al: a guitar is almost always linear to good approximation. Always been curious about all the higher order and cross effects, though.

Edit: if I recall correctly the loudspeaker work focused on the effects of the forcing and restoring terms (the rhs of the diff eq)
"Life is under no obligation to give us what we expect" - Margaret Mitchell

rojarosguitar
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### Re: Guitar Waking Up?

prawnheed wrote:
Mon Mar 26, 2018 5:40 pm
There is a whole branch of mathematics called chaos theory which deals with the fact that even simple, linear systems behave unpredictably based on very small changes to initial conditions.
I don't think linear systems exhibit chaotic behavior, because they can always be analyzed in terms of modes, which essentially means decoupling. I'd be very interested in seeing a counterexample, if you know any. But an ever so slight nonlinearity in a continuous system of at least 3 degrees of freedom can exhibit chaotic behavior, if I remember right...
Music is a big continent with different landscapes and corners. Some of them I do visit frequently, some from time to time and some I know from hearsay only ...

My homepage is: https://www.live-arts.de

prawnheed

### Re: Guitar Waking Up?

rojarosguitar wrote:
Mon Mar 26, 2018 7:27 pm
prawnheed wrote:
Mon Mar 26, 2018 5:40 pm
There is a whole branch of mathematics called chaos theory which deals with the fact that even simple, linear systems behave unpredictably based on very small changes to initial conditions.
I don't think linear systems exhibit chaotic behavior, because they can always be analyzed in terms of modes, which essentially means decoupling. I'd be very interested in seeing a counterexample, if you know any. But an ever so slight nonlinearity in a continuous system of at least 3 degrees of freedom can exhibit chaotic behavior, if I remember right...
The double rod pendulum is an example of a simple, linear, deterministic system that shows chaotic behaviour.

Strings coupled to beams (like a simple guitar) certainly show chaotic behaviour.

chiral3
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### Re: Guitar Waking Up?

You are correct. Physical systems need to be nonlinear to be chaotic.
"Life is under no obligation to give us what we expect" - Margaret Mitchell

chiral3
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### Re: Guitar Waking Up?

Prawnhead, the dp problem usually uses a small angle approximation of sin(theta)=theta.

Edit: sorry if that wasn’t clear, I got pulled into something while I was typing. The dp solution is usually linearized to allow for a solution. This is usually for small angles, i.e, not large swings, which is where the chaotic motion occurs. Physical systems (i.e., finite systems) must be non-linear to to be chaotic.
"Life is under no obligation to give us what we expect" - Margaret Mitchell

prawnheed

### Re: Guitar Waking Up?

chiral3 wrote:
Mon Mar 26, 2018 7:40 pm
Prawnhead, the dp problem usually uses a small angle approximation of sin(theta)=theta.

Edit: sorry if that wasn’t clear, I got pulled into something while I was typing. The dp solution is usually linearized to allow for a solution. This is usually for small angles, i.e, not large swings, which is where the chaotic motion occurs. Physical systems (i.e., finite systems) must be non-linear to to be chaotic.
Understood, you are right. The double pendulum is a simple physical system that shows chaotic behaviour though.

And as I said, I've seen chaotic behaviour demonstrated in a simple string attached to a beam. It was not exactly analogous as the system was continously excited rather than being plucked.

How relevant any of this is to the question of whether a guitar ages or not is another question entirely.

chiral3
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Location: Philadelphia Area, PA / New York.

### Re: Guitar Waking Up?

prawnheed wrote:
Mon Mar 26, 2018 9:23 pm
chiral3 wrote:
Mon Mar 26, 2018 7:40 pm
Prawnhead, the dp problem usually uses a small angle approximation of sin(theta)=theta.

Edit: sorry if that wasn’t clear, I got pulled into something while I was typing. The dp solution is usually linearized to allow for a solution. This is usually for small angles, i.e, not large swings, which is where the chaotic motion occurs. Physical systems (i.e., finite systems) must be non-linear to to be chaotic.
Understood, you are right. The double pendulum is a simple physical system that shows chaotic behaviour though.

And as I said, I've seen chaotic behaviour demonstrated in a simple string attached to a beam. It was not exactly analogous as the system was continously excited rather than being plucked.

How relevant any of this is to the question of whether a guitar ages or not is another question entirely.
One example of what I think you're referring to would be a Duffing oscillator. In a Duffing system you can amp up the amount of restorative force (what I was alluding to before with the loudspeaker chaotic system) and there is a forcing function that "remembers" the recent past. It has period doubling and chaotic solutions. The problem is the string doesn't make much noise in a guitar, the top does. Any kind of complexity that is specific to the string gets "sinked" out via interaction with the bridge rocking back and forth and the string basically becomes a harmonic oscillator (i.e., a linear superposition) that slowly loses energy to the top (as well as some self-interaction via internal damping).
"Life is under no obligation to give us what we expect" - Margaret Mitchell

Bill B
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### Re: Guitar Waking Up?

Alan Carruth wrote:
Mon Mar 26, 2018 5:15 pm
Bill B. wrote:
" I really don't think that its a fault of my hearing, because nobody can demonstrate this phenomenon in reality. "

Again, I do have objective measurements that suggest that 'warming up' is real. As always, I'd like to get better ones, but that sort of thing is time consuming.

There certainly are a lot of legends in this field that have no basis in reality, and your subsequent post brought up several. OTOH, the fact that a lot of what people believe is not so doesn't make everything they believe untrue. The trick, and it's a hard one, is to figure out which is which.

Joe Curtin, one of the better modern violin makers, pointed out years ago that part of the professional qualification of a good player is that they can get the tone they want out of pretty much any fiddle. I'm sure much the same goes for good guitar players. He pointed out that they do this automatically; you can't get them to not do it. His point was that you can't learn much about how good an instrument is by handing it to a good player; they'll just make it sound the way they want. On the other hand, handing it to a poor player won't help either, since they won't be able to make anything sound good. That's why objective tests are necessary.

This brings up an interesting point. If the good players can make any instrument sound good, why do they spend all that money on 'good' instruments? They could just get some imported junker from WalMart and play that.

One answer could be that they find it easier to make the sound they like on a better instrument. That could be part of it, but the 'blind' playing tests with violins have cast some doubt on that: people seemed to find no real difference in that respect between the Old Italians and newer instruments that don't enjoy the exalted reputation. If that's true, then players are paying a lot of money for a placebo effect: they play better because they think they'll play better.

That's certainly one way that the legend of 'warming up' could work. You think the guitar will get better as you play, and it does, but not because of anything that happens with the guitar. But then there's the objective data I've mentioned. A guitar that's being 'plucked' mechanically, using a very repeatable mechanism, is recorded on a computer. When successive plucks are compared the later ones are systematically a little louder than the earlier ones. Other tests using other setups, and measurements of isolated strips of wood show similar changes. So what gives?
I don't know what gives. but heres the thing. I don't think a particularly complicated scientific study is really relevant. at least not to me. I'm less interested in whether a computer can identify differences in the sounds of sleepy versus woke guitars. I'm much more interested in what human listeners think they hear, and what they do hear. there are two different questions here, i suppose. #1. does the potential guitar sound change in any way based on how long its been since its been played. and #2, if there is a change, is it significant enough that even a few humans can reliably perceive it? #1 might require some very complicated setup to study adequately. but number two seems like it should only require a curious guy with too many guitars and one guy with the magic ears, and a little time. If the difference were really significant at all, a guy could easily demonstrate that.
2005 Ramirez R-2

prawnheed

### Re: Guitar Waking Up?

chiral3 wrote:
Tue Mar 27, 2018 12:45 am
prawnheed wrote:
Mon Mar 26, 2018 9:23 pm
chiral3 wrote:
Mon Mar 26, 2018 7:40 pm
Prawnhead, the dp problem usually uses a small angle approximation of sin(theta)=theta.

Edit: sorry if that wasn’t clear, I got pulled into something while I was typing. The dp solution is usually linearized to allow for a solution. This is usually for small angles, i.e, not large swings, which is where the chaotic motion occurs. Physical systems (i.e., finite systems) must be non-linear to to be chaotic.
Understood, you are right. The double pendulum is a simple physical system that shows chaotic behaviour though.

And as I said, I've seen chaotic behaviour demonstrated in a simple string attached to a beam. It was not exactly analogous as the system was continously excited rather than being plucked.

How relevant any of this is to the question of whether a guitar ages or not is another question entirely.
One example of what I think you're referring to would be a Duffing oscillator. In a Duffing system you can amp up the amount of restorative force (what I was alluding to before with the loudspeaker chaotic system) and there is a forcing function that "remembers" the recent past. It has period doubling and chaotic solutions. The problem is the string doesn't make much noise in a guitar, the top does. Any kind of complexity that is specific to the string gets "sinked" out via interaction with the bridge rocking back and forth and the string basically becomes a harmonic oscillator (i.e., a linear superposition) that slowly loses energy to the top (as well as some self-interaction via internal damping).
Not sure if it is a duffing oscillator or not, but the example I saw was of a simple string coupled to a simple beam. When the string, or one of its harmonics, was tuned close to a resonant frequency of the beam the interaction between the two resulted in chaotic motion of the string - even in the simple transverse motion, let alone the other modes of vibration of the string.

The application in this case was more structural (cables and bridges) and I am trying to remember something from more than 10 years ago, but the simple model of a system consisting of a string coupled to a beam would seem to me to apply to a guitar.

chiral3
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Location: Philadelphia Area, PA / New York.

### Re: Guitar Waking Up?

The first thing that pops in my head with your reference is the Tacoma Narrows Bridge. My background is physics, not engineering, but this was always the big engineering example, perhaps incorrectly, attributing it to a periodic pumping of the natural resonance. For a guitar the obvious analogy is a wolf tone, when a note couples to the natural resonant frequency of the guitar and produces a bunch of unintended sympathies. There’s probably a more complex non-linear analogy that nobody really talks about.

Mari Kimura is a violinist that uses a strange bowing technique to produce a subharmonic excitation. She can play a full octave below the violin’s lowest (G) note without tuning down. We probably have all done this with wind instruments by changing how we blow. The fluid dynamical physics analogy would be high Reynolds number flows, which are turbulent, and related to vortex shedding and cascading, which is likely related to the actual reason for the collapse of the Tacoma Narrows Bridge. Similarly, the bowing technique that produces the subharmonic tones are not gentle. By further analogy the double pendulum become chaotic for large angles. I am rambling, but I find this stuff interesting. Guitars are so inefficient and have so much input impedance that finding analogies may be difficult.
"Life is under no obligation to give us what we expect" - Margaret Mitchell

prawnheed

### Re: Guitar Waking Up?

Yes. My background is in engineering, but not on the technical side for many years so I've forgotten a lot of the maths I once knew..

The case I vaguely remember was not related to that bridge, but I think in part inspired by some analysis that had been performed on an Italian bridge. The example I saw demonstrated (at a conference in Beijing that I was attending for a completely different reason) was, in any case, a very simple beam (a steel rod in this case) with a very simple string (a steel wire) stretched between the ends of that beam which was acoustically excited. There was an accompanying mathematical model and numerical analysis that also showed chaotic behaviour. I have done a quick search, but can't find anything published that confirms my memory - it may only be in chinese for all I know.

I really couldn't say how tightly the conditions concerning tension, tuning, coupling, excitation frequencies and amplitudes, the physical properties of beam and strin, and (as your rightly bring up), the damping etc. need to be to demonstrate this behaviour. For all I know, it could be a very special case or it could be that all strings show this under certain circumstances.

Alan Carruth
Luthier
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### Re: Guitar Waking Up?

My understanding is that only non-linear systems display truly chaotic behavior. A linear system will be 'complex'. The difference is (again, in my math-challenged understanding) that chaotic behavior can be wildly divergent, while complex behavior is not. In the case of the guitar, it's complicated enough that predicting the precise response to a given signal in the 'resonance continuum' above, say, 600-800 Hz, is not possible, but neither is it so divergent as to be musically detrimental. Of course, one issue here is that there is probably some non-linearity in the system, although it depends on how you look at it.

The double-rod pendulum may be a good example of that. In order for a pendulum to be truly linear the angle of displacement must be small enough that the sine really changes linearly with displacement. It's 'pretty close' for angles less than 3* or so, but it's still off by enough so that the arc really should be a cycloid. In practice it's hard to make a perfectly linear pendulum.

chiral3 wrote:
"Any kind of complexity that is specific to the string gets "sinked" out via interaction with the bridge rocking back and forth {snip}"

Bridge rocking is a decidedly secondary mechanism in the sound production of the guitar. There are two main signals driving the bridge: 'transverse' and 'tension'. As the string moves the angle change at the saddle pulls the bridge and top with a force that is transverse to the plane of the string vibration. This resolves into two vectors, perpendicular to the top plane and parallel to it. It's the perpendicular transverse force that provides most of the power going into the guitar by pumping the lower bout in a 'loudspeaker' motion. As the string vibrates it also becomes tighter when it is displaced from it's 'rest' position in one direction or the other. Although the magnitude of this force varies from string to string it's seldom more than about 10% of the transverse force with nylon strings, and can be much less. This tugs the top of the saddle toward the nut twice per cycle, causing the area behind the bridge to belly up, and to sink down in front. This is not a very effective way to produce sound. For one thing, we build tops to resist this sort of deformation, so that on most guitars the amplitude achieved for a given force in this way is much lower than the 'loudspeaker' amplitude, at least below about 350 Hz where the top 'long dipole' mode kicks in. Secondly, since half the vibrating area is out of phase with the driving signal, you lose a lot of power through cancellation. In tests where I drove strings in a single direction relative to the top plane 'parallel' driving, which would only drive the bridge rocking mode, produced ~20 dB less output than 'perpendicular' driving that also drove the 'loudspeaker' action of the top. In another series of experiments, looking at the influence of break angle and saddle height off the top, a taller saddle produced a bit more sound in the second partial of the string, due to the greater leverage for bridge rocking. The overall output remained the same, but most listeners were able to pick put the change in carefully made recordings.

Oh, and as to non-linearity: plucked strings also show a longitudinal compression wave, which seems primarily to be driven when the string is plucked off center in it's length. This introduces a signal that also acts through bridge rocking, somewhere between the 7th and 8th partial of most strings. It is also louder when the saddle is taller. I'll note that all of this can be accounted for by strictly linear reckoning, so it's questionable whether it's non-linear or not, but it sure is complex. It also has nothing to do one way or the other with the string being tied to a guitar: I've seen it on a rig that is both massive and quite rigid. '

Bill B wrote:
"I don't know what gives. but here's the thing. I don't think a particularly complicated scientific study is really relevant. at least not to me. I'm less interested in whether a computer can identify differences in the sounds of sleepy versus woke guitars. I'm much more interested in what human listeners think they hear, and what they do hear."

It's a sort of axiom in the instrument acoustics game that there are things that are easy to hear but hard to measure, and things that are easy to measure, but hard to hear. Should we only concentrate on the ones that are easy to hear? The problem with that is that some things are not important until they become important, and then they can be very important. 007 is tied in a chair, and the water is rising: the depth doesn't matter until it reaches his nose.

Several years ago I was talking with a friend about my string measurement experiments. He mentioned that about one D string out of every three that he puts on a guitar buzzes at every fret, and asked why that might be. It turns out that, for whatever reason, the longitudinal wave frequency on Classical D strings tends to be very close in pitch to the seventh partial. When the match is exact the transverse and longitudinal waves couple, and the seventh partial of the transverse wave splits into a pair of closely spaced frequencies. The 'buzz' is the beat frequency of the two seventh partials. This is exquisitely sensitive to the mass and length of the string. Since you can't change the length on an existing guitar a good way to deal with it is to remove the string from the bridge and twist it in a way that tightens up the windings. The small increase in unit mass can be enough to eliminate the problem.

In a larger sense, this bears on the question of whether it's possible to make 'identical' guitars. It would be really nice to be able to make copies of, say, Segovia's '37 Hauser, that sounded exactly like it. It would be nice to be able to duplicate the exact tone of the best one off my own bench! The experiments I've done strongly suggest that, while you can get 'close', 'the same' may not be possible. The complexity of the response in the high range, where your heating is especially sensitive, means that very small differences in initial conditions which are probably impossible to control produce easily audible differences in the tone. Note that I'm not talking about differences in overall 'quality' of the resulting guitars: although some people will prefer one over another, over a range of players/listeners they're judged to be equally 'good'.

In terms of the question at hand (this thread has certainly taken on a life of it's own), from a maker's perspective it would be really nice to settle it. In general it seems to be the 'better' guitars that display some degree of 'warming up' behavior, although too much can be a nuisance. If we knew what was happening in a physical sense we might be able to get some control over it. It's not necessarily something that concerns a player or listener directly, but could be important to makers.

Bill B
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### Re: Guitar Waking Up?

If it's not something you can actually hear in any kind of reliable way then all the talk about players hearing their guitars wake up is nonsense. Our audience has human ears. The example you gave Alan about d strings is interesting but something completely different. You are talking about a clearly observable phenomenon. If this walking up was anything like so easily observed we wouldn't have had a discussion of this length about it at all
2005 Ramirez R-2

prawnheed

### Re: Guitar Waking Up?

Bill B wrote:
Tue Mar 27, 2018 6:35 pm
If it's not something you can actually hear in any kind of reliable way then all the talk about players hearing their guitars wake up is nonsense. Our audience has human ears. The example you gave Alan about d strings is interesting but something completely different. You are talking about a clearly observable phenomenon. If this walking up was anything like so easily observed we wouldn't have had a discussion of this length about it at all
I don't think it is as simple as that. Anecdotally, I have a guitar that I have owned since the early 1970s. I have no idea whether it sounds different now compared to when I bought it. It sounds great to me and to most people who have heard it. But I bought it then because I liked it - it sounded great then. Also my hearing, like everyone else, has deteriorated with age so I no longer hear high frequencies as well as I did. Maybe that makes it seem warmer than I remember it to be. If I remember it to be as it was. Frankly, I've no clue.

So,

For some people, this phenomenon is clearly observed. For others, it is not. There can be many different explanations for that difference in view. The problem is that hearing is not a purely physical phenomenon, it also involves psychological processes that are real and determine a significant part of how we judge and remember sound and especially the quality of sounds.

One scenario is that there is actually a measurable change in a guitar. Is that change discernible? Maybe it is by some and not by others. Maybe that can be explained by differences in their physical hearing, or training of the ear. Maybe it is not discernible at all.

Do some guitars change and not others? Maybe the people who have not heard it have not heard the right guitar.

It could also be that there is no change in the guitar. Some people can still hear it differently at different points of time. If you own the guitar, your familiarity with it affects the way you hear it in a way that another person may not be affected. Your hearing could be the thing that has changed. Certainly the fidelity of recording has changed. Just seeing the guitar can affect your expectations of how it will sound and your perception of that sound will change accordingly (like with cedar versus spruce tops).

So, without a sound basis for understanding the phenomenon, we are working from our beliefs rather than our knowledge. That always creates debate.