Official Luthiers Forum!

"At what point (does) a stack of linear responses acting on each other becomes non-linear?" A stack of truly linear responses will never become non-linear. Of course, in the real world (as opposed to the idealized world of mathematical models) all "linear" systems become non-linear when driven hard enough -- and, in fact, almost all will exhibit some degree of non-linearity even in their "linear" region. But the headroom problem that is heard in some guitars does not seem to be of that subtle nature -- it's, apparently, significant and easily noticeable. So, as you say, Alan, experiments.

There are, to greatly oversimplify, two classes of non-linearities: one results when a system has a smooth, but varying, output in response to input. An example is a gas spring; its resistance to deflection is continuous and smooth but clearly non-linear, the stiffness increasing rapidly as the gas is compressed. If guitar headroom issues were due to deformation of the soundboard, this type of non-linearity would likely be the cause. Alternatively are responses that change due to threshold issues. Clipping, as in an overdriven amplifier is an example, but there are more subtle threshold issues as well, such as found in a system that has a certain linear behavior at low input levels and a different linear behavior above some threshold drive level. An example is a forward-biased silicon diode that has minimal conduction until the voltage across it is about 0.7 volts. A similar sort of non-linearity could occur in a guitar if, for example, the strings "slid" along the saddle when the guitar was strummed vigorously, but, due to static friction, stayed in fixed positions with fingerpicking or light strumming.

So, suggestion for an experiment: starting with a guitar that exhibits limited headroom, install a saddle with small notches at points where the strings cross the saddle. Does the headroom threshold change?

I don't know about strings sliding on top of the saddle or a fret, but they do roll. Woodhouse published a paper on that in the EAA Journal. He noticed a slight pitch 'splitting' and tracked it back to the string having a different length depending on the vibration polarity. I don't think that'a an issue with headroom.

"...all "linear" systems become non-linear when driven hard enough -- and, in fact, almost all will exhibit some degree of non-linearity even in their "linear" region."

Exactly. Everything is linear when the displacements are 'small', but what are the limits of 'small'? We have a hearing apparatus that has evolved over a long time to note 'small' changes: we are all the descendants of the folks who could hear a tiger sneaking through he bushes, and he was trying to be quiet.

The sound of the guitar is shaped by it's resonant structure. If you look closely you'll find something like a dozen resolvable resonances in the top below 1000 Hz, and a similar number in the back, and it the air in the box. Each can be characterized by a central peak frequency and a band width where it can be easily driven. Resonances that have overlapping bandwidths can drive each other easily if there is some method of coupling. This is the basis of the 'Butterworth filter' or the bass reflex speaker enclosure. The output of a properly designed bass reflex cab meanders up and down over a fairly wide frequency range, but never varies from the man level by more than 3 dB or so. Since our hearing is not very sensitive to changes in power we don't notice them, and hear the response as 'flat' although it's not.

On the guitar you start getting into a 'resonance continuum' somewhere between 500-1000 Hz in most cases; there are so many resonances and they overlap so much that is it impossible to say with any certainty that a given peak in the output is, say, a 'top' resonance. You can say that if there are ten peaks in the octave from 500-1000 that there are ten resonance in that range, but it's unlikely that any of them would occur at the pitches of the output peaks in the spectrum if you could isolate them. You can go back and tweak things after the instrument is together and get the output to match some arbitrary model within limits, but there is no way to control what happens here in advance, particularly as you go higher in pitch. In the 2-4 kHz range you're almost totally at the mercy of small local variation in wood stiffness and mass, and this is the range where you're most attuned to small variations in sound.

I've tried several times to make 'matched' guitars that sound the same. I'll keep trying, but I'm pretty well convinced that so long as we're working with wood it won't be possible. The most recent pair had impulse spectra that were 'identical' to all intents and purposes up to about 1000 Hz, but everybody who heard them in a 'blind' listening test could tell them apart.

Remember; each one of those resonances was the outcome of a 'linear' process, and so was the coupling between them. The output is not even close.

Alan: We're on the same page regarding your comments. According to Wikipedia, "The dynamic range of human hearing is roughly 140 dB, varying with frequency, from the threshold of hearing (around −9 dB SPL at 3 kHz) to the threshold of pain (from 120–140 dB SPL)." And the number of identifiable guitar resonances, even below 1000 hz, is astounding.

I'm currently building two guitars that are very similar. They're both made from adjacent cuts of the same source wood and shaped and braced similarly. The major resonances of the two guitars are within 1 hz of each other; the other resonances below 200 hz are somewhat similar between the two guitars; the 40 or so identifiable resonances between 200 and 1000 hz show NO similarity. And above that (where human hearing is most sensitive) it all appears a forest of hair. And, of course, each of the resonances is coupled to all of the other resonances to varying degrees. Though the 200-1000 hz resonance peaks are typically 20 to 30 db lower than the major resonance peaks, our logarithmic hearing, coupled with our frequency dependent sensitivity, assures that these two guitars will not sound the same.

That's why I'll suggest that any experiment regarding perception of guitar tone should be done with a single guitar, recorded and then modified regarding the characteristic of interest, and re-recorded. The recordings played in rapid sequence may provide valid perspective (though human memory for audio details is far from excellent).

"That's why I'll suggest that any experiment regarding perception of guitar tone should be done with a single guitar, recorded and then modified regarding the characteristic of interest, and re-recorded. The recordings played in rapid sequence may provide valid perspective (though human memory for audio details is far from excellent)."

That's what I did when I wanted to check on the relative effects of string height off the top and break angle over the saddle. It was really the only way to do it. There are cases, such as comparing the effects of a side 'port' open or closed where you can do it live, so long as you can switch from one state to the other quickly and silently, but there are really not too many 'interesting' things you can do that way (alas!).

If one could make a 'matched' pair it would certainly open up lots of possibilities; stuff from the effect of different materials to 'playing in'. Since that doesn't appear to be likely the only way to answer those questions would be with big tests involving lots of guitars and statistical treatments, or else computer modeling studies. Neither is likely to satisfy the critics.

For that matter, even recorded tests are problematic. I guess we just have to accept that with such a complex system we're precluded from being able to do much with simple tests. Oh well, we can keep chipping away at it.