I've been mulling replies for several days, and even started typing a few times. After a page or two I give it up. The problem is that I started working with Carleen Hutchins back in 1980, and met Fred Dickens (who was the first person I know of to plate tune on guitars) at about that same time. That's a lot of experience and discussion to summarize in a post. I'll try to describe my current thoughts as briefly as I can, but I make no promises.
Back at the beginning (in the '70s), not much was known about how these things worked, and it was hard to find out. Carleen had $20,000 worth of B&K lab equipment for making spectrum charts on violins, and it was a laborious process. She would run spectra on the 'free' plates, and try to optimize them as best she could, but without any knowledge of that the actual modes were it was a shot in the dark. Laser holography opened that door, and lead to the re-discovery of Chladni testing. Affordable and useful test equipment for that was still hard to get, but at least it was possible on a luthier's income. Now, of course, it's almost trivial.
There were two strands in the research. One was to see what older authorities, such as Savart, had to say. He had actually looked at Chladni patterns on some Strad violins, and gotten pitches. Once in a while somebody got a chance to look at a 'great' instrument and come up with a bit more such data. Needless to say, since it's less common to disassemble guitars, we don't have much of that. Another was to look for correlations in our own work.
'Correlation is not causation', of course, but there is no causation without it. Correlation, backed up with some sort of plausible mechanism can be a place to start.
As has been pointed out, if you knew enough about all of the parts of the instrument, and about how they are coupled to each other, you could predict the final vibration modes of he completed instrument. It's been said that the only branch of physics that's more complicated than acoustics is quantum mechanics, so getting the point A, describing the instrument, to point B, predicting the sound, is extremely difficult, even with the best modern computers. Remember the state of the art in the '80s and '90s? Even now it's hard for computer models to cope with a guitar, even
without factoring in the local variation in the wood.
For me, at least, with my limited math chops, experimentation seemed to be the best way forward. The most productive ones in terms of plate tuning were 'pair' experiments. Instruments were made that were as well matched as I could manage in ways that were thought to be important, and players and listeners were asked to judge the results.
One of the first 'pairs' tested the notion that 'free' plate pitches would predict the final sound. Two classical guitars were made from redwood and mahogany that was cut 'in flitch', and the frequencies of the first ten modes of the tops and backs were matched to within 2Hz. The weights and so on were also very close, and the modes of the assembled guitars also matched well in pitch. The problem was that they didn't sound the same: they were very similar, but most players and listeners preferred one of them over the other.
Part of this outcome was due to a failure on my part: I didn't use perfectly straight grained wood for the fan bracing. There was a wave caused by a distant knot that made the grain in the braces arc vertically from one end to the other very slightly. Some of the fans arced 'up' and others 'down'. This necessitated profiling the braces a bit differently in order to get the pitches of the modes to match well, and the mode shapes were a bit different. The preference was for the top that had the 'better' mode shape for the 'ring+' mode.
Keep in mind that the pitch of the modes 'encodes' information about the overall relationship between mass and stiffness, while mode shapes encode information about their distribution. The absolute pitches of the 'free' plate modes say little about the pitches of the assembled modes of the guitar. What can we learn from the shapes? At this point we're moving into conjecture, but with some justification.
Discussion with other makers who use these methods, and particularly with Mark Blanchard (who is smarter than I am, and keeps better records) has lead to the hypothesis that the free plate modes are indicators of the mass and stiffness distribution in the plate that set up the higher order modes of the assembled guitar.
This fits in with a paradox that was pointed out years ago. Most guitars of reasonable quality work very similarly in the low frequency regime; you have to mess up royally to make one that departs very far from 'normal' in this range. The differences between 'average good' guitars and 'great' ones seem mostly to be in the high frequency area, particularly between 2-4 kHz., where 'normal' hearing acuity is greatest. The problem is that there are so many resonances of the top, back, and air, in that region that they overlap; you're in what is called a 'resonance continuum'. The response is so complex that the maker has no prior control over what happens in any detail. All you can say with any assurance is that, if there are twenty peaks in the output in the octave between 2kHz and 4 kHz, there are twenty resonances, but it's possible (and even likely) that none of them is at the pitch of any peak, and you can't say whether any given peak is primarily an 'air,' a 'top' or a 'back' peak. So, if the maker has no direct control over what it is that makes them sound good how is it that some makers can consistently make better guitars than others?
What we have observed is that guitars that have a larger number of well-formed 'free' plate modes, particularly in the top, tend to be preferred by players and listeners. A recent 'pair' experiment also showed that small differences in the shapes of higher order 'free' top modes were carried over into changes in the shapes of the high frequency modes of the assembled top, and could be seen in the output spectrum of the guitar. In that pair of mahogany/Red spruce OMs the spectra below 1000 Hz were virtually 'identical' but they diverged at higher frequencies. In 'blind' tests listeners were easily able to distinguish them as 'different', but there was no consistent preference for one or the other.
The idea, then, is that bracing is a necessary evil. We could make the top stiff enough to hold up under the string load by making it thicker, but that would be too heavy. We use bracing to get stiffness without commensurate weight, but if it's not profiled carefully it will restrict the ways the top can vibrate at higher frequencies. We can use Chladni patterns to see where we might want to remove material from the braces to get the correct 'balance' of stiffness everywhere. When well done this can enable the top to work better at high frequencies, producing a 'clearer' sound.
The design of the guitar as we have it has been quite well optimized over centuries simply by trial and error. A standard design, carefully made of good wood, will generally end up being a better than average guitar. Some attention paid to getting the 'signature' modes, the 'main top', 'main air', and so on, to work well, with elevate the quality further.
When we made a top plate tuning video a few years ago the producer wanted to bill it as a 'scientific' approach to lutherie, but I wouldn't let him. What we're doing here is just barely 'science', simply because it's a tough row to hoe. 'Free' plate tuning, by itself, is simply a 'tech' version of the old 'tap tone' method, that gives more information. As such, it's useful, although it has the drawback of taking a lot more time than most tap tuners spend.
When we made the video, he wanted me to talk about backs. I'm still having problems getting those to work right.....
