There are five or six patterns that most violin plate tuners look at. Since the violin is pretty standardized they come in the same order on pretty much all violins, and we just number them as #1 through #5 or whatever.
There's a heirarchcy of things that I look at when tuning violin plates. 1) Get the mode shapes 'right', if you can. IMO (and it's only my opinion) the shapes of the patterns are probably more important than the actual frequencies, in some ways. The shape tells you something about the distribution of mass and stiffness throughout the plate, while the frequency tells you about the ratio of mass and stiffness. Note that the stiffness distribution will depend on a lot of things: the character of the wood and the arch height and shape, as well as the thickness distribution.
2) Try to match the frequencies of the #2 modes. If the frequencies are much higher or lower than normal you probably have a plate that is too floppy or too stiff, unless it's much heavier or lighter than usual. I'd expect a balsa top to have very high mode frequencies if it's stiff enough to do the job. We're not entirely sure why this matching works, but it may have to do with establishing strong 'B1-'and 'B1+' modes in the completed fiddle.
3) If you can match the #2 modes, try to match the #5 mode frequencies as well.
4) If possible, get the #2 and #5 mode frequencies into an octave relationship.
5) If you can do that, _and_ get the #1 mode on the top (but not on the back) to be an octave below the #2 mode pitch, that might be even better.
These are the rules that Carleen Hutchins derived from her work. I'll note that other makers have come up with different sets of rules. It's possible that the rules that worked for her might not work as well for other makers, since her archings are a bit different from many other folks'.
I will say that the best sounding fiddle I ever worked on was a cheapo that I regraduated for practice. The archings and wood were such that, when I got the graduations 'right' the tuning was 'perfect'. The top modes were: #1 - 90 Hz, #2 - 180 Hz, #5 - 360 Hz. The back modes were: #1 - 110 Hz, #2 - 180 Hz, #5 - 360 Hz. It was still a wretched looking French cheapo, but the sound was glorious.
In the end, I feel that any stringed instrument is a system with many co-dependant variables. None of them are really independant, and the 'right' value for any of the variables will depend strongly on what the others are. If you get all of them 'right' you've got a great instrument. If you get a strong majority of them 'right' you end up with a good instrument, and it almost doesn't matter at that point which ones you get right or what the rest are so long as they are not too far out. There is a range of possible values for any of them, and values that simply won't work well, no matter what you do. Plate tuning is almost certainly one of the variables, and it was explored early because it was fairly easy to look at. It's not 'The Secret', but it's part of the secret. I think the secret is knowing what the permissable ranges of all the variables are, and how they each effect the others, so that you can tweak them in such a way as to get the best possible balance without putting any of them out of range. In other words, the secret is in knowing what you're doing. Simple, really.
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