Here we go again!
First of all, you cannot tune individual parts. The whole thing vibrates as a whole. If you are tapping one tonebar you are exciting one mode of vibration, and another mode of vibration when you tap the other tone bar. You can see this clearly if you try Chladni patterns. This applies to free plates, partially assembled, or a finished mandolin with the strings on. Siminoff was correct in that it is useful to get the main modes of vibration of the finished mandolin with strings on between notes (from lots of research done on guitars), but the modes are whole body modes, not just individual parts so he got that bit wrong. Problem is, nobody has shown any relationship between modes of vibration of free plates or a partially assembled instrument to modes of vibration of the completed instrument. Note that the tailpiece and bridge will change the frequencies, so it gets complicated, and relationships are very likely to be non linear. There must be some sort of relationship because if you carve a top or back thicker then the frequencies must be higher in a free plate and also in the assembled instrument. The best we can do is to build a database of measurements and note down the sound, and form empirical conclusions without really knowing what the relationships are - i.e. if I do this then I get a certain sound. I did this with oval hole mandolins and published it -
http://petercoombe.com/jaamim7.html
http://petercoombe.com/jaamim8.html
Problem with archtop instruments is that it is difficult to measure modes of vibration on the completed instrument. The best way is with laser interferometry which is very sensitive to frequency, but does not measure amplitude. Unfortunately you do need a laser interferometer which are not cheap so the average Luthier won't have one. Dave Cohen has published papers on this, but the average Joe will find these papers fairly heavy going, but they do indicate reality and not some hair brained theory. Suffice to say, mandolins are similar to guitars, but the frequencies are higher, and the frequencies vary depending on what type of mandolin it is.
At the moment I am measuring free plates and completed instrument measurements on my pancake flattop mandolins to try and understand how this stuff works once the mandolin is assembled - i.e. can I predict what will happen from the free plate measurements. I am using my pancake mandolins because they are about as simple as a mandolin can be, yet still be a decent sounding mandolin, and are much easier to measure than arch top mandolins. This is likely to take years before I can make any sense out of it.
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