Richard Bridge -

You are quite correct that there are additional "resonances" in most instruments. The point, however, is that the hole only affects the Helmholtz resonance of the box, and is only useful for tuning that particular resonance.

If you wish to tune all the other ones, you do it using other methods, as by adjusting the thickness and/or width/length of the panels that vibrate.

I've seen a few guitars where rather fancy sound holes were used to "avoid the bracing" on the theory that the "other resonances" could be enhanced in some way; but never could really detect any significant benefit. Of course, I'm not a guitar whanger so it's possible that the benefit obtained just eluded me.

In a violin, the top is really quite tiny compared to other instruments in similar pitch ranges, and some have theorized that the F-hole "aligns with the arch" to avoid reducing the stiffness in the longer dimension of the box; and that may be an additional - and significant - reason for the absence of (m)any fiddles I've seen with round holes.

For a mando, especially, the top plate is relatively large (compared to a fiddle), making it somewhat less likely that "higher modes" will be strongly enough excited to need "tuning," and the absence of bracing in (most) mandos gives the luthier a great deal of freedom in putting holes - of the proper total size - just about anywhere.

A more difficult geralization would be whether an arch-topped mando is "usually" better than a flat top.

The usual "arch" in a fiddle top actually is quite complex, with convex and concave curves all over the place. A luthier has incredible flexibility in "tuning the plate" since both the amount of wood(?) trimmed and the exact location where it's removed can have fairly large, but different, effects on both the fundamental frequency for one or more modes of vibration and the coupling between modes.

Unfortunately, there are no general rules about "where to cut" and/or "how much" for any particular effect, although there are many theories - a few of which may be "sort of valid." Producing an "exceptional" fiddle depends largely on "working by feel" based on experience.

Mando tops, when arched, generally have much less complex curves, so "working the curvature" - actually adjusting the local thicknesses once the basic form is developed - should, in principal, be a little less difficult than for a fiddle; although I've seen no evidence that most luthiers spend a lot of time in trying to tune the "plate resonances" beyond getting the "tap tone" somewhere "about right."

There also is no general agreement on what any plates should be tuned to, although getting the lowest plate mode near the box resonance, or a third, or fourth, or a fifth from it, seem to be a rough guides used by some. (There is some agreement that the top plate and lower plate should be at different pitches.) It seems that most mando builders usually just "stop when it looks right."

In principle, an arched mando top should offer more "tunability," but I doubt that many luthiers do all that much precise tuning of them. (It's quite obvious that my fairly nice little arch (plywood) topped mando (A-style) has not had any "detail" plate tuning during its assembly.)

John