Sound hole proportions

I wonder if there is some formula for establishing the relative size of the sound hole(s) in proportion to the size of the air chamber. Or is this is all a matter of cut and tap?

I ask because I recently bought a Weber Absaroka mandolin, a model that was discontinued a while back. I am going to have to take it up to the factory in Montana to get a split in one of the sides fixed, and I was wondering if I should have binding put around the oval sound hole while it is there. I think it would improve the look of the dark finished body. But would a minor shrinking change to the tone? If so, how?

Then, I got thinking, how do mando luthiers decide on the "F" hole size? Does it vary by stiffness of the top or back. It is just a design, or does the shape actually make a difference? I noticed that Weber added a much larger "D" shaped sound hole to their model options, and I want to ask about that.

Anybody know?

Size matters, but not as much as some people think. By that, I mean that it takes a relatively large difference in sound hole size to make a relative small difference in the Helmholtz (main) resonance of the enclosure. The area of the sound hole(s) can be mathematically calculated for a specific Helmholtz frequency, but the actual frequency may vary resulting from other factors like top, back or rim stiffness, and other vagaries of wood mass and/or stiffness, neck stiffness and mass etc. etc..
For strongest and best(?) sound, the Helmholtz needs to be somewhere near the main resonant frequencies of the top and back so that good coupling occurs, but that does not mean that the hole/s need to be a specific size nor that the Helmholtz needs to be a specific frequency.

The position of the hole/s has quite a bit of influence over the frequencies of some of the air modes of a mandolin, and less influence over the top plate modes, so mandolins with two holes, one to each side of the top, like f-hole mandolins, tend to sound different from mandolins with a single central hole toward the neck end, like oval hole mandolins.

So what difference would binding make in your sound hole? It would slightly lower the Helmholtz frequency. How would that affect the sound you hear? Maybe none at all, maybe slightly. If the hole were cut bigger by the same amount as the binding thickness before binding is added, there would be no change in size and no change in sound.

How do mandolin luthiers decide on f-hole size? Depends. Some cut them smaller to start, then cut them bigger later to "tune" the air chamber. Others (like me) choose a shape and size that suits them visually and work with that. Any normally sized box and normally sized f-holes (no extreme differences from "standard") will give us a Helmholtz somewhere near D# (IIRC), and if coupling with the top and back is the goal, "somewhere near" is good enough. Specific frequencies are neither needed nor easily achieved.

Thanks, John. That answers many of my questions. I am thinking of adding simple white binding but not enlarging the sound hole.
One more: when you and others refer to the "Helmholtz," what is that exactly? Is that a ratio? I mean I know who Hermann Helmholtz was. I read his book "On the Sensations of Tone" many years ago, and still have a copy on my desk. But it seems he has become personally a reference.

When we misuse the word "Helmholtz" in our ramblings about mandolins, we are referring to the main air resonance of the instrument; the frequency or "note" that you hear when you tap the top. (I say "misuse" because a mandolin body is not a true Helmholtz resonator, but functions as a modified Helmholtz resonator, hence the influence over the frequency from the other parts of the instrument. The enclosure of a true Helmholtz resonator would be immobilized, and the port would have a "neck".)