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The relationship between density, Young's modulus, and the stiffness of the finished top is a bit tricky. Remember that the controlling factor in how thin, and therefore light, we can make the top is the stiffness, an absolute number: if it's not stiff enough to take the bridge torque you're in trouble.
Young's modulus is a measure of how much force it would take to stretch or compress a standard sized piece of the given material by a certain proportion. This might not seem to relate too closely to stiffness, but remember, it's the stretching and compression of the fibers in the wood that resist the bending, and give it it's stiffness.
You have to realize that there's some leverage involved in this. The material at the surface is being stretched/compressed the most, while the stuff toward the middle gets less strain. If the piece is uniform in thickness and properties there's a plane in the center where there is neither stretching or compression; the 'neurtal axis' or 'center of bending moment'. The result o this surface stretching/compression is that the stiffness; the ability of the piece to resist bending, goes not as the thickness, but as the _cube_ of the thickness. Remember the thread on brace height? Same thing. You can think of a top as being a whole lot of little braces glued together side by side. The actual stiffness of the top will be proportional to the Young's modulus of the material, times the cube of the thickness.
Now, it turns out that the lengthwise Young's modulus of all of the 'usual suspect' top woods that I've tested has been pretty well proportional to the density, irrespective of the species. All of the samples that have a density around 350 kg.m^3 have a Young's modulus near 8000 megaPascals, and the stuff at the other end of the scale runs around 500 kg/m^3 and 160000 mPa. There is some scatter in this data, of course, but less than you might think.
The crosswise stiffness is all over the place, which makes a lot of sense when you remember how much of an effect a small change in grain angle can have. I'm assuming, for simplicity's sake, that lengthwise stiffness has the greatest effect on the top's ability to resist folding up under the string load. I know that's not quite true, but you've got to start someplace.
So, suppose we take one of those WRC tops that's down at the bottom of the chart, and a Red spruce from near the top. The WRC top has about 2/3 the density of the spruce, and half the Young's modulus. Let's suppose that we know from past experiance that spruce like this will be stiff enough if it's .100" thick, and that, on the guitar size I'm making, that will give plate weight of 150 grams before bracing.
If we multiply the thickness by itself three times, we get .001, and multiply that times the Young's modulus of 16000, and you get 16. This is just an 'index number', a mathematical way of comparing tops. The number stands for the absolute stiffness, but without any units that make any sense. We need the cedar top to have the same index number, so we divide that by the Young's modulus of 8000, to get .002, which is the cube of the thickness that we will need to have. That comes out to be about .126"; a lot thicker than the .100" of the Red spruce top. If the two had the same density, the WRC top would weigh 189 grams. However, the WRC is actually only 2/3 the density of the Red spruce, so the top weight will be 2/3 of that, or 126 grams. By using the lower density top you've saved almost 25 grams of weight, despite the lower Young's modulus.
That may not seem like much, especially since we had to go to opposite ends of the chart to get it. In practice, there is not much difference between stuff in the middle of the chart: any two tops that are close to 'average' density and Young's modulus will probably end up weighing about the same when you're done. But, for instruments where the weight really counts, like Classical guitars, it's worth while to seek out lower density wood, especially if it has a notably high Young's modulus.
OTOH, it's quite possible that on some guitars we want a certain amount of mass in the top; maybe that's one thing that keeps the tone from 'breaking up' under a hard attack. There's a lot we don't know for sure yet. Again, the best way to figure out what's important for you is to keep track of what works and what doesn't as you build. Eventually, you'll see the relationships that matter if you collect the right data.
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FAQ
) Density and stiffness may have some relationship but it certainly isn't linear or consistent. Neither does grain count relate to stiffness. You can make general assumptions, like stiffer wood would be generally denser, tighter grained wood would be generally stiffer but it certainly is not always the case. And to further make these assumptions harder to hold, one tonewood log is already selected from a huge population of trees. So there is subjectivity in what a tonewood supplier uses to select a log from the hundreds they would look at and that is based on experience that just becomes almost a gut feeling. But it must be processed into tops before you know for sure what you have. I have some tops that are light and wider spaced grain that are among the stiffest I have, and I have some sets that display bearclaw figure (which Cumpiano and others have stated add to the stiffness) with grain counts of near 50 gpi on the tight to about 25 gpi on the loose side that are not that stiff at all. So the variables are huge! Some of my high end customers want wider spaced grain because they believe that will decrease the weight of the top with less "late wood" present, but they still want STIFFNESS. There are lots of thoughts on this but I have found there is no hard fast rule and actually can support what Al has said, at least from my handling of a few thousand sets. So, your deflection test relates to stiffness and not density. It would be interesting for you to check the weights of your various tops once you have them set at the deflection you want and let us know what the weights are. I suspect there will be a difference. And that difference will occur even with consecutive sets from the same tree!
