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What it does change is the cross grain stiffness, particularly in soft woods
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Hardwoods are fundamentally different. The reason cross grain stiffness varies in softwoods is because of the rectangular cell structure. A slight angle of the grain alows the cells to distort into parallelograms when the wood is flexed. Hardwoods have cells that are basically circular, so the cross grain stiffness is not materially affected by the cut.
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If mahogany makes a good top, why not rosewood, locust, wenge, etc.
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I have seen plenty of attempts to use dense woods for soundboards, and they all fail, particularly when it comes to volume.
The important qualities of a soundboard are stiff and light. While some hardwoods like locust or rosewood have similar stiffness/density ratios as spruce, being heavier, the top itself cannot have the same high stiffness/weight ratio.
OTOH, you can use softwoods for the back and sides.....you just won't get the characteristic 'rosewood sound' that comes from a dense, hard surface.