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Place rack in center of oven.
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Preheat oven to 330 degrees F.
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Cream butter with sugars until light and fluffy.
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Add salt, vanilla and cornmeal and beat until combined well.
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Add flour, ¼ cup at a time, beating until just combined.
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Divide dough into 4 balls and on each of 2 baking sheets, press 2 of the balls into 5 inch rounds.
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With fork tines, score each round into 6 wedges and prick edges decoratively.
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Bake for 20-30 minutes, or until shortbread is just firm.
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Cut halfway through each round along prick marks and transfer to racks to cool.