Preheat the oven to 425°F. Grease a 9-inch-square pan generously. In a bowl whisk together the cornmeal, the flour, the baking powder, the baking soda, and the salt. In a small bowl whisk together the eggs, the buttermilk, and the butter, add the buttermilk mixture to the cornmeal mixture, and stir the batter until it is just combined. Heat the greased pan in the oven for 3 to 5 minutes, or until it is very hot, add the batter, spreading it evenly, and bake the corn bread in the middle of the oven for 15 minutes, or until the top is pale golden and the sides begin to pull away from the edges of the pan. Let the corn bread cool for 5 minutes, turn it out onto a rack, and let it cool completely. Cut the corn bread into 9 squares.