For firm polenta use 4 cups of water; for soft polenta use 5 cups of water. Bring water to a boil in a medium-size, heavy sauce pan over high heat. Add 1 teaspoon of salt.
Pour cornmeal slowly into water, stirring with a wire whisk or wooden spoon. Continue stirring as mixture thickens, 2 to 3 minutes.
Turn heat to low. Cook for at least 45 minutes, stirring every 10 minutes or so. If polenta becomes quite thick, thin it with ½ cup water, stir well and continue cooking. Add up to 1 cup more water as necessary, to keep polenta soft enough to stir.
Put a spoonful on a plate, let it cool, then taste. Grains should be swollen and taste cooked, not raw. Adjust salt and pepper if you wish.
For firm polenta, lightly butter a baking sheet or shallow dish, approximately 8½ x 11 inches. Carefully pour polenta into pan. Using a spatula, spread polenta to a thickness of ¾ inch. Cool to room temperature to allow polenta to solidify. Cover and refrigerate for up to 3 days.
For soft polenta, add 6 tablespoons butter to pot and stir well. Serve immediately or transfer to a double boiler set over low heat, cover and keep warm for up to an hour or so. Stir well before spooning into low soup bowls. Sprinkle with Parmesan, if desired.