In bowl of electric mixer with paddle attachment (or hand mixer with beater attachments), beat butter until light, about 3 minutes.
Add brown and granulated sugars. Beat until light and fluffy, about 5 minutes.
Add eggs one at a time, stopping and scraping down sides of bowl after each addition.
Beat in sour cream and vanilla.
Sift together flour, cornmeal, cinnamon, ginger, baking powder, and salt.
On low speed, mix flour batches into batter in 3 batches.
Gently fold in mango and pineapple.
Pour batter into cake pan.
Bake until top is golden brown and a cake tester inserted into center comes out clean, 50-60 minutes.
Transfer to a wire rack to cool completely.
Invert onto a platter to serve.