1. In an extra-large bowl allow egg whites to stand at room temperature for 30 minutes. Add cream of tartar to egg whites. Beat with an electric mixer on medium speed until soft peaks form (tips curl). Gradually add the 1-1/3 cups extra-fine sugar, about 2 tablespoons at a time, beating until stiff peaks form (tips stand straight).
2. Sprinkle orange peel over beaten egg whites. Sift about one-fourth of the flour over beaten egg whites; fold in gently. Repeat, folding in the remaining flour by fourths. Pour batter into an ungreased 10-inch tube pan. Gently cut through batter to remove any large air pockets.
3. Bake on the lowest rack in a 350 degrees oven about 40 minutes or until top springs back when lightly touched. Immediately invert cake (leave in pan); cool thoroughly. Loosen sides of cake from pan; remove cake.
4. Meanwhile, for compote, in a medium saucepan combine water, apricots, cranberries, orange juice, the 1/3 cup sugar, the cinnamon, vanilla, and almond extract. Bring to boiling; reduce heat. Simmer, covered, for 10 minutes. Remove from heat. Discard cinnamon. Transfer the compote to a serving bowl. Serve warm or cool over cake slices.