Preheat oven to 350°F. Mix flour, cocoa powder, cream of tartar, baking soda and salt in medium bowl. Set aside.
Beat butter and sugar in large bowl with electric mixer on medium speed until light and fluffy. Beat in eggs and vanilla. Gradually stir in flour mixture until well mixed. Stir in chopped cherries. Drop dough by tablespoonfuls onto greased baking sheets.
Bake 10 to 12 minutes or until set. Cool on baking sheets 1 minute. Remove to wire racks; cool completely.