Double concerto for violin, cello, and piano, Beethoven's Triple Concerto for Violin, Cello, Piano and Orchestra being perhaps the most notable, but there have been many; see the list of triple concertos for violin, cello, and piano. A notable quadruple concerto is Olivier Messiaen's incomplete "Concert à quatre". Double concerto A double concerto (Italian: "Doppio concerto"; German: "Doppelkonzert") refers to two distinct variations on the concerto. Most often, it refers to a concerto featuring two performers, as opposed to the usual single performer, in the solo role. These two performers' instruments may be of the same type, as in Bach's, Concerto for
Quadruple product Quadruple product In mathematics, the quadruple product is a product of four vectors in three-dimensional Euclidean space. The name "quadruple product" is used for two different products, the scalar-valued scalar quadruple product and the vector-valued vector quadruple product or vector product of four vectors . The scalar quadruple product is defined as the dot product of two cross products: where a, b, c, d are vectors in three-dimensional Euclidean space. It can be evaluated using the identity: or using the determinant: The vector quadruple product is defined as the cross product of two cross products: where a, b, c, d