Pierre Boulez (1971,) described an operation he called pitch multiplication, which is somewhat akin to the Cartesian product of pitch class sets. Given two sets, the result of pitch multiplication will be the set of sums (modulo 12) of all possible pairings of elements between the original two sets. Its definition:
For example, if multiplying a C major chord with a dyad containing C,D, the result is:
In this example, a set of 3 pitches multiplied with a set of 2 pitches gives a new set of 3 × 2 pitches. Given the limited space of modulo 12 arithmetic, when using this procedure very often duplicate tones are produced, which are generally omitted. This technique was used most famously in Boulez's 1955 masterpiece Le marteau sans maître, as well as in his Third Piano Sonata, Pli selon pli, Eclat (and Eclat multiples), Figures-Doubles-Prisms, Domaines, and Cummings ist der Dichter, as well as the withdrawn choral work, Oubli signal lapidé (1952) (Koblyakov 1990; Heinemann 1993 and 1998).
Read more about this topic: Multiplication (music)
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