Anatomy of the six-part all-partition array as used by Milton Babbitt:Preliminary efforts towards a computational method of automatic generation

摘要

For Milton Babbitt (1916–2011), twelve-tone techniques were indispensable forms of musical composition because they are fundamentally derived from mathematical constructs. Perhaps his most complex form of composition is the all-partition array. Its large-scale structure is formed by concatenated permutations of 12-tone rows under the common class of musical equivalencerelations, transposition, inversion, retrograde and retrograde inversion, and then organized into pairs of hexachordally combinatorial rows. The resultant pitch-class space is then partitioned into segment lengths of 12 integers or fewer distributed into a specified number of parts. Both its organization and number of parts are determined in part by the unique mathematical propertiesof all-combinatorial hexachords. Through analyses of two works representative of two types of six-part arrays, initial ground work is established for the automatic generation of the all-partition array structure. Preliminary concerns in the computational process are also addressed.