Multiple Order Functions in Twelve-Tone Music

Abstract

THIS paper is concerned with a particular approach to twelve-tone rhythmic and harmonic structure which is founded upon certain relationships between linear and vertical pitch functions. The pitch functions under discussion are those of the order positions of pitches in twelve-tone sets. The collection of sets under discussion comprises two groups. The sets in each group possess common unique properties. The total number of such sets is somewhat less than the total number of twelve-tone sets. However, the number of such sets' is large enough to suggest a general compositional procedure rather than techniques which can only be sensibly applied in the ways in which I have already applied them. The properties discussed herein have, to my knowledge, never been discussed before. The discovery of these properties was solely the result of a gradual development of similar but less clearly rhythmic techniques in my own compositional activity from 1956 to 1960. Since 1960 my compositions have each involved, in varying degrees, the application of techniques described in this paper. There may be many and various applications of these properties. Of several compositional advantages of the applications described herein, one seems to me to be of particular value. These properties make it possible to compose passages regarding which it is reasonable to interpret each pitch as being in several set instances simultaneously. For example, in an orchestral piece a particular pitch performed on the flute can be reasonably shown to be in order position 3 of the form of the set(s) presented by the flute, in order position 5 of the form of the set(s) presented by all of the woodwinds, and in order position 6 of the form of the set(s) presented by the entire orchestra. This is called multiple order function. When each pitch can meaningfully be shown to function in this way there is a relationship between vertical