Composing with Rotational Arrays

In the later part of Igor Stravinsky’s career (1882 – 1971), in the 1950s, he started writing serial and twelve-tone music by creating a hexachord array and rotating the notes in this array to be used for both melodic lines and structural harmony.  This technique became Stravinsky’s most characteristic element in his compositions.  The hexachord is acted upon by transposition, either up or down to get back to the first note of the original hexachord, which is repeated for each new hexachord until the original melodic line is obtained.  This creates an array of lines that are all related by transposition, and the intervals of the transpositions are complements of mod (12) of the interval within the original hexachord.  For example, if we take the following melody:

A     A#     D     C     E     F

then rotate the A note to the end of the array, then transpose the array to begin on the note A, it would resemble the following diagram:

where the blue arrays represent the new transposed values, and the black arrays represent the precursor to the transposition.  By the way, you can use any size array you want to, it does not have to be a hexachord, the array could consist of any number of notes.  The following piece of music represent each blue array as sets that are used one after another.  When used in this manner, the frequency of a repeated pitch class is useful for situations requiring pitch symmetry.  Please have a look at the following piece of music called, Rotational Translation V, which used the above theory of rotational arrays:

There are other ways to compose using rotational arrays.  For example, you could take the array and apply the retrograde, inverse, and retrograde-inverse and extend the piece of music into more dimensions.