Composer’s Note:

When you write out a succession of related chords, they are structured in two dimensions, as chords and voices, in what I usually refer to as an array. There are a remarkably small number of arrays where both dimensions can be structured with the same relationships between both the chords and the voices. These works are based on sets of tetrachords, or four-note chords, that have these properties. While, according to my theories, there are five families of tetrachords, only four of them can be structured in this way, and each of the etudes is based on a different family of tetrachords. The first and last movements use all seven octaves of the piano, while the second and third use only the more compact range of five octaves. Some passages in the first movement are written on three staves in order to clarify the voices. One of my main concerns in this work has been to make the music playable with a normal hand span for the pianist, and in this I believe that this is successful, although in some passages the fingering may be awkward.