The structure of this matrix is based on 9 points, or units, and the significance of this number becomes more apparent as the matrix evolves. The "traditional" 12-unit matrix evolved from - and was inextricably linked to - the octave, but artificially marrying a 9-unit structure to a 12-tone octave defeats the purpose of creating a matrix whose symmetry is absolute, for what becomes of the remaining 3 notes?

I understood the problem thus: The constraints imposed by the structure of the octave should not interfere with the logic of the matrix itself. Mark Trayle - a composition teacher at the California Institute of the Arts - suggested that I disregard the octave entirely and make use of the overtone series instead. The use of partials would allow me to assign each unit to a specific frequency, providing an effective aural representation of the structure without the constraints of tonality. In addition, I was attracted to the concept of pure sinusoidal waveforms as the basis for the audio material, and this choice would prove to be an important one in later developments of the matrix. Traditionally, the units in a matrix have been allowed to sound in different registers: A#3 was interchangeable with A#4, D5 with D2, and so on. However, due to the absence of any octaval framework in this model, and to maintain consistency - symmetry - throughout the matrix, none of the units are repeated in any other "register". Simply because the frequency of A#4 is twice that of A#3, the frequency of the 4th partial, for instance, is perceived to be characteristically different from that of the 2nd partial. I believe this approach preserves each unit's individuality, and prevents ambiguities from occurring in later stages of development.

This structure is comprised of the fundamental and the first 8 overtones, numbered 1 through 9 respectively. Determining which frequency is to be the fundamental is an arbitrary exercise. In this study, the fundamental is 65.41Hz, or C2.

Audio

In later sections, these frequencies will also be modified by inversion, effectively doubling the overall range of all frequencies occurring in the matrix.

The following section analyzes the arrangement of these units for the construction of the 1-dimensional row.