Symmetry has always been an important element in my work. In the past I have undertaken various approaches in achieving symmetry during the compositional process, and the theoretical model which I am about to demonstrate is aimed at integrating the elements of audio and visual composition into a cohesive whole whose basis is a perfectly symmetrical matrix. The result is not a piece of music, nor is it a work of visual art, but an appreciation of the relationships between audio and visual elements that are created by the inherent symmetry of this model. The process of analyzing and understanding the interaction between these elements is perhaps more important than any tangible piece of art born from its logic.

When I was first exposed to the 12-tone matrix as a lowly undergraduate student in 20th century theory class, it came as no surprise to me that the concept of a matrix was more interesting than the music it shaped. The beauty of the matrix lay in its structure - in its symmetry.

Much later, I started to investigate the matrix on a more numerical basis, in the distant hopes of creating one whose symmetry - whether musically "profitable" or not - was not just a property reserved for individual rows, but was the basis on which the entire structure was built: A symmetry that was absolute.

I soon found that in order for the matrix to be truly symmetrical, the prime row could not contain an even number of "units" because this would preclude the existence of a central point - an axis - around which the opposing sides could "pivot". This axis - as I will demonstrate in The Matrix section - is paramount, and is the essential element in the creation of a symmetrical framework. Initially it was difficult to turn my back on the "traditional" 12-tone structure, but I discovered that the use of nine units instead of twelve was the turning point, and one which would ultimately reveal the possibilities of a perfectly symmetrical structure. This approach will be analyzed in the following sections.

It may have been a product of my visual arts experience, or perhaps just the recognition that there was more at work than simply numbers on a grid, but I began to discover an underlying organization of flat shapes between groups of numbers whose visual properties I did not think could be represented aurally. Or could they? It became clear that the challenge would be to not only create a balanced structure, but to communicate the relationship between the numerical, the audio, and the visual properties derived from them in a suitable environment.

So.

How does one present this kind of material? The possibilities are vast: An installation, a lecture, a guided multimedia performance, a theoretical paper, an interactive CD-ROM, or... the internet. In truth, I cannot think of a better medium in which to present such unusual material to an audience. (I urge you to take note of the recommendations below before continuing with the next section.)

After I had satisfied myself that indeed the numeric properties of the matrix could be represented visually - and perhaps aurally - I made a startling discovery: What if the matrix was "mapped" onto the sides of a three-dimensional cube, and if so, what new audio-visual possibilities would be revealed from this geometric framework? What if the flat, two-dimensional shapes derived from the numeric matrix were used instead of the individual numeric units themselves? A pattern emerged:

The unit - in this case, nine of them being used to construct the prime row - is a point. A point is a geometric element in time or space which has no length, height or depth.

The row, or line, is a one-dimensional element comprised of at least two points, or units - in this case nine. A line only has length.

The matrix has length and height, so it is a two-dimensional element. In this case, the matrix is comprised of nine intersecting lines, or rows which define its dimensions.

Therefore, it should follow:

The cube, which is a three-dimensional element, is comprised of six sides, or matrices. A cube has length, height and depth.

The significance of this approach will become more apparent during the course of my analysis.

It should be understood that it occasionally became neccessary to take some minor liberties - procedural choices, if you will - during this study, and these few instances will be noted as such.

Recommendations

Much of the audio content on this site relies upon repetition, so in those instances where accurate looping was a necessity I opted to use the .WAV audio format, since both PC and Apple platforms support it. The .WAV format can be played from the browser itself, but the looping is not accurate - a serious problem, considering the need for seamless repetition in some instances. This problem can be overcome if the audio files are downloaded and played directly from the hard disk using a simple audio application like Windows Media Player, Real Player or Quicktime (with looping enabled) . Although not required in order to hear the audio on this site, I recommend that looping be enabled in the playback application, as this will assist me in effectively communicating this material.

To download any of these audio files in Windows, right-click on the link and select the "Save as..." or "Download Link to Disk" option. In Macintosh, hold the mouse button down on the link, and select the "Save Link as..." or "Download Link to Disk" option. To save time while reading, none of the files require unstuffing or unzipping. The Quicktime plug-in will be required for playback directly from the browser.

The following section analyzes the evolution and the properties of the unit.