The matrix

Once the prime row (P8) has been established, the matrix is constructed in the "traditional" manner. Starting with the first column and working down, the values for this inversion are found by inverting the progression of addition and subtraction found in the prime row itself. However, since this matrix is comprised of numerical values - and not pitches - its construction hinges on the assumption that anything above unit 9 or below unit 1 is not 0. In the "traditional" matrix, this is not a problem because the pitches extend chromatically in either direction, so any combination of addition or subtraction will always result in one of the twelve pitches. But in this matrix, there are a fixed number of values which are not repeated, so this forces any resulting values that extend above the predefined range - 1 to 9 - to cycle upwards from 1, and any values that extend below the predefined range to cycle downwards from 9. In other words, using modular arithmetic (mod 9).

If one observes the first two units in the prime row (P8), unit 3 is found by subtracting 5 from unit 8 (or subtracting 1 from unit 4 if read from the other direction). The second unit in the first column (I8) - the inversion - should be found by adding 5 to unit 8, which is 13. But 13 extends above the range of 1 to 9, so in order to keep the value within this range, 9 must be subtracted from the total. Therefore, the second value in I8 can be found thus:

if
8 (-5) = 3 (the second value in P8)
then
8 (+5) = 13
and
13 - 9 = 4
therefore
the second value in I8 is 4.

Similarly, to find the second value of the second row (P4), the same rules are applied, but because this is a prime row - not an inversion - the progression of addition and subtraction is not inverted. The total, in this instance, will extend below the range, so 9 must be added to the total, not subtracted:

if
8 (-5) = 3 (the second value in P8)
then
4 (-5) = -1
(-1) + 9 = 8
therefore
the second value in P4 is 8.

The matrix is completed using this method:

It is interesting to note that inverting the progression of addition and subtraction for the construction of the matrix is the same method used to construct the prime row from the individual units in the previous sections. As will be discussed in The Cube section, this same method will be applied to determine the construction of the cube from the individual matrices. In this way, the development of each stage - from a geometric point in space to a 1-dimensional row, to a 2-dimensional matrix, and ultimately to a 3-dimensional cube - is guided by a consistent approach, which provides a clear understanding of both the structure, and the relationships contained therein.

As was mentioned in the previous section, the organization of cell structures found in the prime row (P8) is of great significance in the matrix, particularly those rows forming the outer framework. When the pointer is passed over specific areas in the matrix below, each 3-unit cell - 834, 159, and 672 - is found to be one of two such cells contained within 9 mini-matrices - or sub-matrices - within the larger structure. Two cells of the same cell type are found in each sub-matrix, and each sub-matrix occurs three times.

The symmetrical arrangement of these sub-matrices - and the cells contained therein - are structurally determined by the properties of the matrix whose own symmetry is born from that of the prime row itself. For the purposes of visual analysis, each sub-matrix type has been color-coded.

In the previous section, it was revealed that although unit 8 is the primary axis around which the 3-dimensional cube will be constructed, unit 5 and unit 2 are secondary axes that have equally significant structural roles within the matrix. Unit 5 has already been shown to act as the point of origin around which the prime row (P8) is organized. With reference to the matrix above, unit 5 and unit 2 are found at strategically important points along the edges of the matrix, and work in tandem with unit 8 to solidify the structural framework. If a straight line was drawn between any three of these axes - except diagonally - the same cell structures in P8 can be found here, with only their order changed. P8, for instance, is identical to RI8, as P5 is to RI5, and as P2 is to RI2, but all six of these rows, as well as their inversions and retrogrades, contain the same fundamental cell structures found in the prime row. The outer boundary of the matrix, therefore, in addition to the P2 and I5 rows intersecting in the center, establishes a foundation of three common cells whose structure remains unbroken throughout. The inner rows, whose organization is not as readily evident as it is in the rows forming the outer framework, will play a more important role when the process of inversion is applied to the audio material.

In effect, one could "fold" the entire matrix along an axis perpendicular to the 8-axis - by joining the unit 8 values on the opposite corners of the matrix - and all corresponding numeric values on either side of this theoretical fold would be perfectly superimposed over one another. This small discovery led to another question: What if the same procedure was applied to the other diagonal axis? If this matrix truly is a symmetrical structure, could one not conclude, then, that all corresponding values on opposite sides of either diagonal axis - whether theoretical or not - are equal? Indeed, they are.