11. As Above and Below… If we forget for a minute about counting, and constrain the matrix to a range of 1/2 to 2/1, we see a great widening beam emitted from an undefined center [0/0]. n=32 This beam contains within its median region [.5 < r < 2], all the pitches that appear above or below. This leads us make to an important observation: All the ratios of the matrix defined by H[n,n] (cell[0]+1)/(cell[1]+1) are contained with the submatrix: H[(n/2+1),(n/2+1)] (cell[0]+dim[0]1)/(cell[1]+dim[0]1) Here the reflection: H[n,n] given by (cell[1]+1)/(cell[0]+1) is contained within: (cell[1]+dim[0]1)/(cell[0]+dim[0]1) That is, if we look only at the bottom quarter of a large matrix we find everything within the entire matrix. I offer a corollary to the famous alchemical lemma ’as above, so below’: ‘as above & below, so in the middle.’ Please note, the dimensions of this matrix are made odd to include 1/2 and 2 within the output. This is not necessary, I just like symmetry. When we look at what’s going on here in simple terms, we can see that to produce the next bottom corner matrix, we multiply each element of the previous matrix by two, and then insert the next ratio [in a manner similar to fitting in an infinite number of new guests into the Hilbert Hotel]. This is better seen by example: 1 2/1 1/2 2/2 now multiply each element by 2 2/2 4/2 4/2 4/4 and then inserting the missing ratios, we arrive at: 2/2 3/2 4/2 2/3 3/3 4/3 2/4 3/4 4/4 The first column and top row are redundant. Thus, 3/3 4/3 3/4 4/4 contains all the unique pitches of the first matrix plus the new ones. Again, doubling and inserting, omitting 4 as redundant, we arrive at: 5/5 6/5 7/5 8/5 5/6 6/6 7/6 8/6 5/7 6/7 7/7 8/7 5/8 6/8 7/8 8/8 etc. Basically, we are describing a fractal construction, where each successive iteration of the series contains all parts of the preceding matrix, at each level revealing additional detail or complexity. Using this information we can see how the harmonic series can be reduced to a patterned growing series of odd [n>2] numbers. We can generalize this series as: 1 2 2 3 4 4 5 6 7 8 8 9 10 11 12 13 14 15 16 … And reducing to lowest terms: H[2] 1 2 H[4] 1 3 2 H[8] 1 5 3 7 2 H[16] 1 9 5 11 3 13 7 15 2 H[32] 1, 17, 9, 19, 5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 15, 31, 2 Mathophiles can compare this with the Fibonacci series, and Sierpinski’s Triangle. Take a second look at H[16] reduced to H[916,916] and find the duplicates. While most have been eliminated, there are still a few, found in relation to the power of 3 series 3, 9, 27, 81 … 3 6 9 9 12 15 18 21 24 27 What a strange lonely creature is this number 9. Prime in relation to 2, it shares the same relationship to 3 6 12, as 3 does to 1 2 4. So we begin to see a larger pyramid of relationships forming over top of the foundational 2. This pattern continues on to include 3, 5, 7, etc. Having exposed this interlocking / overlapping series of hyperpyramids as the overarching structure of the harmonic matrix, we draw near the end of this part of the discussion. Doubtless, there are other properties within the structure I have not yet found, and certainly a formal exposition could be articulated in the symbolic language of mathematics [and I would invite anyone with these talents to help out with this]. For my purposes, this symbolic language is secondary to the ability of the ear to discern these relationships. next  12. The Completion Stage
