12. The Completion Stage Most of my life I thought that I had to find an alternative to harmony, but the harmony I was thinking about was the one that had been taught at school. Now I see that everything outside of school is also harmonious.  John Cage We can create variations upon the harmonic matrix by making a simple observation. Let’s return to the dominant and subdominant harmonic series again. P = 1 2 3 4 5 6 7 8 … M = 1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 … By logical extension of the principle, we construct two additional matrices: The Dominant Squared and the Subdominant Squared. H[n,n] = P*P or M*M giving: 1 n n n^2 These numbers become very big [& tiny] quickly, so to keep them within a friendly range, we divide by n: H[n,n] 1/n 1 1 n  MAMA & PAPA   In the P^2 matrix we see the P series ascending on both sides of the matrix now, and in the M^2 matrix the M series descending. Have a listen to these two P^2 and M^2. Each has its own distinct sound. We further see that each matrix is symmetrical across its diagonal but not reducible further. Upon closer examination one finds they are reciprocal of each other. 1/P^2 = M^2 Now, ready for some magic? Start small and watch the note content as the matrix dimensions increase. We’ll use P for now: P^2[2] D P^2[3] G D P^2[4] D A E D P^2[5] Bb F D A C What’s going on? Our Dominant Matrix is being transposed at each step, but it does so in accordance with [should we be surprised?] by the Subdominant Series! In other words, as Pa heads for the stars, he’s held back by Ma. As Ma sweeps the hearth, she’s lifted up by Pa. Now, when the two series are written as powers of 2 we have: P= 2^0, 2^1, 2^(log2(3)), 2^2, 2^(log2(5)), … M= 2^0, 2^1/2, 2^(log2(1/3)), 2^(log2(1/4)), 2^(log2(1/5)), … This is just what’s needed for the universe to make that sleight of hand jump over an undefined [1/0, 0/0, 0/1] to connect the two! We now write both the dominant and subdominant harmonic series as one complete statement, the ‘Sa’ series: n={…3, 2,1, 0, 1, 2, 3 …} if n< 0 then n=1/abs(n) S= 2^(log2(n)) This gives us the Grand Harmonic Matrix GH[n] = S*S
For the sake of completeness, we now present a final few variants, beginning with the arithmetic and geometric equal division matrices. Having already seen the method of dividing the octave into any number of equal parts [12 in ET, etc.], or for that matter the equal division of any interval into any number of parts, we can construct new matrices by squaring these series. Take for an example,12 EDs of the octave, or ET. Multiplying this series by itself reproduces only the twelve pitches we started with. Should we apply an ordering to the original row, we see the matrix sets used in the creation of traditional12tone ET serial music of the late 20^{th} Century. For even more possibilities, we take two different equal division series and put them together. For example, 11 ED of the 7/4 and multiply it by 13 EDs of 17/8, to name only one arbitrary construction. Also, consider the matrix of arithmetic equal divisions of an interval. This special matrix, [and to date I am most familiar with the sound of the arithmetic division of the octave matrix only] has a beautiful shape and sound as seen in its picture. n=16 It is also symmetrical across the diagonal, but not reducible further. Likewise, unequal constructions can be made using two different arithmetic divisions. I have not spoken at all of tempered, or arbitrarily constructed series of unequal parts and their transpositions, but clearly this is another avenue of exploration that some may embark upon should they ever become desirous of an alternative to the endless volumes of mundane 12plane harmonic cube music being pumped out by the universe in future millennia. Here to conclude, is a very special matrix, and one we’ll be seeing quite a lot of in future: the generalized Pythagorean Matrix. The Pythagorean series is given by: T = pow(3/2)^n n={… 2, 1, 0, 1, 2, …} We now know there’s no reason to limit ourselves to the ratio 3/2, as any ratio will do. And there’s no need to divide by two, we can bring the ratios within limits as needed. Using the S series, we grow our matrix S*T to discover that it blasts off in both directions with the dominant heading in one direction, the subdominant in the opposite. One can see that all rows built upon powers of 2 [1, 2, 4, 8, 16 etc.] repeat the fundamental endlessly, but the prime rows [prime in relation to the number two, that is] of the matrix begin those endless spirals we looked at earlier. … [5] spiral of descending major thirds or ascending minor 6ths [4] unison [3] spiral of fourths [2, 1, 0, 1, 2] are all unisons [3] spiral of fifths [4] unison [5] spiral of major thirds or descending minor 6ths … Speaking of spirals, have a look at the big picture when we limit this matrix to between 0.5 and 2.0 and see a turbulent swirl of deterministic complexity. dominant Pythagorean matrix n=64 We animate this series, and watch the waves of time roll merrily along. max patch: Rolling Pythagorean Matrix [RPM] Now hang on to yourself: we can take any two of our general matrices, for example, the Grand Harmonic and Generalized Pythagorean, and multiply [transpose] one by the other: Z = GH*GT A 2dimensional matrix times a 2dimensional matrix produces a 4dimensional hypercube. Have a go at animating this one! Now, consider the Zed Squared Matrix and the entire essence of this theory is seen in one blazing flash of light. Endlessly, higher orderings reveal themselves. When we include the irrational transpositions, and arbitrary divisions of any interval, both geometric and arithmetic, we conclude: no further combinations need be considered. I don’t mean to say they don’t exist, it’s just that I haven’t found them… yet. The implications of all this may cause any practicalminded musician to rush to the brink of despair. The sheer number of possibilities contained within this system are inexhaustible. There’s no possible way to keep track of all this: it’s all just noise. Yes, noise. Now we’re onto something… Perhaps it is better to wish for simple things: nice equally tempered octaves of 12 notes, forever and ever, amen. But I imagine a few of the true, strong and free will hear this call for adventure and turn their minds towards exploring these exquisite sounds and all their relations. Next, we begin looking at methods for managing these vast resources. We will look at the matrix, flip it, stretch it, twist it, turn it upside down, zoom in and out, again and again. Having finally reached this excruciating high level of complexity, it now becomes time to ‘simplify, simplify.’ Here ends an exposition of the prime materials of harmony. Composed at Langside Hermitage Winnipeg, Manitoba, Canada 2008.07.25 revised 2008.09.16
