8. The Reflecting Mirror: Mysterium Contrapunctus During a Counterpoint class at U.C.L.A, Schoenberg sent everybody to the blackboard. We were to solve a particular problem he had given and to turn around when finished so that he could check on the correctness of the solution. I did as directed. He said, "That's good. Now find another solution." I did. He said, "Another." Again I found one. Again he said, "Another." And so on. Finally, I said, "There are no more solutions." He said, "What is the principle underlying all of the solutions?"  John Cage So without further ado, let’s present the 16*16 matrix. One can see right away its most obvious and striking feature: the two inverse series that account for every row and column of the matrix: Rows: The Dominant Harmonic Series [please ignore superfluous intervallic nomenclature from this borrowed web image]
Columns: The Subdominant Harmonic Series Here it is: the Left Hand of Harmony, the Shadow of Sound, the Mistress of the Matrix. We don’t seem to hear the subdominant series in our perception of acoustic harmonics, but evidence for its existence lies within the principle of difference tones. We see it here as integral to an understanding of harmony. Just as the ascending dominant series stretches upwards to the stratosphere, the subdominant lengthens downwards into the depths. The interrelation between the two gives rise to every just harmonic relationship possible. It doesn’t take much logic to show that every Pythagorean ratio is also included within this matrix, eventually… If you haven’t already done so, now would be a nice time to listen to this beautiful series. [You did listen to 7 when it was time, didn’t you?]. One immediately notices that its quality is distinctly minor, yet the ‘tonic’ is not the pitch center, but rather the perfect fifth below. Improvising upon this series, one can’t help but want to resolve to G, just as we’d want to rest on D when playing the dominant series. In other terms, our sense of tonality is not 2, 4, 8 rather it is 3, 6, 12 . I’m afraid to point this out, but in light of this observation, the term ‘tonality’ now falls under suspicion as lacking a true and proper definition. Call it what you like: Yin Yang, positivenegative, up/down, leftright, complementary opposites, heaven and hell, what have you. In Indian solfege, the terms Sa, Ma and Pa correspond to the Western Do, Fa and Sol. Thus Pa corresponds to the dominant, and Ma the subdominant. There is no particular reason the subdominant should descend and the dominant ascend. The matrix is reversible. It can also be rotated. This gives us 8 expressions of this basic matrix. We won’t be dealing with rotation much, simply recognizing it as a property we can use when exploring various ways to manage these resources. The inverse matrix, however, or reflection along the diagonal, though containing exactly the same pitches as its counterpart, is in my estimation, a very important distinction that I am wary of reducing away. Try this: 2 4 6 8 3 6 9 12 15 4 8 12 16 5 10 15 What about: [5, 10, 12, 15] How many times have we heard this? Bass line: 16, 15, 14, 13, 12, (8, 6)! Sometimes, one is simply rendered speechless by the subtlety and consistency contained within the symbols of the human mind. The importance of these two series in music theory cannot be denied, and I’ll wager they apply to many other natural phenomena as well, most notably, colour and light – but that is another paper.
