Sunday, November 6, 2011

Mathematics, as it is

Whenever I am asked whether I make a living as a composer and I have to reveal that no, like so many, I have a day job, and then I'm asked what pray tell might that be, and I say I'm a mathematician or whatever category into which I'm dropping my job as Chief Scientist that day, I brace myself for the inevitable insight that well, Music is Mathematics, isn't it now.  Being a Very Nice Fellow, I smile wanly and nod and then try patiently to explain that no, it is nothing like Mathematics, any more than Cooking is Mathematics, Writing is Mathematics, Painting is Mathematics.

But then, they might say, even those Things are in fact Mathematics. Remember the words to the children's song by Tom Lehrer:
Counting sheep
When you're trying to sleep,
Being fair
When there's something to share,
Being neat
When you're folding a sheet,
That's mathematics!
Like most children's songs, there is a great Truth here, which I capitalize to distinguish it from actual truth. Mathematics is a study of abstract objects, and most would understand that in the sense of modern Platonism: points, lines, ideals, manifolds, rings, lattices, graphs, numbers, cardinals, propositions, sets, symbols. Those abstract objects are fun to study in their own right, but the Truths of Mathematics come into their own and touch our lives when they find a life as models of Real Objects, some of which they model well and some of which they don't. Sometimes we work to arrive at those models, but oftentimes our conceptions of real objects are greatly simplified to match a mathematical theory that we happen to have lying about.

So, let's be more precise. What one might say is that those Fields of Study above have attributes that can be modeled more or less accurately by Mathematical Objects and that one might be able to glean certain knowledge of those Fields of Study by manipulating those Mathematical Objects, assuming all of one's assumptions are more or less correct, that the initial mapping is OK, and that those mappings still remain OK even under the effect of whatever manipulations one might make in the abstract realm.

What is really really hard about applying Mathematics to the Arts, is that manipulation of anything in the Arts might yield something interesting artistically, since there is no absolute arbiter of anything, as Good and Dad and Judgement are of the past, and one can find an audience for any jumble of phonons or photons or smell-ons.  This makes the final judgement as to whether one's model is Right or Wrong well nigh impossible.

But still, limiting ourselves to the artistic area in question, one might ask: how well can available mathematical models map to something in Music that will help us compositionally or analytically?  And, like all fields of study, there are some things that work OK: in my other life, the fields of acoustics and signal processing are based on this. In that day job, I may assume that a sound is modeled by a continuous curve and that I can differentiate and integrate and take limits to infinity. This assumption is far from a Truth, but it's OK, it works OK, I can manipulate all day long and at the end of the day discover something in my Platonist Plane of Existence that I can transcribe into software and drop into an iPad app and voila!, it sells to the masses who want Groupon coupons generated by listening to the ads on the TV. And, in my musical life, I may model a musical event as a note, and further reduce that to some parameters, like a pitch, which is then further reduced to a frequency, and that to a number which, in a ratio with other numbers, can inform me as to how to tune my guitar.

We all saw many attempts in the heady days of the post war academy to model musically related parameters like crazy, to manipulate them like crazy, and to come up with maps that we hoped we might follow to some Heavenly Abode where - well, I'm not exactly sure what.  Where we might find the Perfect Music?  Or the next Perfect Musical Publication? The serialists' attempts to construct a set-theoretic world based on a small set of discrete parameters is in my opinion a model-of-a-model far removed from the world of a sound, where it's hard enough to pin down music into discrete anythings, e.g. where a musical event starts and stops, or what its please-pick-one pitch is. If notes had single pitches, Auto-tune wouldn't exist. The funny thing is that there is a lot of fabulous serialist music, but, in my opinion, I don't think the models were helpful in getting there any more than any other kick in the pants.

Of course I know the models, and I use them, in a crafty way, to solve problems that I hit here and there, just like the Painter knows the models and may compute the Golden Ratio from time to time, not knowing whether it really is Good or Bad but whatever. And, if my inquisitor by this point hasn't run for the table with the potato chips, I would then set my hand on her shoulder and explain further that the joy of Music Composition, for me, is the impossible-to-quantify or at least the uninteresting-to-quantify ineffable aspects that I don't really even want to understand: how one writes when one has stayed up all night; how one allows God and his Angels, dark or light or their familiars, to speak through us; how it is that there is that one passage of Boulez's Le soleil des eaux that gives me chills; how I can find my way to a piece of music that, when listened to later, I don't understand in the slightest. There is an aspect of the endless, of eternity here, and I might remind my partner in conversation of the end of the B section of the tune above:
If you could count for a year, would you get to infinity,
Or somewhere in that vicinity?
The answer is no, as Wittgenstein said in his Philosophical Remarks:
Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small one. ... The infinite is that whose essence is to exclude nothing finite. 
A model, no matter how finely developed, is, like the runner in Zeno's paradox, no closer to reality than the model before it. Music is not Mathematics, no how and no way. 
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