Desperately Seeking Relevance: Music Theory Today [4]

"Modes of Imagining"
A few thoughts on rules, definitions, common notions,
requests, postulates, axioms, hypotheses,
and other stuff.

I will not run through all the modern axioms laid down by Russian boys on the subject, which are all absolutely derived from European hypotheses; because what is a hypothesis there immediately becomes an axiom for a Russian boy, and that is true not only of boys but perhaps of their professors as well, since Russian professors today are quite often the same Russian boys. And therefore I will avoid all hypotheses. – Ivan Karamazov[1]

I often think of music as a game. There may be a large, even infinite, number of ways to play, but a limited, relatively small set of rules defines any game. Over a long period of time, since originality is often an important goal, some of the rules might be altered so that boredom doesn't set in (possibilities tend to get used up as the history of play is extended). But the question then arises, which rules are you allowed to change and how much can you alter them before you find you are playing a different game altogether?

It doesn't take long before tic-tac-toe becomes boring. So you change the rule that says it has to be played on a two-dimensional surface. Is tic-tac-toe in three dimensions still tic-tac-toe? I think most people would say yes. But let's say you stick to two dimensions, expand the grid, change a few other rules, expand the grid some more and so on. Over the years, possibly centuries if you live so long, you come to realize you're playing a game that looks an awful lot like a game in another culture called Go. But are you still playing tic-tac-toe? Does it matter?

If we ask if Wagner was playing the same game as Josquin, it seems the answer would be both no and yes, depending on which rules you look at. Some of the rules in Wagner's music game would be unrecognizable to Josquin, and some of Josquin's would be anathema to Wagner. Others (very deep ones, I think) would be unchanged between the two – if not, we couldn't say that both composers were composing music that sounds enough alike such that no one is surprised when we call them both music.

The game metaphor suggests a few related knotty issues as well. When is a rule change called for? Is it simply a matter of avoiding boredom as I suggested at first, or are there other compelling reasons for rule changes? This raises issues of authority. Who decides a rule change is called for and what rule or rules to change? The composer? The performer? Who gets to decide whether or not to accept a rule change? The audience? The critic? History?

Now I suppose I could continue in this vein to segue into a critique of musical developments and experiments in the early-to-mid 20th century. It now seems to many, not without reason, that rule-changing itself became the game in those years. I can't deny this happened, but neither can I say that many of these rule changes were not desirable or necessary. The old game was getting awfully stale and a little too easy. But that's not the turn I want this metaphor to take. At least not yet. Instead, I'd like to move away from music a bit to see how players in another game have dealt with radical rule changes.

"In the nineteenth century, geometry, like most academic disciplines, went through a period of growth verging on cataclysm."[2]  (One might say that in the twentieth century, music, like most other arts, went through a period of cataclysm verging on growth.)

While there was an explosion of activity in many branches of mathematics, I think most would agree that a main cause for all the foment was a problem that kept bubbling up from just below the surface for centuries: the Fifth Postulate in Euclid's Elements. But the "parallel postulate" is not the issue here. The issue I'd like to dwell on is the word postulate per se, and how mathematics as a discipline learned to handle the problem of its "primitive concepts."

I just recently discovered that the Greek word Euclid used, which is generally translated as "postulates," was αιτεματα [aitemata], more accurately rendered as "requests." This is significantly different from the old, and still common, sense of postulates and axioms as self-evident propositions. It would seem that Euclid's Elements does not really begin with five postulates, nor with five axioms. Euclidean geometry begins with five requests. So then is the reader free to accept or reject any of these requests? Yes. But the consequence of rejecting any one of them is that Euclidean geometry doesn't work. – Well, that's not entirely true. As it turns out, the Fifth Request, concerning "parallel lines," can be left out, leaving much of the Euclidean edifice still standing, but the space it is left standing in need not be "flat" as common sense would dictate. And if the Euclidean Fifth Request is denied such that space is no longer "flat," some very strange objects and relations begin to appear – logically valid but playing havoc with our human sense of the way things are and the way they ought to be.

[N]o mathematician would invent something new in mathematics just to flatter the masses.... He who really uses his brain for thinking can only be possessed of one desire: to resolve his task. He cannot let external conditions exert influence upon the results of his thinking.... An idea is born; it must be moulded, formulated, developed, elaborated, carried through and pursued to its very end.[4]

[V]ery little of mathematics is useful practically, and ... that little is comparatively dull. The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas. Thus a serious mathematical theorem which connects significant ideas, is likely to lead to important advances in mathematics itself and even in other sciences.[5] 

Still, it was not always so. Mathematicians are human, and tend to cling to their inherited realities like anyone else. It took centuries (and the development of a more safely liberal culture) for them to face up to the problem fully. In 1733, Giovanni Saccheri, having proven many theorems in hyperbolic geometry, dismissed his own work simply because it contradicted Euclid. To this day, outside the world of mathematics and science generally, the common sense (consensus: literally "feeling together") tells us that everything in Euclidean geometry is obviously true. Imagining the world otherwise, even "in theory," is nearly impossible. Outside of science fiction (NB!), suggesting a world where parallel lines meet is an abomination. To explain what our collective sense tells us, therefore, we accept Euclid's request because we need it to prove (justify) what we already know is true. We knowingly commit to this invalid argument because we need to get along in a world we can understand. "We do, doodley do, ... what we must, muddily must...." That's all. Is this beginning to sound familiar??

Despite being locked in by collective wisdom, in the nineteenth century mathematics found a way to stop begging this ancient question. Nikolai Lobachevsky devised an alternative system of geometry based on the negation of the Fifth Postulate (we may as well return to this commonly accepted translation now, my point having been made (I think)). Lobachevsky called the geometry he built an "imaginary" geometry.[6]  Around the same time, Janos Bolyai simply deleted the Fifth Postulate and termed what could then be deduced from the definitions and the first four postulates alone the "absolute geometry."[7] In 1854, Bernhard Riemann (who Milton Babbitt liked to call "the good Riemann") built a spherical geometry where there are no parallel lines (paving the way to the general theory of relativity). There were now two non-Euclidean geometries. Mathematicians had finally freed themselves from trying to make the worlds of reality and imagination conform to the Euclidian sacred text, and they did this essentially by populating the world with other geometries – not to improve upon, and certainly not to take the place of Euclid, but to join him. Euclidean geometry became one among many valid geometries, and this caused a different problem. With more geometries appearing on the scene, what, if anything, was the connection between them?

Enter Felix Klein and his Erlangen Program of 1872 which was a synthesis of many of the then-existing geometries (including Cartesian, projective, and others – the exception was spherical) as models of the same "abstract geometry."[8] Remember that term.

Neither reverence for the past nor the common sense of reality prevailed. Euclid remained, but the Euclidean lock was broken.

I'm going to break off this ultima Thule of a digression here because a musical Erlangen Program is precisely the place where I would like to resume my commentary on music theory today (MTT) with a challenge that will take two more posts. In my next post I'll review a list of what I understand to be MTT's basic, mostly unexamined assumptions. These assumptions I see as tracking the early history of geometry described above. Full of potential, MTT has stopped short of Lobachevsky & company and settled into the safe holding mode of a brilliant but frightened Saccheri. Just what the Erlangen Program is in math and what it might suggest to music theory should then become clear with what I intend to be my final post in this thread. But I'm more than willing to extend it if the challenge is taken up in a meaningful way.

The recognition of frontiers implies the possibility of crossing them. It is just as urgent for musical theory to reflect on its own procedures as it is for music itself. It is the bitter fate of any theory worthy of the name that it is able to think beyond its own limitations, to reach further than the end of its nose. To do this is almost the distinguishing mark of authentic thinking.

––Theodor Adorno (Quasi una Fantasia)

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[1] Fyodor Dostoevskii. The Brothers Karamazov, Part 2, Book 5, Chapter 3. Tr. Richard Pevear and Larissa Volokhonsky. San Francisco: North Point Press, 1990. p. 235.

[2][3][6][7] ("Nineteenth Century Geometry," Stanford Encyclopedia of Philosophy)

[4] Arnold Schoenberg. "New Music, Outmoded Music, Style and Idea" (1946) (In Style and Idea)
[5] G.H. Hardy. A Mathematician's Apology (p.89)
[8] Felix Klein. "A Comparative Review of Recent Researches in Geometry."