Variations on a theme

It's Thanksgiving here in the US, and so we thought we'd illustrate our usual subject -- complexity --  much more pleasantly today, while we eat turkey rather than complain about the ones that get published.

In that holiday spirit, here's a video depicting Johan Sebastian Bach's "Crab Canon."  We stumbled across it on YouTube; it's attributed to Jos Leys and Xantox.



The mechanics of this canon are beautifully illustrated in this video rendering.  Musical canons come in a variety of forms but are essentially compositions in two or more voices, in which each voice can stand alone but they also weave together harmoniously.  Bach's Crab Canon, with its double counterpoint, is an example of one of the most challenging of canonical structures to compose.  And when it came to canons, Bach was the mad genius at following rules so intricate that it makes our ordinary heads spin.  As the video shows, this piece is Möbius strip-like in the way the two voices interweave.  (Indeed, here are instructions for making your own Möbius strip of this piece.)

But we wanted to understand more about canons, and specifically Bach's canons, so we asked our violinist daughter, Amie, some further questions, and she in turn sent us on to her polymath musician friend, Ben Grow.  He's a conductor, composer/arranger, teacher and multi-instrumentalist in New York City.

Ben writes:

"Bach is s a composer who, even during improvisations, could squeeze canons out of even the most unlikely-sounding musical material. Much of his music is imitative even if it isn't described as a canon, so when he set out to write a proper canon his powers were huge. That said, the "Crab Canon" is unusual even for Bach.

"A defining characteristic of canons is that their voices enter at less predictable time intervals than simple rounds and they often enter at intervals besides the unison. Also, canons can be imposed over an independent bass line. The canons of the Goldberg Variations, for example, occur at every diatonic interval through the major 9th and all but one have a separate bass that doesn't participate in the canon (the canon at the 9th is only two voices, so both participate; the bass line outlines the same chord progression as the aria that is used in every movement).



"The 'Crab Canon' is unusual because all the dissonances must be functional in both directions. Because we expect dissonances to resolve a certain way, it's difficult to give them a double role – it's even more difficult to make the thing sound like real music!

"The way a canon works is built on certain laws of tonality (the function of notes in dissonance and consonance), which I guess are slightly different between musical styles/eras, but are generally the same. Certainly, Bach's artistry is what's most impressive about the canon because the form seems academic and dry, but I don't think he stretched the rules. Canons either work or they don't, but the real test is if they sound like music or or something similar. This is where most others would follow the rules of tonality and counterpoint but create something awkward. Bach's mastery is that his ear and taste are completely connected to the more mathematical elements of music – I think his filters only showed him the beautiful options. Since he was such a master improviser the mere time it took to write music down was more than enough to edit the work to perfection. Canons are also rare, and I can't think of other composers of the time who wrote so many separately in addition to fragments nested within bigger works."

So, there are strict rules of musical form but whether or not a piece of music works, and beautifully does not depend on the rules being followed.  Indeed, following the rules is easy.  As Ben says:

"If you'd like to try to write a canon, just write a measure of music into one voice and then copy it a measure later into the second voice. The second measure of the first voice will need to fit over the second voice's first measure (which is the same as the first voice's first measure). You can trial-and-error stitch together a canon this way if you don't have the rules of harmony to guide you – you can just use your ear!"

So I did just that.  I have never composed music of any sort before, so this really was just following the rules to see what happened.  This was not, of course, the experience true composers are said to have, of simply transcribing the music they hear in their heads!  To make it easier, and in keeping with the genetic-y theme of this blog, Amie suggested I use just the letters A, C, G, and T -- though, there not being a note T, I substituted E instead (to stretch this analogy to the breaking point, we could say that's rather like RNA using U, uracil, instead of T.)

The result?  The ACGT Canon, composed by me and recorded by Amie.  She decided to play it pizzicato, and here, if you'll forgive the hiss in the recording because she recorded it on her computer, is what we got.



You'll agree that it follows the rules.  But, ahem, you'll also agree that clearly not just anyone could produce the kind of beauty that Bach did, over and over and over; music that transcends time and touches souls.  But if his greatness did not come from stretching the rules in ways that others hadn't thought of but instead he wrote within the rules in ways that others didn't or couldn't do, what made him great?  Perhaps that is an emergent property made up of, but not explained by, an assemblage of parts: in Bach's case, the result of a combination of instruments or voices, conception, inspiration and genius, the whole being greater than the sum of these parts.

That is, just like life, it's complex, it's irreducible.  But achingly beautiful.  An organism is an assemblage of atoms rigidly following the universal rules of chemical interaction.  Slap some nucleotides together, following the rules, and you'll get something, but the rules themselves don't tell you what it will be.

These atoms make up gargantuan molecules, like DNA, that, too, interact rigidly following chemical rules.  In turn, massive complexes of DNA, RNA copies, and proteins literally by the thousands interact following physical rules but in locally organized regions, and surrounded by millions of other molecules that make a cell membrane -- again, each rigidly following the rules of molecular interaction.  And if that weren't enough, a human is made of countless billions of cells that interact mechanically.

In all this, there is a chaotic buzz of chance factors, variations on the theme. But it is not chance that organizes these elements into the emergent, unpredictable, awesome, beautifully organized trait that is you -- you who, using your assemblage of countless molecules, can appreciate beautiful music.

Despite our hubris, whether we will ever truly understand how this happens is quite a thought to chew thankfully on.  So, whether it's a holiday for you today or not, we hope you simply enjoy this venture into the beauty of Bach.  Thanks, Ben.

Slot machines and thoughts: neural determinism?

Coin flips are probabilistic for all practical purposes (unless you learn how to "predetermine" the outcome, here).  By 'probabilistic' we mean that the outcome of any given flip can only be stated as a probability, such as 50% chance of Heads: we can't say that a Heads will or won't occur.  This is for all practical purposes, since if we knew the exact values of all the variables involved, standard physics can predict the outcome with, with complete certainty.  Machines have been built to show this, as we've posted about before (e.g., here).

Slot machines are (purportedly) random dial-spinners that stop in ultimately random ways (that are adjusted for particular pre-set overall payoff levels, but not individual spins).  In this sense, the slot machine is nearly a random device, but even the computer-based random number generator of modern slot machines is not 100% random and, in a sense, every spin could be predicted at least in principle.

So, as far as anybody can tell in practice, each flip or each jerk of the one-armed bandit, is random.  We still can say much about the results:  We can't predict a given coin flip or slot-pull, but we can predict the overall net result of many pulls, to within some limits based on statistical probability theory--though never perfectly.

On the other hand, a casino is a collection of numerous devices (roulette wheels, poker tables, slot machines, and so on).  Each is of the same probabilistic kind.  Nobody would claim that the take of a casino was not related to these devices, not even those who believe that each one is inherently probabilistic.  To think that would be to argue that something other than physical factors made up a casino.

But the take of a casino on any given day cannot be predicted from an enumeration of its devices! The daily take is the result of how much use was made of each device, of the decision-making behavior of the players, of the particular players that were there that day, of how much they were willing to lose, and so on.  The daily take is an 'emergent' property of the assembled items.  Interestingly, nonetheless, the pattern of daily takes can be predicted at least within some limits.  This is the mysterious connection between full predictability and emergence, and it is a central fact of the life sciences.

Genes exist and they do things.  On average, we can assess what a gene does.  Clearly genes underlie what a person is and does.  But each gene's net impact on some trait depends not just on itself, but on the rest of the genes in the same person's genome, and countless other factors.  A particular individual's particular action is simply not predictable with precision from its genome (or, for that matter, its genome and measured environmental factors).  There are simply too many factors and we can't assess their individual action in individual cases, except within what are usually very broad limits.

Brain games
A common current application of the issues here is to be found in neurosciences.  There is a firm if not fervid belief that if we enumerate everything about genes and brains we'll be able to show that, yes, you're just a chemical automaton.  Forget about the delusion of free will!

Location of the amygdala; Wikipedia

A story last week in the NY Times largely asserts that behavior is going to be predictable from the 'amygdala', a section of the brain.  There is also a story suggesting that psychopaths can be identified early in life.  And there are frequent papers about let's call it 'econogenomics', claiming they will save the day by showing how our genomes determine our economic behavior.

Day after day, in the media and in the science journals themselves, the promise is made of ultimate (often, of imminent) predictability even of complex emergent phenomena, from examination of their parts.  If we just have enough sequencers, fMRI machines, and other kinds of technology, everything will work out.  Not to worry!

So the Human Connectome Project, exploits the 'omics idea that if we mindlessly enumerate every single little thing we can understand every single big thing, is funded and off and enumerating.....every connection between every neuron in the brain (starting, we think, with 'the' mouse, whatever that means).  Mindlessly is the right word, because the investigators of such things often proudly proclaim that they are not testing any hypothesis about Nature: this is pure Baconian empiricism, something we've discussed in earlier posts:  collect all the facts and the theory will emerge automatically.  There seems to be a feeling of imminent triumph that, like the priests of old, we The Scientist will be able to see inside your very soul to see what you are really like, no matter how much you may delude yourself that you are a free agent.

Clear-cut cases of prediction in complex systems from specific identified elements do exist, due to individually very strong factors.  They are usually rare, but they addict us to the idea that all cases--all behaviors or even all thoughts, will be predictable by enumerating all causal factors and their effects.  But this is, at best, not practicable.  Is it an ultimate illusion?

So why the persistent belief to the contrary?
Could it be that really, truly, and ultimately when so many countless probabilistic factors interact to generate a net result, our ability to predict them other than in a few special cases is inherently limited?  Could it be that our claims to do otherwise are, in fact, no more than a current version of Delphic mumbo-jumbo that has always existed in society?  Whether or not that is true, science, like religion, is not likely to agree to that.

Why is there such reluctance to simply accept limits to our knowledge, or perhaps even to our ability to know things by applying current methods?  Is it just arrogance, careerism, profit-chasing?  Is it ignorance of the landscape?

One thing is that of course we cannot apply scientific methods that we haven't yet discovered.  There are programs and even organizations, like the Santa Fe Institute of which Ken is an external faculty member, that are dedicated to working out an understanding of complexity.  We think it's fair to say that they haven't solved the problem!

At present, a nay-sayer may be viewed as someone who is anti-science, or perhaps even being mystical.  After all, either things are material or they aren't!  If they are material, should we not be able to understand them?  If they are numerous or individually small, doesn't the history of science show that instrumentation and technology need to be brought to bear on the problem?

The answer to these questions is certainly 'yes'.  We're not mystics. But physical problems need not be amenable to the kinds of solutions we currently have, any more than astrology solved problems when observing the stars and planets was the technology of the time.   Our society certainly believes in technology and even more so, perhaps, in the idea that technology is for making a profit.  That's the often explicitly stated that the point of science is its application, that we do this for our careers and labs, or for patients, or for society at large.

But it is not defeatism to ask whether the current approaches, based on 400 year-old Enlightenment-derived methods and concepts, are obsolete for the kinds of questions we are now asking (no matter how powerful they were for lesser questions that were successfully answered).  It could potentially help to withdraw resources from business as usual as a way of trying to force more creative thinking--but there's no guarantee that, if, or when, it would work to stimulate the next Darwin or Einstein.

It is similarly not out of line to ask, as regular readers know we ask regularly, whether much of what is being supported in science is on wrong trails, even if good for maintaining funding and other sorts of momentum, by diverting funds from things more likely soluble with traditional approaches, like diseases that really are genetic and for which genetic treatments would be fantastic.

And it is not out of line to ask whether when there are so many really serious human ills in the world, that have nothing to do with genes (or, for that matter, with science), that resources are often wrongly being used to maintain an academic welfare system, the way passing the plate maintains religious establishments on the promise of Things to Come.

As we have often also said often,  triggered by yet more grandiose claims in the news or journals, complexity due to multiple interacting but individually small factors is the challenge of the day.  It is even more challenging to the extent that really, or for all practical purposes, these factors are probabilistic results of large numbers of interacting, individually minor, factors.

If that's the case, we are back in the 1800s, when it was discovered that every year a predictable number of people will commit suicide, and by predictable arrays of methods, but yet this can rarely be predicted for individuals.  That kind of problem was perhaps first recognized more than a century ago, but is still with us.

And it's a no-brainer to recognize that.