Friday, 8 January 2021

Recursion and Function

I attended an interesting talk before Christmas on music and structure at Liverpool's PhD music discussion group (which is great fun!).  The talk was really about the "function" of musical chords, and how this function related to the structure of the music. What was being argued was that there was a kind of recursive relation between the deep structure and the function on the surface. This isn't a new idea, of course - Heinrich Schenker was the first to talk about this over 100 years ago - but it was interesting because it muddied the distinction between structure and function: function was something that is "done" by some component (a chord) which in aggregate articulates a structure. Or does it? Is it structure which determines the function of its surface components? The problem is that when "structure" is discussed, it is always comprised of some fundamental unit (in this case, a "function"), but one could not talk objectively about the structure if one were to consider a different function - a different way of carving up the structure. 

Our understanding of "function" is rather confused more generally - not just in music. It does appear to be the case that music has some kind of recursive structure - units of homologous processes repeat at different orders of scale. It is a fractal in this sense. But fractals in nature are not made from simply repeating some kind of pattern; they emerge through interference of multiple variables at different orders of scale. It is the difference between an L-system fractal (see L-system - Wikipedia), which can produce something that looks like a tree, and a real tree. Photographic holograms provide the real clue: these are images of 3D objects encoded into a 2D pattern by virtue of the interference between the beams of laser light which are used to encode the image. The fractal pattern that emerges is actually an encoding of space and time, since space and time are the fundamental values in the frequency of light.

We might look at a hologram produced in this way and say that it is comprised of a number of features which "repeat". We might even argue that there are particular functions which repeat. And yet these are some surface features which resut from a deeper process of interference which we cannot fully comprehend. Ascribing "function" to some element is our way of dealing with this uncertainty. But it can lead us astray.

What is the process of interference that leads to the fractal in which we identify "functional units"?

Ultimately, I cannot see how it is other than physical and physiological. In other words, it involves (quantum) mechanics and cells. Music is clearly within this realm: vibrations in the air and physiological responses. What is remarkable, given the variety of physiological possibilities, is that there can be any agreement about function at all. This suggests to me that the social function of coordination and agreement serves in some way to establish coherence and pattern in the fractal between social groups - and that indeed, this may be fundamental to inter-human (or indeed, inter-organism, or inter-cell) coordination.  

We are given to believing that there are fundamental surface units of functionality which produce coordination. Yet, we are continually reminded that this is not the case. From this we conclude:

  1. There is nothing fundamental in nature that unites us, or that can be harnessed to orchestrate our minds better: it's just "culture";
  2. There remains to be found some structural method which with sufficient force, vigour or control can mobilise collective action;
  3. It is the job of education to programme the young to reinforce the functions of existing ways of thinking, and to teach them that this is as good as it gets.

Point 1 will lead to destruction of our environment (as it is doing!). Point 2 leads to totalitarianism. Point 3 leads to the enslavement of education to those in power rather than an authentic inquiry into nature. 

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