I recently saw Christopher Nolan‘s latest blockbuster Inception. I thought it was exceptional and it may well have jumped straight into my favourite films’ shortlist. But, imagine the compounded joy when I began to realise it had some links with one of my favourite books of all time, namely Douglas Hofstadter‘s incredible Pulitzer Prize-winning Gödel, Escher, Bach: An Eternal Golden Braid, hereafter GEB . Hofstadter’s book weaves together the works and ideas of these three men in a beautiful way, touching on many many themes throughout, the overall complexity of which is some sort of metaphor for or representation of “how cognition and thinking emerge from well-hidden neurological mechanisms”.
The analogies in this post are tenuous at best and the braid I weave is thin and frayed in comparison but it’s all fun, so let’s weave away.
It began for me when I heard that Hans Zimmer had read (in fact re-read [give the man some credit]) GEB for research and inspiration before writing the score for Inception. In the book, there is a great deal said about canon, counterpoint and specifically the fugues of J.S. Bach. A fugue is a technique of composition (warning: I’m definitely not an expert) in which the piece is based on a short, often very simple theme called the ‘subject’. The essence of the idea is that the subject is repeated all over the fugue, introduced at different stages and in different voices. You’ll see what I mean if you watch the first minute or so of this video.
There is a cute, if somewhat naive, way of thinking about fugues mathematically. Think of my stave as a half-strip in a two-dimensional plane. So, it is like a strip which goes on forever to the right.
Now suppose I have a subject, i.e. a few notes at the start of my piece. Like I said, the subject is repeated all over the fugue. Once the piece is more developed, the subject is introduced at later stages, in different keys and in different voices. These are called the ‘entries’ of the subject.
Suppose that after a while the subject is repeated exactly. On the page, the new entry is therefore exactly the same as the old one except that it is further to the right. So, taking the original theme and shifting it to the right would map it onto the new entry. If I shift the original theme to the right by exactly the right amount it would be superposed on top of the new entry. This type of shift is called a translation in mathematics.
More often than not though, and as in the video I linked earlier, one does not repeat the subject exactly, but repeats it in a different key, that is to say, played at a higher or lower pitch than before. Such an entry would be to the right of and slightly above or slightly below the original. The shift in question would again be a translation.
This is the most basic way one can repeat the subject in a fugue. There are more devious ways to layer the fugue with your subject. For example, at some later stage in the fugue, I could bring the subject in at half the speed that it is originally played at, so that it is drawn out and lasts twice as long. On the page, the notes would be spaced out more. The pattern will have been stretched out horizontally. This kind of stretching motion is called a dilation (this particular dilation is ‘inhomogeneous’). So if I took the original subject and moved it to the right by the right amount and then stretched it out horizontally by the right amount, it would lie on top of the new entry.
One can go much further and introduce inversions of the subject. In an inversion I am basically playing the subject upside down. I take a starting note and whenever I went up in the original (from a lower note to a higher note) I go down in the inversion by the same amount. The geometric transformation which is relevant here is reflection. And similarly for the even more ambitious retrogrades. A melody, such as the subject, played backwards is said to be the retrograde of the original. I needn’t say what retrograde-inversions are. Reflection in a vertical line gives you a retrograde; reflection in a horizontal line gives you an inversion.
There are lots of cute little mathematical ideas in music, though I cannot speak for the merits (or otherwise) of proper music theory and the role of mathematics therein. So, Mr. Zimmer what’s the result? The most obvious manifestation of these ideas is here, but I suppose there could be others:
The time-dilation effect of the dreaming is thus represented by a dilation of the music. Simple but effective.
The relevance of M. C. Escher‘s art to the film ought to be obvious to anyone who knows his work. For those who do not but have seen the film, I need only show you two of his drawings. Just compare Escher’s Relativity,
And think about her lesson with Arthur on paradoxical architecture. Escher’s surreal architecture is a clear inspiration. If you are unfamiliar with M.C. Escher then do seek out more of his work. He was brilliant!
In the next post, once I’ve got my head round his ideas better, we’ll begin to move on to how on earth Gödel fits in.