The Artful Universe
- Extra Reading
A look at art and science in which we explore the aesthetics of fractals, the abstract expressionism of Jackson Pollock, musical appreciation, the biological basis of landscape appreciation and some contrasts between art and science.
THE ARTFUL UNIVERSE
Professor John D Barrow
We’re going to look at a subject that’s really rather large, and in some ways it’s rather too large. It’s a subject that might be the course of an entire series of lectures. I want to talk about some of the intersections between art and science, to show you just a selection of one or two points of intersection where on the one hand science might shed some interesting light on works of artistic creation, and vice versa, where if you’re a scientist interested in the complexity that works of art offer; something that’s novel and unusual that’s being systematically processed by the human mind.
There are various simplistic and rather reductionistic ways perhaps of looking at art and science. If you want to take this link between complexity and science, you might regard complexity with reductionism, complexity that you attempt to manipulate, you attempt to understand. You attempt to reduce complexity to simpler things and this characterises the process that we sometimes call science, but complexity without reductionism, perhaps complexity that has no rules attached to it, is close to what we call art.
There are some things that people are in the habit of calling art that are perhaps just reactions to things that other people do. There’s a hearty tradition of just providing nothing as a work of art. If you’re old enough, you may remember this was started in book form by an English footballer called Len Shackleton who was England ’s centre forward and was a rather cantankerous person. He wrote a rather nice autobiography in the days when footballers didn’t do such a thing, and it has a chapter entitled “What the average director knows about football”, and it consists of a number of completely blank pages!
If you’re a musician, you’ll know of John Cage’s famous work, 4 minutes and 33 seconds of silence. This is performed by a pianist arriving in full evening dress, and he sits down on his piano stool, raises the lid and is poised to attack the keyboard, but he sits poised for 4 minutes and 33 seconds and then leaves. I was once at a conference where some musicians and musicologists were present, and I asked them if they knew why it was 4 minutes and 33 seconds of silence, and to my astonishment, nobody knew, they thought it didn’t matter, but if you convert 4 minutes and 33 seconds to seconds, it’s 273, and remember minus 273 was absolute zero temperature, and so Cage’s idea was that this was absolute zero of sound.
Well, this type of blank canvas has found its way into books. There was a book published in ‘74 called “The Nothing Book” which was a set of completely blank pages; interesting, because just a couple of years afterwards there was a legal action brought against the publishers when they published another book of completely blank pages!
An interesting way to look at the pattern of design that we go in for in artistic creation is to think of art as the creation of certain types of structure in space and time. We might imagine that a structure has a certain dimensionality in space, so if it’s a line, it’s just length to the power one; if it’s an area, it’s length to the power two, and it can either take place in time and be as one or it can be static. Our creations really fill all the possibilities. If you have no time aspect, then in one dimension of space you have things like friezes, design patterns around the wall; in two dimensions, you cover an area, as in painting; if you have a static three dimensional structure, then you’re in the world of sculpture; if you introduce a time dimension, in the one dimensional patterns changing in time, you might have music, two dimensional pictures changing in time, you’re in the world of film, and if three dimensional things move in time, then we’re in an area like theatre or opera. So we’ve filled out all these possibilities, and if you’re higher dimensional, you might imagine what things might look like in four dimensions or with two dimensions of time.
Something that we looked at once upon a time in one of my earlier lectures that’s quite instructive in this connection is the case of a sand pile. If you start dropping sand on to a flat surface, it gradually builds up, an ever-steepening pile, but only to a point. When you reach a certain steepness, what happens is that the sand just produces more and more avalanches and the pile gets no steeper. So when it’s in that state, it’s in what physicists call a critical state: the stability of the overall shape and pattern of the sand is maintained by avalanches, by instabilities, and this critical slope is a state of maximum sensitivity in some sense. So the sand will build up a little, then there’ll be an avalanche, build up a little more, another avalanche. Although it’s always unstable, there’s another equilibrium state very, very close by.
I think there’s an interesting lesson here for works of artistic creation. If you ask people what makes a good book or a good play, you could argue for hours, but one test that you might apply is what type of listening or watching does it induce. Do you want to read this book again? Do you want to see this play again? Do you want to listen to this piece of music again? So John Cage’s 4 minutes and 33 seconds of silence might be amusing to see once, but you probably wouldn’t want to see it twice, and you wouldn’t want to go and see it every night of its run. But great works of art, whether they’re plays of Shakespeare or symphonies of Beethoven, pictures by Leonardo and so on, one of the characteristics is that you want to see them on many occasions, so you never have the attitude that, oh, I’ve heard Beethoven’s Fifth Symphony, or I’ve seen Macbeth or Hamlet, I’ve ticked it off in my I-spy book, I don’t need to see it again. I think you want to see it again because it’s rather like the sand pile; a great work of art has the property that very small changes in its staging, in the direction, in the musicians, in the style of production, produces a completely different experience. In this sense, many great works of art are like critical systems: a little change in the way it’s done, artistic direction in the theatre, produces a completely new experience and so it’s worth going again to see Hamlet, because it’s going to be different, whereas John Cage’s 4 minutes and 33 seconds of silence is not going to be significantly different.
I want to look at a number of “art and” topics, and first I want to say a few things about art and symmetry. I’ll use the word “art” in a rather general way, sometimes I’ll be talking about music, sometimes I’ll be talking about painting, and here I’m going to talk about decoration. This may be the decoration that you have in your house, or it may be the decoration on the Elgin marbles, or a frieze around the Institute of Directors or somewhere like that. Symmetry is something that mathematicians like and understand rather deeply and know how to classify it. It’s rather interesting to compare what’s been done intuitively by artists in many cultures and what we know from mathematics. If we think about friezes, a repeating pattern in one dimension that might go around a border of our wallpaper, we know there are just 7 distinct patterns. You can use different shapes to characterise them, you can change the colours. If you move to two dimensions to make what mathematicians call wallpaper patterns, so you make a repeating pattern in two dimensions, there are only 17 types of wallpaper.
How does that work? Let’s look at the friezes first. There are really only 4 things you can do with a frieze: you can change the little pattern that you use to repeat; you can have a vertical reflection; you can have a horizontal reflection; you can have a rotation and you can have a glide reflection, so you move the pattern along and then you reflect it. These are the only 4 ingredients that you have at your disposal. If you combine them, there are just 7 results that you can come up with. What little shape you use as the model is up to you and Sanderson or whoever makes your wallpaper, but there are just the 7 distinct possibilities that you could make with a basic shape. There are no others.
Some years ago I wrote a book called The Artful Universe. I had a look at examples from all cultures. It’s rather interesting, humanity has discovered those 7 frieze patterns intuitively, in many, many cultures, long before mathematicians ever thought about that problem. There are examples from Asia Minor , China and from the Renaissance. All over the world we find people have intuitively discovered these patterns. The African design is particularly interesting. Seven possibilities, that’s all there is. You can change the colours, but that’s all you’ve got to do. What about the wallpaper? Well, here the situation is more complicated. You could imagine doing the same thing in one dimension, and you’ve got the same possibilities in the other dimension, you can combine them, you just have 13 possibilities. Again there are examples from all over human cultures, from Pompeii , from the Indus culture, French stained glass, Persia , and so forth. Human civilisation discovered all of the logically possible combinations, and lots of cultures discovered them.
There’s an honourable tradition of geometry having intimate contact with art. We know the discovery of perspective, Satchio and others, a remarkable combination of people’s geometrical intuition and talent. But geometers discovered only rather late in the day, the 19 th Century, what we would now call Euclidean, and non-Euclidean, geometries. There was great debate for centuries about whether it was logically possible for there to be geometries that didn’t contain Euclid ’s famous parallel postulate that two parallel lines would never meet. What would happen if you abandoned that? There was great argument about whether it was possible for there to be a logically consistent geometry without the parallel postulate.
Well, what would the alternatives be? To demonstrate non-Euclidean geometry, imagine an inverted vase. Up at the top of the vase, there’s positive curvature, and if you take 3 points on the surface and join them by the shortest lines that you’re able to draw on the curved surface, you will make so-called spherical triangles, and you add up the internal angles and they’ll be more than 180 degrees, and that’s what you mean by positively curved. Down in the lip, it is negatively curved, and if you mark out 3 points on the vase, join them up by the shortest lines you can draw on the surface, then you will get a rather concave triangle, and the three internal angles would add up to be less than 180 degrees. That’s what you mean by negatively curved. Somewhere in between, there’s a sort of British compromise point, where there is an ordinary triangle, where the angles add up to 180 degrees. What’s interesting is that mathematicians took a very long time to discover that non-Euclidean geometries are possible, and there were deep philosophical consequences of that, because philosophers regarded Euclidean geometry as being the way the world was. It was part of the absolute truth of things, and if other geometries were possible, it really introduced a type of relativism that could spread all over human thinking.
But what’s instructive is to look back. There’s a famous painting done by Jan van Eyck in 1434. It’s Arnolfini Wedding, the one with the little dog at the bottom. It’s in the National Gallery, if you want to look at it. The interesting thing is the little mirror in the back. If you see a blow up of the mirror, it shows the rear view of the unhappy couple. What’s interesting about this is that the mirror is curved, and if you look at the image in the mirror, you have a distorted picture of what’s going on in the foreground. In order to do this, you have to rather painstakingly copy or create the curve geometry in the curve mirror. Now a right thinking artist or geometer of the time could have realised that there had to be logically consistent non-Euclidean geometries from this picture, just by looking in a curved mirror, because if you had the ordinary Euclidean world obeying Euclid’s laws of geometry and you looked at it in the curved mirror, you had another curved world where there was a one-to-one correspondence between the two, so there had to be a logically consistent set of axioms and rules to govern the curve geometry of the world in the mirror. So the looking glass world of non-Euclidean geometry was staring at people all the time. Artists were drawing it and recreating it, but nobody twigged that this was really evidence that it was a mathematical logical possibility.
The next artistic thing I want to mention is something about fractals and Jackson Pollock. Pollock is the biggest selling modern artist in the world, in the sense that his works command far and away the highest prices. The last time I think a big one went on sale, in 1998/99, it reached 40 million dollars. So if you’ve got one at home, let me know! What are fractals? Suppose that you have a complicated pattern that’s wiggling all over the place, and you want to compare this in some way with a rather simple old fashioned straight line. One way to characterise it is to somehow ask how much of a square does the curve cover. How many places does it visit? A rather simple straight line would cut through some bits of the square, but the wriggly line goes almost everywhere, so the amount of visitation of the curve to parts of the square is a measure of how complicated it is. Mathematicians can codify this rather straightforwardly. Suppose you decide to divide up your picture into squares? So, if there were 4, then you split them all in half again, then split them in half again, and so on, and ask how many of the squares get visited by the wandering curve, it’s still visited nearly every square, there might be just one square that doesn’t get visited. If we made the squares smaller and smaller, there would be more and more that didn’t get a visit from the curve. So if we have a rather boring straight line, the number of squares that it visits is just proportional to the length of the curve.
Fractals exploit the fact that wiggly curves can behave as though they are more complicated than lines. A straightforward straight line curves rather little of the plane, where a squiggly wiggly line could if it was squiggly enough in effect behave like it was an area because it could cover the whole of the plane. If we ask how many squares get covered by our curve, a fractal is a structure where the answer to that question is some power of the size of the square. If the line is a simple one, the number just changes like the size to the power minus one, it’s one over the size as the size gets bigger, or as the size gets smaller, it increases, like one over the size of the square, but the very wiggly line can increase much faster in the number of squares it cuts as the squares get smaller. Wiggly curves can behave in an in between way, and this number can be anything between one and two. It can be 1.5, 1.3, 1.4, so the curves can behave as thought they have fractional dimension, and hence the name fractal that was introduced by Mandelbrot.
In complicated patterns, the human eye and visual system tends to start to behave rather differently. So if you have a crowded scene with a message hidden in it, when the whole scene has a fractal dimension more than about this number, you won’t be able to pick out the message, but when it’s smaller, you’re looking at a photograph with hidden runways and so forth from an aerial photograph, you will probably be able to pick it out, below that critical dimension. What you should notice about this is that fractals are really pictures which are scale invariant. We say that fractals are scale invariant. So if you have a picture of a fractal, you can’t tell whether you are looking at a reduced image or an enlarged image. It has the same appearance on any scale. It’s for this reason you’ll find fractals in nature, trees, look at the head of a cauliflower or something, the idea of the design, you take the same picture, you repeat it on a smaller and smaller scale. A tree has branches, branches on branches, and so on and so on. So fractal design is something that living systems use a lot, and we have co-evolved with an environment that has a fairly impressive fractal structure, and no doubt this plays a role as to why we like it.
Well, what’s Jackson Pollock got to do with this? Look at Jackson Pollock’s pictures Blue Poles or Blue Poles 2, the bigger version, in the Canberra Gallery in Australia . It was bought for a couple of million US Dollars rather controversially in 1972. Fantastic investment, it’s now worth probably about 50 million. Like most Pollocks, it’s gigantic; it’s 5 metres by 2 metres. The way Pollock created these pictures, why he became known as Jack the Dripper, was that he had just two techniques: he would put a huge canvas on the floor of his studio, and he would drip paint from the end of a stick locally, and he would throw paint, and at the end of a very long exercise, often he would return to a painting over many years, it would be cut out of the large canvas on the floor to remove edge effects, and that would be the final work.
What’s been discovered about Pollock’s work by Richard Taylor, now in Oregon , who I have quite a lot of contact with on this problem, is that fractals were in effect discovered intuitively by Pollock. Pollock’s works are precise self-similar fractals really to very high precision. You can’t tell what the size of a Pollock is by taking pictures of it. You wouldn’t guess that it was a 5 metre picture. He perfected the art of creating a pattern that had scale invariance. If you think that your children, by throwing paint in the air would be able to make a Pollock, or that you would be able to go home and make a picture like a Pollock, you wouldn’t be able to do it. You wouldn’t be able to create a scale invariant picture. It’s very difficult. So Pollock’s pictures are fractals, scale invariants, of very high precision, and as his career went on, the fractal dimension of his work became higher and the complexity of the curves became greater. Copies of his pictures have been made, photographs have been made, with different colour screenings, and you can, as it were, overlay your grid on the picture to fantastically fine precision, compute the fractal dimension of the picture in all colours. What you discover is that the picture is a fractal double fractal in some sense, to really rather high precision.
Now this produced a very interesting situation a couple of years ago. Richard Taylor was approached by an owner of a drip painting, a Texan inevitably. This was not signed, and he liked to believe that this was a Pollock, because if it was a Pollock, it’s going to be worth 30 million, something like that, and if it’s not it’s probably worth sixpence. There’d been much argument by art critics over the years about whether this really was a Pollock or not, and so Taylor did a fractal analysis of it, and the results are very interesting, because it’s not a fractal. It doesn’t have the characteristic transition, down on the 1.8 centimetre scale that all Pollock’s pictures have, so you can’t fit it by a single fractal or by two, and you end up with a rather ambiguous notion of a transition, on a completely different scale. So people were rather satisfied by this that this is not a Pollock, it’s not a fractal, and even if it was forced to be a fractal, it would be a different type of fractal to Pollock’s.
On discovering that Pollock’s works were all fractals, so they’re scale invariant, they look the same on every scale, the advice I always give to fund raisers and university vice-chancellors is that your fund raising strategy should be to buy a Pollock and cut it up into pieces and sell the pieces. There would be no static loss because they all have statistically the same impression, but there will be a massive financial gain!
What about astronomy? What’s astronomy got to say to us about art? What interesting things might we learn from that? Well, something that always intrigued me over many years, and so much so that I started to look into it, was if you look at Van Gough’s Starry Night, this is a painting I think painted about 1899, about the end of the 19 th Century. He painted many, many versions of it. The Art Library at Sussex had about 21 versions – much change of the foreground, of the church and so on. But if you’re an astronomer, there’s something very odd about this picture. It’s not about stars; it’s got a great galaxy in the middle of it, swirling around with a spiral arm, and a little companion. This is very odd. No one has ever seen that type of galaxy with the naked eye. Why is this image in Van Gogh’s picture? A little bit of detective work I think gets to the bottom of it. A few years before Van Gogh painted this work, the Earl of Ross, with his great telescope in Blair Castle in Ireland , was able for the first time to look at galaxies like M51, the whirlpool galaxy. Here is Ross’ sketch that he made from the view he got from his great telescope. Ross, his real name was William Parsons before he was enobled, and Parsons subsequently became a great optics company. The M51, the whirlpool, has a companion, so there is a small companion galaxy. The image that we see in Van Gogh is the swirling spiral galaxy with the small companion. How did that happen? What I think happened here was that Ross’ picture was big news at the time, a spectacular thing. No one had seen images quite like that before, and it found its way into a best selling astronomy book of the time Flammarion’s Popular Astronomy, a “Brief History of Time” of its period, and it went through countless editions all over Europe , and undoubtedly this is where Van Gogh would have seen this picture. It was published just a few years before he painted his picture, and I think what’s happened here is that image has formed the centrepiece of the sky. So neither Van Gogh nor anybody else had seen a galaxy. There’s only one galaxy you can see with the naked eye, as a tiny dot, and then only from the southern hemisphere, and that’s Andromeda.
The other odd thing about astronomy and art and culture generally is the role of the night sky on the development of human culture, of myths and legends and literature. It was relatively late in the day that commentators on these issues came to appreciate the way in which the appearance of the night sky at different latitudes alters the way that different ancient cultures incorporated their visions and interpretation of the sky into their legends, into their literature. So the sky at high latitudes shows things rather differently to viewers of the sky near the equator.
What would you see if you lived at different latitudes? Let’s suppose you lived at the equator, what determines what you see in the sky? Remember the Earth is spinning on its axis; it’s inclined to its orbital plane. If you stand on the surface of the Earth, as the Earth rotates every 24 hours, you will see the sky rotate above you. But if you’re down at the equator, you see something very different to what you see if you live up near the pole. If you live near the equator, you see the stars at night rise, pass overhead, and then set on the other side. So you see yourself really as the centre of the world, everything goes up and over, the stars are extremely useful for navigation, because they tend to just have a vertical motion above you, there’s no lateral motion. You’re in a boat, you’ve set a course, and you just follow that trajectory as the star rises. But if you live at Stonehenge or in Holborn High Street , then you see a very different type of sky. You’re much more northerly in latitude, and there is a north celestial pole pointing towards Polaris, our north star at the moment, and as the night goes on, you see the stars rotate around that pole star, and as you look away, they will take longer and longer to rotate round. Eventually, you will find stars which will rise and they will set down. You see a much more complicated sky; you don’t just see vertical motion, you see some lateral motion as well, but your view is dominated by the stars circling around that Pole Star. If you live up near the North Pole, you’ll see a rather odd sky; you’ll see the stars just circling around above you.
If you went out to a dark spot at our latitude, and you had a long exposure picture of the sky, what would you see? The sky star trails over one night and produces a dramatic effect. You would see how the North Star acts as a centre for the rotational motion of all the other stars.
Some years ago, people started to investigate the impact that this idea might have had on people’s perceptions of the sky and what they wrote about it in ancient times. Von Dechend and Santillana, two historians, wrote a famous and perhaps even infamous book called Hamlet’s Mill, and what this looked into was that how in northern latitudes there are a whole host of sky legends in different countries about a great millstone in the sky, and similar ideas, which were being influenced by the fact that your view of the sky made you see this great swirling motion of stars above you, whereas when you went to southern latitudes, you didn’t have that type of sky legend, you had something very different, we ourselves at the centre of some great palatial lattice with the stars moving above us as though we were at the centre of things. So the simple message is that the appearance of the night sky varies with where you’re located on Earth, and that influences your picture of who you are, where you are, and what the world is like.
It is interesting to notice that the Pole Star is not something that will be around forever. When the Earth rotates every 24 hours, the North Pole is pointing in a particular direction in the sky; at the moment, that points towards this convenient star called Polaris, but actually the Earth when it rotates wobbles a little bit, like a top on one of those little Eiffel Towers . As it spins, it precesses, and the precession takes 26,000 years to go all the way round, but it means that at any epoch in our history, our rotation axis was pointing in a different direction on the sky. At the moment, the past few hundred years, we’ve really been rather lucky, it’s been pointing towards a star, but for large parts of human history, celestial North Pole has not been pointing towards a convenient star, and it’s not been so easy to navigate and point out where the north is, and in the future, this will again be the case.
If you read Shakespeare’s Julius Caesar, there’s a line in there where Caesar says that he is “constant as the northern star”. This is a complete anachronism. Shakespeare is writing in 1599. In Caesar’s time, there was no northern star, so there wasn’t a Pole Star as there was in Shakespeare’s time.
As we’re in London , one mustn’t miss the chance to talk about something that you might not really sort of think of as art, but I think is a rather wonderful thing, and that’s Harry Beck’s London Underground map. This is a unique thing in many ways as a piece of design. Before Beck, maps were geographical maps, and the picture that you had of the things on the map bore complete correspondence with their relative positions on the Earth’s surface. Beck, extremely controversially, produced the first example of what we might now call a topological map, so what it emphasised were the connections between things. The London Underground map is not a good guide to walking around up on the surface, and this has all sorts of sociological implications. It brought in completely remote places like Rickmansworth and Bromley and made you think that they were part of London . Of course they’re miles away, further away than Cambridge probably! This happened about 1930. Before Beck came along, on a typical Underground map, (a geographical map) the place names were located close to where they really were located.
Beck was from the world of electronics, and was used to drawing electronic circuit diagrams, and so what you have with the London Underground map is a plan that’s laid out like an electronic circuit diagram. It changed every few years, changed with colouration, typography and so forth, but it’s remained remarkably invariant, and it’s a defining characteristic, I think it’s a design classic of its sort, but it was inspired by the world of electronics.
It is the time to say something about biology, and here we’re on to a rather unusual world of trying to understand the way in which the process of evolution has altered and conditioned our aesthetic senses. Why do we like certain things? A whole subject has developed by the name of evolutionary psychology which tries to understand why we like certain things in terms of the survival value that that liking would have had perhaps one or two million years ago in the environments in which our earliest ancestors developed. They spent vastly longer periods of time in African Savannah type landscapes, What type of sensitivity for pattern, for what we call aesthetics, has a survival value, and we would expect to be engendered? In one of my earlier lectures I talked about symmetry and why we like it that one rather good biological reason why we like symmetry is it’s a very rough and ready way to tell whether something is alive or not. If you look into a crowded scene in the jungle and you see something that has right/left symmetry, it’s very likely to be something that’s alive, something that’s either contemplating having you for lunch, or you might have it for lunch, or it’s a potential mate and you can both have lunch together! So you can see having this rough and ready lateral pattern recognition is something that would tend to survive, and it might even help to be over-sensitive, because if you rush home telling your family that you’ve seen tigers in the bushes, and there are no tigers in the bushes, they’ll just think you’re a bit paranoid, but if you don’t rush home and tell them that and there are tigers in the bushes, then you’re all going to be dead, and that sort of lack of pattern recognition will not tend to survive.
In the typical Savannah landscape in which our pre-human ancestors spent huge periods of time in their evolution, there are small clumps of bushes and trees, and then open prospects in between. The characteristic is that it’s an environment where you can see and not be seen, and we rather like environments like that, and the whole image that they conjure up in literature and stories, the concept for Little House on the Prairie, the inglenook fireplace, the secret garden, the tree house and so forth, they’re all environments where you can see without being seen. This type of landscape has become known as prospect and refuge, so you have a refuge where you can avoid being seen and you feel safe, and yet you have an open prospect and you can see things around you.
If you go and walk round some of our art galleries in London , and you don’t look at modern art but you look at landscapes, you’ll find they’re completely dominated by prospect and refuge scenes. One could give hundreds of examples: the castle, with the great open terrain, invites you to explore, so the environment invites you in, where there is an obvious way of entry, there’s an open prospect, it looks safe. We don’t like environments where there’s dense forest and hidden corners where somebody might jump and surprise you, and of course that’s why we don’t like shopping centres like that. So these are environments that don’t invite you to enter them. There is no obvious way to enter. They don’t have that sort of prospect and refuge structure. If you look at works of computer generated art, what distinguishes computer-generated landscapes from landscapes that are created by human artists is often this lack of the refuge symbol. They have lots of prospect, but they don’t have refuge symbol.
You can take this argument on to other areas. Huge numbers of works of art are about the things that we eat, fruit, still life, flowers. Why do we like flowers? You can’t eat flowers, but if you look in a floral scene, flowers tell you whether things are ripe or not. They’re a sense where you can use colour. Colour sends important signals about an environment, and we like flowers perhaps because of our evolutionary history of the information that they give us. They enable us to distinguish things, those things which can be eaten from those which cannot. Landscape art, the artistic representation of things that we might eat, things that we might drink, carries something of an inheritance from this pre-history where we’re programmed with certain types of response.
Well, let’s say a little bit about music, a subject about which I know nothing, but I’ll tell you what little things I might know. If we take the point of view of an acoustic engineer, it’s quite interesting to look for example at a picture of sound intensity against the frequency of the sound. If you stroll outside of a particular range of intensity, you will either hear nothing because the intensity is too low, or you will be deafened because the sound intensity will burst your ear drums or produce other types of damage. If we go too far in the high frequency range, we’ll become ultrasonic; we’ll be outside of the receiving band for our audio system, and similarly if we go infrasonic. You can plot where we can receive sound, where we can hear things comfortably without pain and without straining ourselves, and in the middle is the area of speech.
It is interesting to compare the range of the human audio system with the range that we’ve exploited and filled out in the creation of music. As time has gone on, the development of new instruments, also the development of particular types of auditoria in which to perform music, the development of large symphony orchestras, has expanded the domain of this intensity frequency pitch which is exploited by modern music. As time has gone on, the general trend has been to try to fill out the whole of the allowed plane. We’re really quite a way towards filling the whole thing.
If you also take the view of an acoustic engineer, you regard all music simply as noise, and noise is interesting. It has an infinite number of characteristics, but let’s just pick on one of them, and this is what’s called the spectrum. When plotting intensity against frequency, how does the intensity vary statistically with the frequency? Noises can have a scale-free character that we saw earlier with fractals, and the intensity might vary like one over the frequency to some power, some number. As you change that number, the sound will have a different quality.
When the number is zero, there’s no dependence on the frequency at all, and we have what people call white noise. This is truly random in the sense that there are no correlations in the sound signal over any time intervals. There’s no structure. This is the sort of thing that you get if you add together in a random way many different sound sources. If you stand in the middle of a forest and you listen to trees falling down and birds singing and the wind in the trees and so forth, the net result is probably quite close to white noise. And if you come from California , you probably have a machine that generates the sound of breaking waves. These are white noise machines, and they’re supposed to send you off to sleep or relax you. The reason they do it is because white noise has no correlations in pattern over any time intervals, your brain’s pattern recognition software rather quickly gives up trying to find correlations, switches off, and so you feel rather relaxed.
If this number increases, so we go one over the frequency squared, we have what people call brown noise. It’s nothing to do with colour, it’s browny in noise, so called because Brown discovered what we now call really atomic motion, the motion of atoms or diffusive motion has the statistics of what’s called Brownian motion. Brown was I think a Scottish botanist. If you want to hear Brownian motion, I think the way to do it is you put your hand on your ear or you have a shel. You’re supposed to hear the waves, but what you’re hearing there are the molecules of air bouncing around inside your ear drum.
If beta went to three, you would have a very strong correlation, so as beta increases, you get more and more correlation, and in between, there is a compromise type of noise, beta is one, that’s sometimes called pink noise or it’s just called one over F noise. This is the happy medium between random white noise and strongly correlated coloured noise. One over F noise has correlations on all scales, so it’s the happy medium between surprise and predictability.
A few years ago, two physicists in the department where I was in Berkeley, John Clark and Richard Vass, carried out some studies of information that was coming out of the radio in the physics laboratory while they were doing something else, and they thought it was interesting enough that they started systematically analysing it. If you were seeing white noise, your signal would be horizontal, one over F noise, it’s going down at 45 degrees, if you’re seeing one over F squared, it would be steeper. They got very excited because they looked at all sorts of different type of sound signal: Scott Joplin, classical music, radio station, rock station, news and music, and so forth – and they claimed that all these signals had the one over F slope, that the spectrum followed this characteristic slope. They then did the same thing looking at non-Western music. At Berkeley in the Bancroft Library I think there’s probably the world’s biggest sound library of music from different human cultures, and so they were well placed to do this, and again, you see the same type of general message. So for a while people thought that this was really rather significant, that somehow we like music that has an optimal amount of predictability versus unpredictability.
Unfortunately, the problem is, for a physicist, that if you look at any sound signal over a long enough period, it always tends to give you this one over F signal. They were being really tricked by this sensitivity, so the fact that they were looking at such long snatches of music and averaging always produced the one over F spectrum. Then people started looking in much more detail at what happened in short pieces of music, which were much more designed to be single pieces of performance, rather than averaging over many. The result that appeared there was that everything they looked at, had a much stronger correlation. So instead of beta being one, it was between about 1.7 and 1.9 or so – 2 would be the black noise. If you played the music backwards, it would have the same spectrum. If you turned it upside down, it would have the same spectrum. So in no sense does the spectrum encompass all the information in the signal. There are other ways to look at the complexity of the music, which might break this degeneracy and tell you more interesting things.
Finally, something amusing about art and statistics. I think it was Gulliver’s Travels that first introduced the idea of a collection of individuals who just type at random and eventually the works of Shakespeare will pop out. Usually they’re monkeys, for some reason. Gulliver came across some academy somewhere, where the chief scientist had armies of people just randomly writing letters and words, and the research strategy was that eventually all forms of knowledge would emerge from this process. In the early years of this century, Arthur Eddington immortalised this picture of the typing monkeys producing works of Shakespeare.
Well, there is on the web a monkey typing simulation which all of you can take part in. It began on the 1 st of July 2003 , and you can join it and the computer produces a whole random typing sequence, and each time somebody joins, their computer helps double the output. So far, I think the total output has amounted to 10 to the 35 pages of random output, each with a couple of thousand keystrokes on each. An interesting question is how many bits of Shakespeare actually appear in that? They have a random search algorithm that looks for snatches of Shakespeare’s plays in the random output. There are many, many productions of about 18 and 19 letters, so almost every day, you’ll get a new 18, 19 letter string. The record when I looked last week was 21. A random output suddenly appears, and it’s peculiar, you suddenly get some English, a full stop, a snatch of meaningful words, and then it carries on in a gibberish random way. There is a little 18 symbol snatch from A Midsummer Night’s Dream. “Us now for” and then we fade. But here’s the record: Love’s Labours Lost, have this section KING in capital letters with a full stop, so it’s the king who’s going to speak, it’s a bit of the stage directions, “Let fame that” and then it goes back into gibberish. This matches a nice little stretch here from Love’s Labours Lost. It’s quite encouraging, that after just a year, we’re already up to 21 letters from one of the plays. I find this really quite spooky….but good time to end!
© John D Barrow, Gresham College, 14 December 2004
This event was on Tue, 14 Dec 2004
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