The Science of Discworld II - The Globe tsod-2

by Terry Pratchett

  However, that's not what we want anyway. 'Natural' is an illusion. Desmond Morris made a lot of money selling paintings done by apes. The apes clearly enjoyed the whole business, and so did Morris, and presumably so did the people who bought them and looked at them in art galleries.

  There is also an elephant that paints, and signs its paintings. Sort of. There's a segment of modern painting whose philosophy seems to relate to this quest for the genuinely primitive. One side is the tackiest, painting by children, which clearly demonstrates the stepwise effects of the culture -the extelligence -on their burgeoning intelligence. To our inexpert eyes, though, these paintings demonstrate only the enormous gratification achieved by some parents in response to minimal effort by their children.

  Another aspect, more intellectual, is the move towards apparently real-world constraints, like cubism, or attempts to develop styles that force us to re-evaluate how we see, like Picasso's profile faces but with the two eyes on one side. There is a very common modern form that arranges rectangles of paper with different textures, or sprays sparse paint droplets according to some minimal rule, or scatters charcoal dust on a bold swirly bright oil-paint background and then combs it into the texture and pattern of the whole canvas. All of these can give pleasure to the eye. Why? How do they differ from natural objects, some of which also give considerable pleasure?

  Now we want to make a giant leap and bring Mozart, jazz, paper-texture and charcoal-swirl oil paintings into the same frame. We think that this frame naturally includes ancient cave-paintings, which we know to be early, so have more claim to being genuinely primitive, if we could only look at them with the eyes and minds of viewers contemporary with the artist. The same problem occurs with Shakespeare, too: we no longer have the ears or minds - the extelligence - of the first Elizabethan age.

  We have to be more than a bit scientific here. We have to consider how we perceive light, sound, touch -what our sense organs tell us. For a start, they don't, and this is the first lesson. In his book Consciousness Explained, Daniel Dennett is very critical of the Cartesian Theatre[67] picture of consciousness. In this picture, we imagine ourselves sitting in a little theatre in our minds, where our eyes and ears pipe in pictures and sounds from the outside world. In school we all learned that the eye is like a camera, and that a picture of the world is imaged in the plane of the retina, as if that was the difficult bit. No, the difficult bit starts there, with different elements of that picture taking different routes into different parts of the brain.

  When you see a moving red bus, the features 'moving', 'red' and 'bus' are separated fairly early in the brain's analysis of the scene ... and they don't just get put together again to synthesise your mental picture. Instead, your picture is synthesised from lots of clues, lots of bits, and nearly all of what you 'see' as you look around the room is only 'there' in your brain. It's not at all like a TV

  picture. It is not picked up instantly and updated, but nearly all of that 'detailed' surround is invented as a kind of wallpaper around the little bit that has your attention. Most of the details are not present as such in your mind at all, but that's the illusion that your mind presents to you.

  When we see a painting ... except, again, we don't. There are several ways to convince people that they invent what they 'see', that perception is not simply a copy of the eye's image on the retina. There is, for example, a blind spot on the retina where the optic nerve leaves it. This is big. It's as big as 150 full moons (that's not a misprint: a hundred and fifty). Not that the moon is as big, to our eyes, as we usually think -and certainly not as big as Hollywood repeatedly shows it. We 'see' the full moon as much bigger than it 'is' (sorry, we have to use some trick to separate what's in your mind from reality out there), especially when it's near the horizon. The best way to appreciate that is to demonstrate to yourself that the moon's image is the size of your little fingernail at arm's length. Hold out your arm, and the tip of your littlest finger more than covers the moon. So the blind spot is smaller than our description may have suggested, but it's still a big chunk of the retinal image. We don't notice any hole in the picture we get of the outside world, though, because the brain fills in its best estimate of what's missing.

  How does the brain know what's missing from right in front? It doesn't, and it doesn't have to: that's the point. Although 'fills in' and 'missing' are traditional terms in this area of science, they are, again, misleading. The brain doesn't notice that anything is missing, so there isn't a gap to be filled in. The neurons of the visual cortex, the part of the brain that analyses that retinal image into a scene that we can recognise and label, are wired up in elaborate ways, which reinforce certain perceptual prejudices.

  For example, experiments with dyes that respond to the brain's electrical signals show that the first layer of the visual cortex detects lines -edges, mostly. The neurons are arranged in local patches, 'hyper-columns', which are assemblies of cells that respond to edges aligned along about eight different directions. Within a hypercolumn, all connections are inhibitory, meaning that if one neuron thinks it has seen an edge pointing along the direction to which it is sensitive, then it tries to stop the other neurons from registering anything at all. The result is that the direction of the edge is determined by a majority yote. In addition, there are also long-range connections between hyper-columns. These are excitatory, and their effect is to bias neighbouring hypercolumns to perceive the natural continuation of that edge, even if the signal they receive is too weak or ambiguous for them to come to that conclusion unaided.

  This bias can be overcome by a sufficiently strong indication that there is an edge pointing in a different direction; but if the line gets faint, or part of it is missing, the bias automatically makes the brain respond as if the line was continuous. So the brain doesn't 'fill in' the gaps: it is set up not to notice that there are gaps. That's just one layer of the visual cortex, and it uses a rather simple trick: extrapolation. We have little idea, as yet, of the inspired guesswork that goes on in deeper layers of the brain, but we can be sure that it's even more clever, because it produces such a vivid sensation of a complete image.

  What about hearing? How does that relate to sound? The standard lie-to-children about vision is that the cornea and lens make a picture on the retina, and that allegedly explains vision.

  Similarly, the corresponding lie-to-children about hearing centres on a part of the ear called the cochlea, whose structure allegedly explains how you analyse sound into different notes. In cross- section, the cochlea looks like a sliced snail-shell, and according to the lie-to-children, there are hair-cells all the way down the spiral attached to a tuned membrane. So different parts of the cochlea vibrate at different frequencies, and the brain detects which frequency -which musical note -it is receiving, by being told which part of the membrane is vibrating. In support of this explanation, we are told a rather nice story about boiler-makers, whose hearing was often damaged by the noise in the factories where they worked. Supposedly, they could hear all frequencies except ones near the frequency that was most common in making boilers. So just one place on their cochlea was burnt out, and the rest worked OK. This proved, of course, that the

  'place' theory of hearing was correct.

  Actually, this story tells you only how the ear can discriminate notes, not how you hear the noise. To explain that, it is usual to invoke the auditory nerve, which connects the cochlea to the brain. However, there are as many connections, or more, that go in the other direction, from brain to cochlea. You have to tell your ear what to hear.

  Now that we can actually look at what the cochlea does when it's hearing, we find not one place vibrating for each frequency, but more like twenty. And these places move as you flex your outer ear. The cochlea is phase-sensitive, it can discriminate the kind of difference that makes an 'ooh'

  sound different from an 'eeh' at the same frequency. This is the kind of change to the sound that you make when you change the shape of your mouth as you speak
. And surprise, surprise, that's just the difference that the cochlea -after your outer ear and your own particular auditory canal, and your own particular eardrum and those three little bones -can best discriminate. A recording from someone else's eardrum, played back up against yours, makes little sense. You have learned your own ears. But you have taught them, too.

  There are about seventy basic sounds, called phonemes, that Homo sapiens uses in speech. Up to about six months old, all human babies can discriminate all of these, and an electrode on the auditory nerve gives different patterns of electrical activity for each. At about six to nine months old, we start talking scribble, and it very soon becomes English scribble or Japanese scribble. By a year old the Japanese ear cannot distinguish 'l' from 'r', because both phonemes send the same message from cochlea to brain. English babies can't discriminate the different clicks of the !Kung San, nor the differences between the distinct 'r's in French. So our sense organs do not show us the real world. They stimulate our brains to produce, to invent if you like, an internal world made of the counters, the Lego™ set, that each of us has built up as we mature.

  Such apparently straightforward abilities as vision and hearing are far more complicated than we usually imagine. Our brains are much more than just passive recipients. An awful lot is going on inside our heads, and we project some of it back into what we think is the outside world. We are conscious only of a small part of its output. These hidden depths and strange associations in the brain may well be responsible for our musical sensibilities.

  Music exercises the mind; it's a form of play. It seems probable that our liking for music is linked to other things than our ears. In particular, the brain's motor activity may be involved, as well as its sensory activity. In primitive tribes and advanced societies, music and dance often go together. So it may be the combination of sound and movement that appeals to our brains, rather than one or the other. In fact, music may be an almost accidental by-product of how our brains put the two together.

  Patterns of movement have been common in our world for millions of years, and their evolutionary advantage is clear. The pattern 'climb a tree' can protect a savannah ape from a predator, and the same goes for the pattern 'run very fast'. Our bodies surround us with linked patterns of movement and sound. Like music, they are patterns in time, rhythms. Breathing, the heartbeat, voices in synch with lips, loud bangs in synch with things hitting other things.

  There are common rhythms in the firing of nerve cells and the movement of muscles. Different gaits - the human walk and run, the walk-trot-canter-gallop of the horse -can be characterised by the timing with which different limbs move. These patterns relate to the mechanics of bone and muscle, and also to the electronics of the brain and the nervous system. So Nature has provided us with rhythm, one of the key elements of music, as a side-effect of animal physiology.

  Another key element, pitch and harmony, is closely related to the physics and mathematics of sound. The ancient Pythagoreans discovered that when different notes sounded harmonious, there was a simple mathematical relationship between the lengths of the strings that produced them, which we now recognise as a relation between their frequencies. The octave, for example, corresponds to a doubling of frequency. Simple whole number ratios are harmonious, complicated relationships are not.

  One explanation for this is purely physical. If notes with frequencies that are not related by simple whole numbers are sounded together, they interfere with each other to produce 'beats', a jarring low-frequency buzz. Sounds that make the sensory hairs in our ears vibrate in simple patterns are necessarily harmonious in the Pythagorean sense, and if they aren't, we hear the beats and they have an unpleasant effect. There are many mathematical patterns in musical scales, and they can be traced, to a great extent, to the physics of sound.

  Overlaid on the physics, though, are cultural fashions and traditions. As a child's hearing develops, its brain fine-tunes its senses to respond to those sounds that have cultural value. This is why different cultures have different musical scales. Think of Indian or Chinese music compared to European; think of the changes in European music from Gregorian chants to Bach's Well-Tempered Clavier.

  This is where the human mind is situated: on the one hand, subject to the laws of physics and the biological imperatives of evolution; on the other, as one small cog in the great machine of human society. Our liking for music has emerged from the interaction of these two influences. This is why music has clear elements of mathematical pattern, but is usually at its best when it throws the pattern book away and appeals to elements of human culture and emotion that are -for now, at least - beyond the understanding of science.

  Let's come down to Earth and ask a simpler question. The wells of human creativity run deep, but if you take too much water from a well it runs dry. Once Beethoven had written the opening bars of his Symphony in C Minor -dah-dah-da DUM -that was one less tune for the rest of us.

  Given the amount of music that has been composed over the ages, maybe most of the best tunes have been found already. Will the composers of the future be unable to match those of the past because the world is running out of tunes?

  There is, of course, far more to a piece of music than a mere tune. There is melody, rhythm, texture, harmony, development ... But even Beethoven knew you can't beat a good tune to get your composition off the ground. By 'tune' we mean a relatively short section of music - what the cognoscenti call a 'motif' or a 'phrase', between one and thirty notes in length, say. Tunes are important, because they are the building blocks for everything else, be it Beethoven or Boyzone.

  A composer in a world that has run out of tunes is like an architect in a world that has run out of bricks.

  Mathematically, a tune is a sequence of notes, and the set of all possible such sequences forms a phase space: a conceptual catalogue that contains not just all the tunes that have been written, but all the tunes that could ever be written. How big is T-space?

  Naturally, the answer depends on just what we are willing to accept as a tune. It has been said that a monkey typing at random would eventually produce Hamlet, and that's true if you're willing to wait a lot longer than the total age of the universe. It's also true that along the way the monkey will have produced an incredible amount of airport novels[68]. In contrast, a monkey pounding the keys of a piano might actually hit on a reasonable tune every so often, so it looks as though the space of acceptably tuneful tunes is a reasonable-sized chunk of the space of all tunes. And at that point, the mathematician's reflexes can kick in, and we can do some combinatorics again.

  To keep things simple, we'll consider only European-style music based on the usual twelve-note scale. We'll ignore the quality of the notes; whether played on a piano, violin, or tubular bells, all that matters is their sequence. We'll ignore whether the note is played loudly or softly, and more drastically -we'll ignore all issues of timing. Finally, we'll restrict the notes to two octaves,

  25 notes altogether. Of course all these things are important in real music, but if we take them into account their effect is to increase the variety of possible tunes. Our answer will be an underestimate, and that's all to the good since it will still turn out to be huge. Really, really huge, right? No - bigger than that.

  For our immediate purposes only, then, a tune is a sequence of 30 or fewer notes, each chosen from 25 possibilities. We can count how many tunes there are in the same way that we counted arrangements of cars and DNA bases. So the number of sequences of 30 notes is 25 x 25 x ... x

  25, with 30 repetitions of that 25. Computer job, that: it says that the answer is

  867361737988403547205962240695953369140625 which has 42 digits. Adding in the 29-note tunes, the 28-note ones, and so on we find that T- space contains roughly nine million billion billion billion billion tunes. Arthur C. Clarke once wrote a science fiction story about the 'Nine billion names of God'. T-space contains a million billion billion billion tunes for every one of God's names. Assume that a million compos
ers write music for a thousand years, each producing a thousand tunes per year, more prolific even than The Beatles. Then the total number of tunes they will write is a mere trillion. This is such a tiny fraction of that 42-digit number that those composers will make no significant inroads into T- space at all. Nearly all of it will be unexplored territory.

  Agreed, not all of the uncharted landscape of tune-space consists of good tunes. Among its landmarks are things like 29 repetitions of middle C followed by F sharp, and BABABABABABABABABABABABABABABA, which wouldn't win any prizes for musical composition. Nevertheless, there must be an awful lot of good new tunes still waiting to be invented. T-space is so vast that even if good-tune-space is only a small proportion of it, good-tune-space must also be vast. If all of humanity had been writing tunes non-stop since the dawn of creation, and went on doing that until the universe ended, we still wouldn't run out of tunes.

  It is said that Johannes Brahms was walking along a beach with a friend, who was complaining that all of the good music had already been written. 'Oh, look,' said Brahms, pointing out to sea.

  'Here comes the last wave.'

  Now we come to what may well be the chief function of art and music for us -but not for edge people or chimpanzees, and probably not for Neanderthals. This, if we are right, is what Rincewind has in mind. When we look at a scene we see only the middle five to ten degrees of arc. We invent the rest all around that bit, and we give ourselves the illusion that we're seeing about ninety degrees of arc. We perceive an extended version of the tiny region that our senses are detecting. Similarly, when we hear a noise, especially a verbal noise, we set it in a context.

  We rehearse what we've heard, we anticipate what's coming, and we 'make up' an extended present, as if we'd heard the whole sentence in one go. We can hold the entire sentence in our heads, as if we heard it as a sentence, and not one phoneme at a time.