The Science of Discworld II

by Terry Pratchett

  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.2 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 × 25 × … × 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 composers 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. ‘O
h, 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.

  This is why we can get the words of songs completely wrong and not realise it. The Guardian newspaper ran an amusing section on this habit, with examples such as ‘kit-kat angel’ for ‘kick-ass angel’ – bit of a generation gap there, which underlines how our perceptions are biased by our expectations. Ian recalls an Annie Lennox song that really went ‘a garden overgrown with trees’, but always sounded like ‘I’m getting overgrown with fleas’.

  Holding a whole sentence, or a musical phrase, in our minds is what we do with time when we watch a TV or a cinema-screen. We run the frames together into a series of scenes, as well as making up all the spatial stuff that we’re not actually looking at. The brain has so many tricks that its owner is not conscious of: as you sit there in the cinema, your eyes are flicking from place to place on the screen, as they are doing while you read these words. But you turn off your perceptions as your eyes move, and re-jig your invented image so that your new retinal image is consistent with the previous version. That’s why you get seasick or car-sick: if the outside image jumps about and isn’t where you expect it to be, then that upsets your sense of balance.

  Now think about a piece of music. Isn’t the construction of an extended present precisely the exercise that your brain ‘wants’ to do with a series of sounds, but without the complication of the meanings? As soon as you get used to the style of a particular kind of music, you can listen to it and grasp whole themes, tunes, developments, even though you’re hearing only one note at a time. And the instrumentalist who is making the noise is doing the same kind of thing. His brain has expectations of what the music should sound like, and he fulfils those. To some extent.

  So it seems that our sense of music may be tied to a sense of an extended present. Some possible scientific evidence for this proposition has recently been found by Isabelle Peretz. In 1977 she identified a condition called ‘congenital amusia’. This is not tone-deafness, but tune-deafness, and it should give us some insight into how normal people recognise tunes, by showing how that goes wrong. People with this condition cannot recognise tunes, not even ‘Happy Birthday To You’, and they have little or no sense of the difference between harmony and dissonance. There is nothing physically wrong with their hearing, however, and they were exposed to music as children. They are intelligent and have no history of mental illness. What seems to be wrong is that when it comes to music, they have no sense of an extended present. They cannot tap their feet in rhythm. They have no idea what a rhythm is. Their sense of timing is impaired. Mind you, so is their sense of pitch; they cannot distinguish sounds separated by an interval of two semitones – adjacent white keys on a piano. So the lack of an extended present is not the only problem. Congenital amusia is rare, and it affects males and females equally. Its sufferers have no difficulty with language, however, suggesting that the brain’s music modules, or at least those affected by amusia, differ from its language modules.

  The same kind of interpretational step takes place in the visual arts, too. When you look at a painting – a Turner, say – it evokes in you a variety of emotions, perhaps nostalgia for a nearly forgotten holiday on a farm. That may give you a little burst of endorphins, chemicals in the brain that create a sense of well-being, but presumably you’d get much the same from a photograph or even a verbal description or a bit of pastoral poetry. The Turner painting does more than that, perhaps because it can be more sentimental, more idealised than a photo, however idyllic. It evokes the memory on a more personal level.

  What about other kinds of painting: the paper textures, the charcoal smear? Jack went to an art gallery, as an innocent in art appreciation, and tried the ‘context’ trick that any novice is always told to try. You’re supposed to sit in front of the picture, and gaze at it, and kind of sink into it and feel how it relates to its surroundings. The result was instructive. When he paid attention to a small part of the canvas, he found that he could match the context that his brain had invented with the one that the artist had actually provided. The charcoal smear was particularly good for this: each part implied something of the pattern of the whole. However, there were intriguing differences from part to part. There were variations on the theme, as in music, superimposed on the brain’s expectations. Jack’s brain enjoyed comparing the picture that it was inventing with the progressively different one that the artist was forcing his brain to construct.

  Art goes back a long, long way; the further back we look, the more controversial the evidence is. The ‘Dame à la Capuche’, a 1.5-inch (3.5-cm) high statuette of a woman, exquisitely carved from mammoth-tusk ivory, is 25,000 years old. Some of the most elegant cave paintings, with simple, sweeping lines that depict horses, bison and the like, are found in the Grotte Chauvet in France, and in 1995 they were dated at 32,000 years old. The oldest art that undoubtedly is art is about 38,000 years old: beads and pendants, found in Russia. And some beads made from ostrich egg shells in Kenya, which may be 40,000 years old.

  Further back, it all gets less certain. Ochre is a common pigment in rock drawings, and ochre ‘crayons’ found in Australia are 60,000 years old. There is a lump of rock from the Golan Heights, whose natural crevices have been worn deeper, presumably by a human hand wielding another lump of rock. It bears a vague resemblance to a woman, and it is about 250,000 years old. But maybe it’s just a lump of rock that a child idly scratched, and the shape is accidental.

  Imagine yourself in the cave as the artist paints bison on the wall. He (or she?) is creating a picture for your brain that differs progressively from the one that your brain expects: ‘Now let’s put a female woolly rhinoceros under him …’ There have been several ‘artists’ on television, doing precisely that trick. Rolf Harris was surprisingly good at drawing animal sketches before your very eyes. And they were iconic animals, too: sly fox and wise owl.

  There it all is, tied up in a bundle. Our perceptions are tied to our expectations, and we do not segregate sensations from each other, or from memories. They are all played off against each other in the seclusion of our minds. We absolutely do not program our brains with direct representations of the real world. From the beginning we’re instructing our brains what to make of what we see, hear, smell and touch. We put spin on everything, and we anticipate, compare and contrast, construct lengths of time from successive instants, construct areas of picture from focused observation. We’ve been doing this, layer upon layer, taking more subtle nuances from conversation, from flirting glances, to ‘Will she come to look like her mother does now?’ assessments of the real world, all the time.

  That’s what our brains do, and what edge people’s brains don’t.

  We suspect that Neanderthals didn’t do that kind of thing much, either, because there’s an alternative, and it’s consistent with their cultural torpor. The alternative is to live in a world that you’ve set up to ensure that nothing is unexpected. All the events follow your expectations from previous events, so habit engenders security. Such a world is very stable, and that means it doesn’t go anywhere much. Why try to leave the Garden of Eden? Gorillas don’t.

  Tribal life could be like that for Homo sapiens, except that reality al
ways intrudes, for instance those barbarians up on the hill. But Neanderthals, maybe, weren’t afflicted by barbarians. Certainly, nothing seems to have provoked big changes in their lifestyles, even over tens of thousands of years. Art does provoke changes. It makes us look at the world in new ways. The elves like that, it gives new ways for them to terrify people. But Rincewind has seen further than the elves are capable of seeing, and he’s worked out where art takes us. Where? You’ll soon find out.

  1 Cartesian, again, because of Descartes, whose cogito ergo sum and mind-is-different-stuff-from-matter still influence pop philosophy.

  2 Though Ian has a friend, an engineer named Len Reynolds, whose cat managed to type ‘FOR’ into his computer by walking on the keyboard. Three more letters, ‘MAT’, and the cat would have wiped his hard disc.

  TWENTY-FIVE

  PARAGON OF VEGETABLES

  THE WINE-DARK SEA lapped the distant shore. Nice country, Rincewind thought. A bit like Ephebe. Grapes, olives, honey and fish and sunshine.

  He turned back to his group of proto-actors. They were having difficulty grasping the idea.

  ‘Like the priests do in the temples?’ said a man. ‘Is that what you mean?’

  ‘Yes, but you can … expand the idea,’ said Rincewind. ‘You can pretend to be the gods. Or anything else.’

  ‘Wouldn’t we get into trouble?’

  ‘Not if you did it respectfully,’ said Rincewind. ‘And people would … sort of see the gods. Seeing is believing, eh? Besides, children pretend to be other people all the time.’

  ‘But that is childish play,’ said the man.

  ‘People might pay to see you,’ said Rincewind. There was an immediate increase in interest. Human-shaped creatures were the same everywhere, Rincewind thought; if you got money for doing it, it had to be worth doing.