No More Learning

T. Hardin's 1988 book, Color for Philosophers: Unweaving the Rainbow. I am embarrassed to say that I discovered this book, and in particular its excellent subtitle, only after mine had gone off to the publishers. Doctor Dolittle, by the way, may be hard to find, as he is now often banned by pompously correct librarians. They worry about the racism in The Story- of Doctor Dolittle but this was all but universal in the 1920's. In any case it is offset by the Doctor's splendid fight against the slave trade in Doctor Dolittle's Post Office, and, more profoundly, by the stand that all the Doctor Dolittle books make against the vice of speciesism which is as unquestioned today as racism was in earlier times.
We know now in some detail how the eye informs the brain about the wavelengths of light. It is a three-colour code, as used in colour television. The human retina has four kinds of light-sensitive cell: three kinds of 'cones' plus the 'rods'. All four are similar and have surely diverged from
a common ancestor. One of the things it is easy to forget about any sort
of cell is how intricately complicated even a single cell is, much of the complexity being built up of fine-folded internal membranes. Each tiny rod or cone contains a deep stack of membranes, packed like a tall column of books. Threaded back and forth through each book is a long, thin molecule, a protein called rhodopsin. Like many proteins, rhodopsin behaves as an enzyme, catalysing a particular chemical reaction by providing a correctly shaped place for particular molecules to slot in.
It is the three-dimensional form of an enzyme molecule which gives it its catalytic property, serving as a carefully shaped, albeit slightly flexible, template for other molecules to fit in and meet one another - otherwise they'd have to rely on bumping into each other by chance (which is why enzymes so dramatically speed up chemical reactions). The elegance of this system is one of the key things that makes life possible, but it does raise a problem. Enzyme molecules are often capable of coiling into more than one shape, and usually only one of them is desirable. Much of the work of natural selection over the millions of years has been to find 'decisive' or 'single-minded' molecules whose 'preference' for their
favoured shape is much stronger than their tendency to coil into any other shape. Molecules with two alternative shapes can be a tragic menace. 'Mad cow disease', sheep scrapie and their human counterparts Kuru and Creutzfeldt-Jakob disease, are caused by proteins called prions which have two alternative shapes. They normally fold into one of the two shapes, and in this configuration they do a useful job. But occasionally they adopt the alternative shape. And then a terrible thing happens. The presence of one protein with the alternative shape induces others to come over to the rogue persuasion. An epidemic of misshapen proteins sweeps through the body like a cascade of falling dominoes. A single misshapen protein can infect a new body and trigger a new domino run. The consequence is death from spongy holes in the brain, because the protein in its alternative shape cannot do its normal job.
Prions have caused some confusion because they spread like self- replicating viruses, yet they are proteins, and proteins aren't supposed to be self-replicating. Textbooks of biology would have it that self-replication is the unique privilege of polynucleotides (DNA and RNA). However, prions are self-replicating only in the peculiar sense of one misshapen rogue molecule 'persuading' its already existing neighbours to flip into the same shape.
In other cases, enzymes with two alternative shapes turn their switchability to good account. Switchability is, after all, the essential property of transistors, diodes and the other high-speed electronic gates that make the logical operations of computers - IF, NOT, and, OR and the like - possible. There are 'allosteric' proteins that flip from state to state in a transistor-like way, not through infectious 'persuasion' by a neighbour, as in prions, but only if some biologically useful condition is met, AND NOT under certain other conditions. Rhodopsin is one of these 'transistor' proteins which make good use of their property of having two alternative shapes, like a photocell, it flips from one state to the other when it is hit by light. It automatically clicks back to the previous shape after a brief recovery period. In one of its two shapes it is a powerful catalyst, but not in the other. So, when light causes it to snap into its active shape, this initiates a special chain reaction and a rapid turnover of molecules. It is as if the light had switched on a high-pressure tap.
The end product of the resulting chemical cascade is a stream of nerve impulses which are relayed to the brain via a series of nerve cells, each of which is a long thin tube. Nerve impulses, too, are rapidly catalysed chemical changes. They sweep along the long thin tubes like fizzing trails of gunpowder. Each fizzing sweep is discrete and separate from the others, so they arrive at the far end of the tube like a series of short, sharp reports - nerve impulses.
The rate at which the nerve impulses arrive - which may be hundreds per second - is a coded representation of (in this case) the intensity of light falling on the rod or cone cell. As far as a single nerve cell is concerned, the difference between strong stimulation and weak is the difference between a high-speed machine gun and intermittent rifle fire.
So far, what I have said applies to the rods and all three kinds of cone. Now for the ways in which they differ. Cones respond only to bright light. Rods are sensitive to dim light and are needed for night vision. Rods are found all over the retina, and are nowhere particularly crowded, so they are no good for resolving small detail. You can't read with them. You read with cones, which are extremely densely packed in one particular part of the retina, the fovea. The denser the packing, of course, the finer the detail that can be resolved.
Rods are not involved in colour vision because they all have the same wavelength sensitivity as each other. They are most sensitive to yellow light in the middle of the visible spectrum, less sensitive nearer the two ends of the spectrum. This does not mean that they report all light to the brain as yellow. That isn't even a meaningful thing to say. All nerve cells report to the brain as nerve impulses, and that's all. If a rod fires rapidly, this could either mean that there is a lot of red or blue light, or that there is somewhat less yellow light. The only way for the brain to resolve the ambiguity is to have simultaneous reports from more than one kind of cell, differentially sensitive to different colours.
This is where the three kinds of cone come in. The three kinds of cone have three different flavours of rhodopsin. All of them respond to light of all wavelengths. But one kind is most sensitive to blue light, another is most sensitive to green light, and the third is most sensitive to red light. By comparing the firing rates of the three kinds of cone - in effect, subtracting them from each other - the nervous system is able to reconstruct the wavelengths of light hitting the retina. Unlike the case of vision by rods alone, the brain is not confused between dim light of one colour and bright light of another colour. The brain, because it receives reports from more than one kind of cone, is able to compute the true colour of the light.
As I said when recalling Doctor Dolittle on the moon, the colours that we finally think we see are labels used for convenience by the brain. I used to be disappointed when I saw 'false colour' images, say, satellite photographs of earth, or computer-constructed images of deep space. The caption tells us that the colours are arbitrary codes, say, for different types of vegetation, in a satellite picture of Africa. I used to think false colour images were a kind of cheat. I wanted to know what the scene 'really' looked like. I now realize that everything I think I see, even the
colours of my own garden through the window, are 'false' in the same sense: arbitrary conventions used, in this case by my brain, as convenient labels for wavelengths of light. Chapter 11 argues that all our perceptions are a kind of 'constrained virtual reality' constructed in the brain. (Actually, I am still disappointed by false colour images! )
We can never know whether the subjective sensations that different people associate with particular wavelengths are the same. We can compare opinions about what colours seem to be mixtures of which. Most of us agree to find it plausible that orange is a mixture of red and yellow. Blue-green's status as a mixture is conveyed by the compound word itself, though not by the word turquoise. It is controversial whether different languages agree on how they partition the spectrum. Some linguists aver that the Welsh language does not divide the green and blue region of the spectrum in the same way as English does. Instead, Welsh is said to have a word corresponding to part of green, and another word corresponding to the other part of green plus part of blue. Other linguists and anthropologists say that this is a myth, no more true than the equally seductive allegation that the Inuit ('Eskimos') have 50 different words for snow. These sceptics claim experimental evidence, obtained by presenting a large range of coloured chips to native speakers of many languages, that there are strong universals in the way humans partition the spectrum. Experimental evidence is, indeed, the only way to settle the question. It matters nothing that, at least to this English speaker, the story about the Welsh partitioning of blue and green feels implausible. There is nothing in physics to gainsay it. The facts, whatever they are, are facts of psychology.
Unlike birds, which have excellent colour vision, many mammals have no true colour vision at all. Others, including certain kinds of partially colour-blind humans, use a two-colour system based on two kinds of cones. High-quality colour vision with a three-colour system may have evolved in our primate ancestors as an aid to finding fruits in the green forest. It has even been suggested, by the Cambridge psychologist John Mollon, that the three-colour system 'is a device invented by certain fruiting trees in order to propagate themselves': an imaginative way of calling attention to the fact that trees benefit from attracting mammals to eat their fruits and spread the seeds. Some New World monkeys go in for weird arrangements in which different individuals within a species have different combinations of two-colour systems, and are thereby specialized to see different things. Nobody knows whether or how this benefits them, but it may be suggestive that bomber crews in the Second World War liked to include at least one colour-blind member, who could penetrate certain kinds of camouflage on the ground.
Unweaving the wider rainbow, moving to other parts of the electromagnetic spectrum, we separate station from station on the radio dial, we insulate conversation from conversation in the network of cellular telephones. Without sensitive unweaving of the electromagnetic rainbow, we'd hear everybody's conversation simultaneously, and all the frequencies on the broadcasting dial, in a white babel of noise. In a different way, and with the assistance of special-purpose computers, unweaving the rainbow underlies Magnetic Resonance Imaging, the spectacular technique by which doctors today can discern the three- dimensioned structure of our internal organs.
When a source of waves is itself moving relative to its detector, something special happens. There is a 'Doppler shift' of wavelengths as detected. This is easy to notice in the case of sound waves because they travel slowly. A car's engine note is of a distinctly higher pitch when it is approaching than when receding. This is why we hear the characteristic dual-tone eee-aaah when a car whizzes past. The Dutch scientist Buys Ballott in 1845 first verified Doppler's prediction by hiring a brass band to play on an open railway wagon as it sped past his audience. Light waves travel so fast that we notice the Doppler shift only if we are moving very fast towards the source of light (in which case the light is shifted towards the blue end of the spectrum) or away from it (in which case the light is red-shifted). This is true of distant galaxies. The fact that they are fast receding from us was first discovered because of the Doppler shift in their light. It is redder than it should be, shifted consistently towards the longwave, red end of the spectrum.
How do we know that the light coming from a distant galaxy is red- shifted? How do we know that it wasn't red when it set out? You can tell by using the Fraunhofer lines as markers. Each element, remember, signs its name in a unique barcode of lines. The spacing between the lines is characteristic like a fingerprint, but so is the precise position of each line along the rainbow. Light from a distant galaxy shows barcodes that have familiar spacing patterns. This very familiarity is what tells us that other galaxies are made of the same range of stuffs as our own. But the whole pattern is shifted a fixed distance towards the longwave end of the spectrum: it is redder than it should be. In the 1920s, the American astronomer Edwin Hubble (after whom the Hubble Space Telescope is now named) discovered that distant galaxies have red-shifted spectra. Those galaxies with the most pronounced red shift are also the most distant - as estimated from the faintness of their light.
Hubble's famous conclusion (although it had been suggested by others before) was that the universe is expanding, and from any given observation point the galaxies seem to recede at ever-increasing speed.
When we look at a distant galaxy, we are looking far back into the past, for the light has taken billions of years to reach us. It has become faint, which is how we know it has come a great distance. The speed with which our galaxy is racing apart from the other galaxy has had the effect of shifting the spectrum towards the red end. The relationship between distance and velocity of receding is a lawful one (it obeys 'Hubble's law). By extrapolating this quantitative relationship backwards we can estimate when the universe began expanding. Using the language of the now prevailing 'Big Bang' theory, the universe began in a gigantic explosion between ten billion and 20 billion years ago. All this we infer from unweaving the rainbow. Further developments of the theory, supported by all available evidence, suggest that time itself began in this mother of all cataclysms. You probably don't understand, and I certainly don't, what it can possibly mean to say that time itself began at a particular moment. But once again that is a limitation of our minds, which were only ever designed to cope with slow, rather large objects on the African savannahs, where events come well behaved and in order, and every one has a before. An event that has no before terrifies our poor reason. Maybe we can appreciate it only through poetry. Keats, thou shouldst be living at this hour.
And are there eyes out there in the galaxies, looking back at us? Back is the word, for they can see us only in our past. The inhabitants of a world 100 million light years distant might at this moment see, if they could see anything at all on our planet, red-shifted dinosaurs lunging over the rose-tinted plains. Alas, even if there are other creatures in the universe, and even if they have eyes, it is unlikely that, however powerful their telescopes, they will have the resolving power to see our planet, let alone individual inhabitants of it. We ourselves have never seen another planet outside our solar system. We didn't even know about all the planets in our own solar system until recent centuries. Neptune and Pluto are too faint to be seen by the naked eye. The only reason we knew which way to point the telescope was by calculations from minute perturbations in the orbits of nearer planets. In 1846, two mathematical astronomers, J. C. Adams in England and U. J. J. Leverrier in France, were independently puzzled by a discrepancy between the actual position of the planet Uranus and where it theoretically should have been. Both calculated that the perturbation could have been caused by the gravity of an invisible planet of a particular mass in a particular place. The German astronomer J. G. Galle duly pointed his telescope in the right direction and discovered Neptune. Pluto was discovered in the same way, as late as 1930 by the American astronomer C. W. Tombaugh, alerted by its (much smaller) gravitational effects on the orbit of Neptune. John Keats would have appreciated the excitement those astronomers felt:
Then felt I like some watcher of the skies When a new planet swims into his ken; Or like stout Cortez when with eagle eyes He stared at the Pacific - and all his men Look 'd at each other with a wild surmise - Silent upon a peak in Darien. 'On First Looking into Chapman's Homer' (1816)
I have had a special affection for these lines ever since they were quoted to me by a publisher on first reading the manuscript of The Blind Watchmaker.
But are there planets orbiting other stars? An important question, this, whose answer affects our estimate of the ubiquity of life in the universe. If there is only one star in the universe that has planets, that star has to be our sun and we are very very alone. At the other extreme, if every star is the centre of a solar system, the number of planets potentially available for life will exceed all counting. Almost whatever the odds of life on any one planet, if we find planets orbiting one other typical star as well as the sun, we feel sensibly less lonely.
Planets are too close to their suns, and too smothered by their brightness, for our telescopes normally to see them. The main way we know that other stars have planets - and the discovery waited till the 1990s - is, once again, through orbital perturbations, this time detected via Doppler shifts in coloured light. Here's how this works. We think of the sun as
the centre about which planets orbit. But Newton tells us that two bodies orbit each other. If two stars are of similar mass - they're called a binary pair - the two swing round each other like a pair of dumb-bells. The more unequal they are, the more it seems the lighter one orbits the heavier one, which almost stays still. When one body is much larger than the other, for instance the sun versus Jupiter, the heavier one just wobbles a bit while the lighter one whizzes round like a terrier circling its master on a walk.
It is such wobbles in the positions of stars that betray the presence of otherwise invisible planets orbiting them. But the wobbles themselves are too small to be seen directly. Our telescopes cannot resolve such small changes in position; less so, indeed, than they can resolve the planets themselves. Again, it is unweaving the rainbow that comes to the rescue, As a star wobbles back and forth under the influence of an orbiting planet, the light from it reaches us red-shifted when the star is moving away, blue-shifted when it is moving towards us. Planets give themselves away by causing minute, but measurable, red/blue oscillations in the light reaching us from their parent stars. In the same way, inhabitants of distant planets might just detect the presence of Jupiter by watching the sun's rhythmic changes of hue. Jupiter is probably the only one of our sun's planets large enough to be detectable
in this way. Our humble planet is too tiny to make gravitational ripples for aliens to notice.
They might, however, be aware of us through unweaving the rainbow of radio and television signals that we ourselves have been pumping out for the past few decades. The swelling spherical bubble of vibrations, now more than a light-century' across, has enveloped a significant number of stars though an insignificant proportion of those that populate the universe. Carl Sagan, in his novel Contact, has darkly noted that in the vanguard of images announcing earth to the rest of the universe will be Hitler's speech opening the 1936 Olympic Games in Berlin. No reply has so far been picked up, no message of any kind from any other world.
We have never been given any direct reason to suppose that we have company. In very different ways, the possibility that the universe is teeming with life, and the opposite possibility that we are totally alone, are equally exciting. Either way, the urge to know more about the universe seems to me irresistible, and I cannot imagine that anybody of truly poetic sensibility could disagree. I am ironically amused by how much of what we have discovered so far is a direct extrapolation of unweaving the rainbow. And the poetic beauty of what that unweaving has now revealed, from the nature of the stars to the expansion of the universe, could not fail to catch the imagination of Keats; would be bound to send Coleridge into a frenzied reverie; would make Wordsworth's heart leap up as never before.
The great Indian astrophysicist Subrahmanyan Chandrasekhar said, in a lecture in 1975:
This 'shuddering before the beautiful', this incredible fact that a discovery motivated by a search after the beautiful in mathematics should find its exact replica in Nature, persuades me to say that beauty is that to which the human mind responds at its deepest and most profound.
How much more sincere that sounds than Keats's better-known expression of a superficially similar emotion:
'Beauty is truth, truth beauty,'- that is all Ye know on earth, and all ye need to know.
'Ode on a Grecian Urn' (1820)
Keats and Lamb should have raised their glass to poetry, and to mathematics, and to the poetry of mathematics. Wordsworth would have needed no encouragement. He (and Coleridge) had been inspired by the
Scottish poet James Thomson, and might have recalled Thomson's 'To the Memory of Sir Isaac Newton' (1727):
. . . Even Light itself, which every thing displays. Shone undiscovered, till his brighter mind Untwisted all the shining robe of day;
And, from the whitening undistinguished blaze. Collecting every ray into his kind,
To the charmed eye educed the gorgeous train Of parent colours.
First the flaming red
Sprung vivid forth; the tawny orange next; And next delicious yellow; by whose side
Fell the kind beams of all-refreshing green
Then the pure blue, that swells autumnal skies.
Ethereal played; and then, of sadder hue,
Emerged the deepened indigo, as when
The heavy-skirted evening droops with frost;
While the last gleamings of refracted light
Died in the fainting violet away. These, when the clouds distil the rosy shower,
Shine out distinct adown the watery how;
While o 'er our heads the dewy vision bends
Delightful, melting on the fields beneath.
Myriads of mingling dyes from these result,
And myriads still remain - infinite source
Of beauty, ever flushing, ever new.
Did ever poet image aught so fair,
Dreaming in whispering groves by the hoarse brook?
Or prophet, to whose rapture heaven descends?
Even now the setting sun and shifting clouds,
Seen, Greenwich, from thy lovely heights, declare
How just, how beauteous the refractive law.
4
BARCODES ON THE AIR
We shall find the Cube of the Rainbow, Of that, there is no doubt But the Arc of a Lover's conjecture Eludes the finding out.
EMILY DICKINSON (1894)
On the air, in contemporary English, means on the radio. But radio waves have nothing to do with air, they are better regarded as invisible
light waves with long wavelengths. Airwaves can sensibly mean only one thing and that is sound. This chapter is about sound and other slow waves, and how they, too, can be unwoven like a rainbow. Sound weaves travel about a million times more slowly than light (or radio) waves, not much faster than a Boeing 747 and slower than a Concorde. Unlike light and other electromagnetic radiation, which propagates best through a vacuum, sound waves travel only through a material medium such as air or water. They are waves of compression and rarefaction (thickening and thinning) of the medium. In air, this means waves of increasing and then decreasing local barometric pressure. Our ears are tiny barometers capable of tracking highspeed rhythmic changes of pressure. Insect ears work in another way entirely. In order to understand the difference, we need a small digression to examine what pressure really is.
We feel pressure on our skin, when we place our hand over the outlet of a bicycle pump, for example, as a kind of springy push. Actually, pressure is the summed bombardments of thousands of molecules of air, whizzing about in random directions (as opposed to a wind, where the molecules predominantly flow in one particular direction). If you hold your palm up to a high wind you feel the equivalent of pressure - bombardment of molecules. The molecules in a confined space, say, the interior of a well-pumped bicycle tyre, press outwards on the walls of the tyre with a force proportional to the number of molecules in the tyre and to the temperature. At any temperature higher than -275? C (the lowest possible temperature, corresponding to complete motionlessness of molecules), the molecules are in continuous random motion, bouncing off each other like billiard balls. They don't only bounce off each other, they bounce off the inside walls of the tyre - and the walls of the tyre 'feel' it as pressure. As an additional effect, the hotter the air, the faster the molecules rush about (that's what temperature means), so the pressure of a given volume of air goes up when you heat it. By the same token, the temperature of a given quantity of air goes up when you compress it, i. e. raise the pressure by reducing the volume.
Sound waves are waves of oscillating local pressure change. The total pressure in, say, a sealed room is determined by the number of molecules in the room and the temperature, and these numbers don't change in the short term. On average, every cubic centimetre in the room will have the same number of molecules as every other cubic centimetre, and therefore the same pressure. But this doesn't stop there being local variations in pressure. Cubic centimetre A may experience a momentary rise in pressure at the expense of cubic centimetre B, which has temporarily donated some molecules to it. The increased pressure in A will tend to push molecules back to B and redress the balance. On the much larger scale of geography, this is what winds are - flows of air from high-pressure areas to low-pressure areas. On a smaller scale sounds
can be understood in this way, but they are not winds because they oscillate backwards and forwards very fast.
If a tuning fork is struck in the middle of a room, the vibration disturbs the local molecules of air, causing them to bump into neighbouring molecules of air. The tuning fork vibrates back and forth at a particular frequency, causing ripples of disturbance to propagate outwards in all directions as a series of expanding shells. Each wavefront is a zone of increased pressure, with a zone of decreased pressure following in its wake. Then the next wavefront comes, after an interval determined by
the rate at which the tuning fork is vibrating. If you stick a tiny, very fast-acting barometer anywhere in the room, the barometer needle will swing up and down as each wavefront passes over it. The rate at which the barometer needle oscillates is the frequency of the sound. A fast- acting barometer is exactly what a vertebrate ear is. The eardrum moves in and out under the changing pressures that hit it. The eardrum is connected (via three tiny bones, the famous hammer, anvil and stirrup, sequestered in evolution from the bones of the reptilian jaw hinge) to a kind of inverse harp in miniature, called the cochlea. As in a harp, the 'strings' of the cochlea are arranged across a tapering frame. Strings at the small end of the frame vibrate in sympathy with high-pitched sounds, those at the big end vibrate in sympathy with low-pitched sounds. Nerves from along the cochlea are mapped in an orderly way in the brain, so the brain can tell whether a low-pitched or a high-pitched sound is vibrating the eardrum.
Insect ears, by contrast, are not little barometers, they are little weather- vanes. They actually measure the flow of molecules as a wind, albeit a queer kind of wind which travels only a very short distance before reversing its direction. The expanding wavefront which we detect as a change in pressure is also a wave of movement of molecules: movement into a local area as the pressure goes up, then movement back out of that area as the pressure goes down again. Whereas our barometer ears have a membrane stretched over a confined space, insect weather-vane ears have either a hair, or a membrane stretched over a chamber with a hole. In either case, it is literally blown back and forth by the rhythmic, backward and forward movements of the molecules.
Sensing the direction of a sound is therefore second nature for insects. Any fool with a weather-vane can distinguish a north wind from an east wind, and a single insect ear finds it easy to tell a north-south oscillation from an east-west oscillation. Directionality is built into insects' method of detecting sound. Barometers aren't like that. A rise in pressure is just a rise in pressure, and it doesn't matter from which direction the added molecules come. We vertebrates therefore, with our barometer ears, have to calculate the direction of sound by comparing the reports of the two
ears, rather as we calculate colour by comparing the reports of different classes of cones. The brain compares the loudness at the two ears and separately it compares the time of arrival of sounds (especially staccato sounds) at the two ears. Some kinds of sounds lend themselves to such comparisons less readily than others. Cricket song is cunningly pitched and timed so as to be hard for vertebrate ears to locate, but easy for female crickets, with their weather-vane ears, to home in upon. Some cricket chirps even create the illusion, at least to my vertebrate brain, that the (in fact stationary) cricket is leaping about like a jumping squib.
Sound waves form a spectrum of wavelengths, analogous to the rainbow. The sound rainbow is also subject to unweaving, which is why it is possible to make any sense of sounds at all. Just as our sensations of colour are the labels that the brain slaps on to light of different wavelengths, the equivalent internal labels that it uses for sounds are the different pitches. But there is a lot more to sound than simple pitch, and this is where unweaving really comes into its own.
A tuning fork or a glass harmonica (an instrument favoured by Mozart, made from fine glass bowls tuned by the depth of water they contain, sounded by a wetted finger drawn around the rim) emit a crystalline pure sound. Physicists call these sine waves. Sine waves are the simplest kind of waves, sort of theoretical ideal waves. The smooth curves that snake along a rope when you wiggle one end up and down are more or less sine waves, although of much lower frequency than sound waves, of course. Most sounds are not simple sine waves but are more jagged and complicated, as we shall see. For the moment we shall think of a tuning fork or glass harmonica, singing out its smooth, curvaceous waves of pressure change that race away from the source in concentrically expanding spheres. A barometer ear placed at one spot detects a smooth increase in pressure followed by a smooth decrease, rhythmically oscillating with no kinks or wiggles in the curve. With every doubling in frequency (or halving in wavelength, which is the same thing) we hear a jump of one octave. Very low frequencies, the deepest notes of the organ, shudder through our bodies and are hardly heard by our ears at all. Very high frequencies are inaudible to humans (especially older humans) but audible to bats and used by them, in the form of echoes, to find their way about. This is one of the most enthralling stories in all natural history, but I've devoted a whole chapter to it in The Blind Watchmaker so will resist the temptation to expand.
Tuning forks and glass harmonicas aside, pure sine waves are largely a mathematical abstraction. Real sounds are mostly more complicated mixtures, and they richly repay unweaving. Our brains unweave them effortlessly and to astonishing effect. It is only with much labour that our mathematical understanding of what is going on has caught up, clumsily
and incompletely, with what our ears have effortlessly unwoven - and our brains rewoven - from childhood on.
Suppose we sound one tuning fork with an oscillating frequency of 440 cycles per second, or 440 Hertz (Hz). We shall hear a pure tone, the A above middle C. What is the difference between this and a violin playing the same A, a clarinet playing the same A, an oboe, a flute? The answer is that each instrument includes admixtures of waves whose frequencies are various multiples of the fundamental frequency. Any instrument playing the A above middle C will deliver most of its sound energy at the fundamental frequency, 440 Hz, but superimposed will be traces of vibration at 880 Hz, 1520 Hz and so on. These are called harmonics, although the word can be confusing since 'harmonies' are chords of several notes that we hear as distinct. A 'single' trumpet note is actually a mixture of harmonics, the particular mixture being a kind of trumpet 'signature' that distinguishes it from, say, a violin playing the 'same' note (with different, violin signature harmonics). There are additional complications, which I shall ignore, around the onset of sounds, for example the lippy irruption of a trumpet blast or the zing as a violin bow hits the string.
These complications aside, there is a characteristic trumpet (or violin, or whatever it is) quality to the sustained part of a note. It is possible to demonstrate that the apparently single tone of a particular instrument is a rewoven construct of the brain, summing up sine waves. The demonstration works as follows. Having decided which sine waves are involved in, say, trumpet sound, select the appropriate 'tuning fork' pure tones and sound them one at a time. For a brief period you can hear the separate notes, as if they really were a chord of tuning forks. Then, quite eerily, they click into focus with each other, the 'tuning forks' disappear, and you hear only what Keats called the silver, snarling trumpets, sounding the pitch of the fundamental frequency. A different barcode combination of frequencies is needed to make the sound of a clarinet, and "again you can fleetingly distinguish them as separate 'tuning forks' before the brain homes in on the illusion of one 'woody' clarinet note. The violin has its own barcode signature, and so on.
Now, if you watch a tracing of the pressure wave when a violin is playing some note, what you see is a complicated wiggly line repeating itself at the fundamental frequency but with smaller wiggles of higher frequency superimposed. What has happened is that the different sine waves that constitute violin noise have summed up to make the complicated wiggly line. It is possible to program a computer to analyse any complicatedly repeating pattern of wiggles back into its component pure waves, the separate sine waves that you would have to sum up to make the complicated pattern. Presumably, when you listen to an instrument, you
are performing something equivalent to this calculation, the ear first unweaving the component sine waves, then the brain weaving them together again and giving them the appropriate label: 'trumpet, 'oboe' or whatever it is.
But our unconscious feats of unweaving and weaving are greater even than this. Think what is happening when you listen to a whole orchestra. Imagine that, superimposed on a hundred instruments, your neighbour in the concert is whispering learned music criticism in your ear, others are coughing and, lamentably, somebody behind you is rustling a chocolate wrapper. All these sounds, simultaneously, are vibrating your eardrum and they are summed into a single, very complicated wriggling wave of pressure change. We know it is one wave because a full orchestra, and all the noises off, can be rendered into a single wavy groove on a phonograph disc, or a single fluctuating trace of magnetic substance on
a tape. The entire set of vibrations sums up into a single wiggly line on the graph of air pressure against time, as recorded by your eardrum. Mirabile dictu, the brain manages to sort out the rustling from the whispering, the coughing from the door banging, the instruments of the orchestra from each other. Such a feat of unweaving and re-weaving, or analysis and synthesis, is almost beyond belief, but we all do it effortlessly and without thinking. Bats are even more impressive, analysing stuttering volleys of echoes to build up, in their brains,
detailed and fast-changing three-dimensional images of the world
through -which they fly, including the insects which they catch on the wing, and even sorting out their own echoes from those of other bats.
The mathematical technique of decomposing 'wiggling waveforms into sine waves which can then be summed again to make the original wiggly line is called Fourier analysis, after the nineteenth-century French mathematician Joseph Fourier. It works not just for sound waves (indeed, Fourier himself developed the technique for a quite different purpose) but for any process that varies periodically, and it doesn't have to be high- speed waves like sound, or ultra-high-speed waves like light. We can think of Fourier analysis as a mathematical technique which is convenient for unweaving 'rainbows' where the vibration that makes up the spectrum is slow compared with that of light.
To go to a very slow vibration indeed, I recently saw, on a road in the Kruger National Park in South Africa, a wiggly wet line which followed the course of the road and apparently traced out some kind of complicated repeat pattern. My host and expert guide told me that it was a trail of urine from a male elephant in musth. When a bull elephant enters this curious state (perhaps the elephantine equivalent of an Australian on 'walkabout') he dribbles out urine more or less continuously, apparently for scent-marking purposes. The side-to-side
waving of the urine trail on the road was presumably produced by the long penis acting as a pendulum (it would be a sine wave if the penis were a perfect, Newtonian pendulum, which it is not) interacting with the more complicated periodicity of the lumbering four-footed gait of the whole animal. I took photographs with the vague intention of later performing a Fourier analysis. I am sorry to say I have never got around to doing it. But in theory it could be done. A tracing of the photographed urine line could be laid over squared paper and its coordinates digitized for feeding into a computer. The computer could then perform a modern version of Fourier's calculations and extract the component sine waves. There are easier (though not necessarily safer) ways to measure the length of an elephant's penis, but it would have been fun to do, and Baron Fourier himself would surely have been delighted at such an unexpected use of his mathematics. There is no reason why a urine trail might not fossilize, as footprints and worm casts do, in which case we could in principle use Fourier analysis to measure the penis length of an extinct mastodon or woolly mammoth, from the indirect evidence of its urine trail in musth.
An elephant's penis swings at a frequency much slower than sound (although in the same ballpark as sound when you compare it with the ultra high frequencies of light). Nature offers us other periodic waveforms, of much lower frequency still, with wavelengths measured in years or even millions of years. Some of these have been subjected to the equivalent of Fourier analysis, including the cycles of animal populations. Since 1756, the Hudson's Bay Company kept records of the abundance
of pelts brought in by Canadian fur trappers. The distinguished Oxford ecologist Charles Elton (1900-1991), who was employed as a consultant by the company, realized that these records could provide a read-out of fluctuating populations of snow-shoe hares, lynxes and other mammals persecuted by the fur trade. The figures rise and fall in complicated mixtures of rhythms, which have been much analysed. Among the wavelengths that have been pulled out by these analyses is a prominent one of approximately four-year periodicity, and another of around 11 years. One hypothesis that has been suggested to account for the four- year rhythms is a time-lagged interaction between predators and prey (a glut of prey feeds a plague of predators, who then nearly wipe out the prey; this in turn starves the predators, then the consequent drop in predator population allows a new boom in the prey population, and so on). As for the longer rhythm of 11 years, perhaps the most intriguing suggestion connects it with sun spot activity, which is known to vary on an approximately 11-year cycle. How the sun spots affect animal populations is open to discussion. Perhaps they change the earth's weather, which affects abundance of plant food.
Wherever you find regular cycles of very long wavelengths, they are likely to have astronomical origins. They stem from the fact that celestial objects often rotate on their own axis, or follow repetitious orbits around other celestial objects. Twenty-four-hour rhythms of activity pervade almost all the fine details of living bodies on this planet. The ultimate reason is the rotation of the earth about its own axis, but animals of many species, including humans, when isolated from direct contact with day and night, continue to cycle on with a rhythm of approximately 24 hours, showing that they have internalized the rhythm and can free-run it even in the absence of the external pacemaker. The lunar rhythm of 28 days is another prominent component of the mix of waves in the bodily functions of many creatures, especially marine ones. The moon exerts its rhythmic influence via the succession of spring and neap tides. The earth's orbital rhythm of slightly more than 365 days contributes its slower pendulum to the Fourier sum, manifesting itself via breeding seasons, seasons of migration, patterns of moulting and growth of winter coats.
Perhaps the longest wavelength picked up by the unweaving of biological rhythms is a suggested 26-million-year cycle of mass extinctions. Fossil experts reckon that more than 99 per cent of the species that have ever lived have become extinct. Fortunately, the rate of extinction is, over the long term, roughly balanced by the rate at which new species are formed by the splitting of existing ones. But this doesn't mean they stay
constant in the shorter term. Far from it. Extinction rates fluctuate all over the place, and so do the rates at which new species come into existence. There are bad times when species disappear, and good times when they burgeon. Probably the worst of the bad times, the most devastating Armageddon, occurred at the end of the Permian era, about a quarter of a billion years ago. Around 90 per cent of all species became extinct in that terrible time, including on land many mammal-like reptiles. Earth's fauna eventually bounced back on to the denuded stage, but with a very different cast list: on land the dinosaurs stepped into the range of costumes left by dead mammal-like reptiles. The next largest mass extinction - and the most talked about - is the famous Cretaceous extinction of 65 million years ago, in which all the dinosaurs, and many other species with them both on land and in the sea, were wiped out, instantaneously as far as the fossil record can tell. In the Cretaceous event, perhaps 50 per cent of all species went extinct, not as many as in the Permian but nevertheless this was a fearful global tragedy. Once again, our planet's devastated fauna bounced back and here we are, we mammals, descended from a few fortunate relicts of the once rich mammal-like reptile fauna. Now we, together with the birds, fill gaps left by the dead dinosaurs. Until, presumably, the next great extinction.
There have been many episodes of mass extinction, not as bad as the Permian and Cretaceous events, but still noticeable in the chronicles of the rocks. Statistical paleontologists have gathered the numbers of fossil species over the ages and fed them into computers to perform Fourier analysis and extract such rhythms as they can find, as if listening for the flutter of preposterously deep organ notes. The dominant rhythm that has been claimed (albeit controversially) is a periodicity of about 26 million years. What could cause rhythms of extinction with such a formidably long wavelength? Probably only a celestial cycle.
Evidence is accumulating that the Cretaceous catastrophe was caused when a large asteroid or comet, the size of a mountain and travelling at tens of thousands of miles per hour, scored a direct hit on our planet, probably somewhere around what we now call the Yucatan peninsula in the gulf of Mexico. Asteroids hurtle round the sun in a belt which lies inside the orbit of Jupiter. There are plenty of asteroids out there - small ones are hitting us all the time - and a few of them are large enough to cause cataclysmic extinctions if they were to hit us. The comets have larger, eccentric orbits around the sun, mostly well outside what we conventionally think of as the solar system, but occasionally coming inside it, as Halley's comet does every 76 years and the Hale Bopp comet every 4,000 years or so. Perhaps the Permian event was caused by an even larger comet strike than the Cretaceous one. Perhaps the suggested 2 6-million-year cycle of mass extinctions is caused by a rhythmic boost in the rate of comet strikes.
But why should comets become more likely to hit us every million years? Here we launch ourselves into deep speculation. It has been suggested that the sun has a sister star, and the two orbit each other with a periodicity of about 26 million years. This hypothetical binary partner, which has never been seen but which has nevertheless been given the dramatic name Nemesis, passes, once per orbital rotation, through the so-called Oort Cloud, the belt of perhaps a trillion comets which orbits the sun beyond the planets. If there was a Nemesis that passed close to, or through, the Oort Cloud, it is plausible that it would disturb the comets, and this might increase the likelihood of one of them hitting earth. If this all happened - and the chain of reasoning is admittedly tenuous - it could account for the 26-million-year periodicity of mass extinctions that some people think the fossil record shows. It is a pleasing thought that mathematical unweaving of the noisy spectrum of animal extinctions might be the only means we have of detecting an otherwise unknown star.
Starting with the ultra high frequencies of light and other electromagnetic waves, we passed, via the intermediate frequencies of sound and the swinging elephant's penis, to ultra low frequencies and
the alleged 26-million-year wavelength of mass extinctions. Let's return to sound, and in particular that crowning feat of the human brain, the weaving and unweaving of speech sounds. The vocal 'cords' are really a pair of membranes which vibrate together in the breathing passage like a pair of woodwind reeds. Consonants are produced as more or less explosive interruptions of the air flow, caused by closure and contact of the lips, teeth, tongue and back of throat. Vowels vary in the same kind of way as trumpets differ from oboes. We make different vowel sounds rather as a trumpeter moves a mute in and out, to shift the preponderant sine waves summing into the composite sound. Different vowels have different combinations of harmonics above the fundamental frequency. The fundamental frequency itself, of course, is lower for men than for women and children, yet male vowels sound similar to the corresponding female vowels because of the pattern of harmonics.
Each vowel sound has its own characteristic pattern of frequency stripes, like barcodes once again. In the study of speech, the barcode stripes are called 'formants'.
Any one language, or dialect within a language, has a finite list of vowel sounds, and each of those vowel sounds has its own formant barcode. Other languages, and different accents within languages, have different vowel sounds which are made by holding the mouth and tongue in intermediate positions, again as a trumpeter disposes the mute in the
bell of the instrument. Theoretically there is a continuous spectrum of vowel sounds. Any one language employs a useful selection, a discontinuous repertoire picked out from the continuous spectrum of available vowels. Different languages pick out different points along the spectrum. The vowel in the French tu and the German uber, which doesn't occur in (my version of) English, is approximately intermediate between oo and ee. It doesn't too much matter which landmark points along the spectrum of available vowels a language picks on, so long as they are spaced far enough apart to avoid ambiguity within that language.
The story for consonants is more complicated, but there is a similar range of consonant barcodes, with actual languages employing a limited subset from those available. Some languages employ sounds which are far off the spectrum of the majority of languages, for example the clicks of some southern African tongues. As with vowels, different languages parcel up the available repertoire differently. Several of the languages of the Indian subcontinent have a dental sound which is intermediate between the English 'd' and 't'. The French hard 'c' as in comme is intermediate between the English hard 'c' and hard 'g' (and the 'o' is intermediate between the English vowels in cod and cud). The tongue, lips and voice can be modulated to produce an almost infinite variety of consonants and vowels. When the barcodes are patterned in time to form
phonemes, syllables, words and sentences, the range of ideas that can be communicated is unlimited.
Stranger yet, the things that can be communicated include images, ideas, feelings, love and exultation - the kind of thing that Keats does so sublimely.
My heart aches, and a drowsy numbness pains My sense, as though of hemlock I had drunk, Or emptied some dull opiate to the drains One minute past, and Lethe-wards had sunk: Tis not through envy of thy happy lot, But being too happy in thy happiness - That thou, light- winged Dryad of the trees, In some melodious plot Of beechen green, and shadows numberless, Singest of summer in full-throated ease.
'Ode to a Nightingale' (1820)
Read the words aloud and the images tumble into your brain, as if you really were drugged by a nightingale's song in a leafy summer beechwood. At one level it is all done by a pattern of air pressure waves, a pattern whose richness is first unwoven into sine waves in the ear and then rewoven together in the brain to reconstruct images and emotions. Stranger yet, the pattern can be broken down mathematically into a stream of numbers, and it retains its power to transport and haunt the imagination. When a laser disc (CD) is made, say, of the Saint Matthew Passion, the rising and falling pressure wave, with all its wiggles and kinks, is sampled at frequent intervals and translated into digital data.