CHAPTER 5

‘All things known have number’

Sixth Century B.C.

The Pythagorean discovery that ‘all things known have number – for without this, nothing could be thought of or known’ – was made in music. It is well established, as so few things are about Pythagoras, that the first natural law ever formulated mathematically was the relationship between musical pitch and the length of a vibrating harp string, and that it was formulated by the earliest Pythagoreans. Ancient scholars such as Plato’s pupil Xenocrates thought that Pythagoras himself, not his followers or associates, made the discovery.

Musicians had been tuning stringed instruments for centuries by the time of Pythagoras. Nearly everyone was aware that sometimes a lyre or harp made pleasing sounds, and sometimes it did not. Those with skill knew how to manufacture and tune an instrument so that the result would be pleasing. As with many other discoveries, everyday use and familiarity long preceded any deeper understanding.

What did ‘pleasing’ mean? When the ancient Greeks thought of ‘harmony’, were they thinking of it in the way later musicians and music lovers would? Lyres, as far as anyone is able to know at this distance in time, were not strummed like a modern guitar or bowed like a violin. Whether notes were sung together at the same time is more difficult to say, but music historians think not. It was the combinations of intervals in melodies and scales – how notes sounded when they followed one another – that was either pleasing or unpleasant. However, anyone who has played an instrument on which strings are plucked or struck knows that unless a string is stopped to silence it, it keeps sounding. Though lyre strings may not have been strummed together in a chord, more than one pitch and often several pitches were heard at the same time, the more so if there was an echo. Even when notes are played in succession and ‘stopped’, human ears and brains have a pitch memory that causes them to recognise harmony or dissonance. In truth, the ancient Greeks, including Pythagoras, heard harmony both ways, between pitches sounding at the same time and between pitches sounding in succession.

The instrument Pythagoras played was probably the seven-stringed lyre. He tuned it with four of the seven strings at fixed intervals. There were no options about what these intervals would be. The lowest- and highest-sounding of the fixed-interval strings were tuned to sound an octave apart. The middle string on the lyre (the fourth of the seven strings) was tuned to sound a fourth above the lowest string, and the one next higher was tuned to sound a fifth above the lowest string.[1] The intervals of the octave, fourth, and fifth were considered concordant, or harmonious. A Greek musician could adjust the other three strings on the seven-stringed lyre (the second, third, and sixth string), depending on the type of scale desired.

A seven-stringed lyre

Pressing a string exactly halfway between the two ends produces a tone one octave higher than the open, unpressed string plays. The ratio of those string lengths is 2 to 1, and they always produce an octave. But the octave is not something a musician creates by pressing the string. Plucking an open string without pressing it at all causes it to vibrate as a whole, sounding the ‘ground note’, but various parts of the string are also vibrating independently to produce ‘overtones’. Even without the string being pressed at the halfway point to play an octave, the octave is present in the sound coming from the open string. Pressing the string releases tones at the octave, fifth, fourth, and so on – depending on where you press it – that were always there in the ground note but more difficult to hear.[2]

Tradition credits Pythagoras with inventing the kanon, an instrument with one string, and using it to experiment with sound. He would have found that the notes that sounded harmonious with the ground note were produced by dividing the string into equal parts. Dividing it into two equal parts produced a note an octave higher than the open string. Pressed so as to divide it into three equal parts, the string played a note a fifth above that octave; in four equal parts, it played a note a fourth above that. The series goes on to a major third, then a minor third, then smaller and smaller intervals, but there is no indication the Pythagoreans took the process any further than the interval of the fourth.[3]

Looking beyond the task of getting good, practical results from a musical instrument to ask more penetrating questions about what was going on, and whether it could have wider implications, required an unusual turn of mind. Though with hindsight a shift of focus from useful knowledge to recognising deeper principles can look simple, it is not a trivial change. A lyre sounded pleasant used one way and not another way . . . but why? Often, in writings about the Pythagoreans, a clause added to that question has them asking whether there was any meaningful pattern? . . . any orderly structure? but they were not necessarily looking for pattern or order yet, for no precedent would have led them to expect it. Nevertheless, they were about to discover it.

When Pythagoras and his associates saw that certain ratios of string lengths always produced the octave, fifth, and fourth, it dawned on them that there was a hidden pattern behind the beauty they heard in music – a pattern that they were able to understand, but that they had not created or invented and could not change. Surely this pattern must not be an isolated instance. Similar mathematical and geometrical regularities must lie concealed behind all the everyday confusion and complexity of nature. There was order to the universe, and this order was made of numbers. This was the great Pythagorean insight, and it was different from all previous conceptions of nature and the universe. Though the Pythagoreans hardly knew what to do with the treasure they had found – and modern mathematicians and scientists are still learning – it has guided human thinking ever since. Pythagoras and his followers had also discovered that there apparently was a powerful link between human sense perceptions and the numbers that pervaded and governed everything. Nature followed a fundamental, rational, beautiful logic, and human beings were tuned in to it, not only on an intellectual level (they could discover and understand it) but also on the level of the senses (they could hear it in music).

There are other mathematical relationships hidden beneath the experience of music that neither Pythagoras nor others of his era had any way of discovering. The ratios he found represent the rate at which a string vibrates, but there was no way he could have studied the vibrations. However, after the initial discovery using a kanon or a lyre, Pythagoras and/or his early associates may well have begun listening for octaves, fourths, and fifths in other sounds and attempted to discover what could, and what could not, produce the intervals. Perhaps it is the memory of some of their experiments that lies behind several puzzling early stories in which Pythagoras made the discovery of the relationship in ways that he could not possibly, in fact, have made it.

According to one tale Pythagoras was passing a blacksmith’s shop and noticed that the intervals between the pitches the hammers made as they struck were a fourth, a fifth, and an octave. That part of the story is possible, but the next part is not: The only differences between the hammers were their weights, and Pythagoras found that those weights were related in the ratios 2:1, 3:2, and 4:3, presupposing that the vibration and sound of hammers are directly proportional to their weight, which is not the case. Pythagoras then took weights equalling those of the hammers and hung them from strings of equal length. He plucked the taut strings and heard the same intervals – another supposed discovery based on false premises, for the account incorrectly assumes that the frequency of vibration of a string is proportional to the number of units of weight hanging from it. However, it is easy to imagine Pythagoras, or his followers, or both, performing such experiments and considering, with more understanding and skill than those who later ignorantly repeated the tales, what could be learned from the successes and failures. The manner in which these stories came down in history as the way Pythagoras made the discovery could be an example of how knowledge is sometimes preserved while the manner of its discovery, and true understanding of it, are lost. Such a loss would be explained if, as some have supposed, the more sophisticated knowledge of Pythagoras was largely forgotten with the breakup of Pythagorean communities after his death.

Aristoxenus told a story having to do with another harmonic ratio experiment that involved Hippasus of Metapontum, and this experiment has particular significance because it is one of the reasons scholars are willing to attribute the discovery of the musical ratios to Pythagoras and his immediate associates. Hippasus, himself a contemporary of Pythagoras, made four bronze disks, all equal in diameter but of different thicknesses. The thickness of one ‘was 4/3 that of the second, 3/2 that of the third, and 2/1 that of the fourth’. Hippasus suspended the disks to swing freely. Then he struck them, and the disks produced consonant intervals. This experiment is correct in terms of the physical principles involved, for the vibration frequency of a free-swinging disk is directly proportional to its thickness. Whoever designed and executed this experiment understood the basic harmonic ratios, or learned to understand them from doing the experiment, and the way the story was told suggests that the musical ratios were already known and Hippasus made the four disks to demonstrate them. According to Aristoxenus, the musician Glaucus of Rhegium, one of Croton’s neighbouring cities, played on the disks of Hippasus, and the experiment became a musical instrument.

To Walter Burkert, a meticulous twentieth-century scholar, the blacksmith tales make ‘a certain kind of sense’. In ancient lore, the Idaean Dactyls were wizards and the inventors of music and blacksmithing. According to Porphyry, Pythagoras underwent the initiation set by the priests of Morgos, one of the Idaean Dactyls. A Pythagorean aphorism stated that the sound of bronze when struck was the voice of a daimon – another connection between blacksmithing and music or magical sound. ‘The claim that Pythagoras discovered the basic law of acoustics in a smithy’, writes Burkert, may have been ‘a rationalisation – physically false – of the tradition that Pythagoras knew the secret of magical music which had been discovered by the mythical blacksmiths.’1

When the Pythagoreans, with their discovery of the mathematical ratios underlying musical harmony, caught a glimpse of the deep, mysterious patterned structure of nature, the conviction became overwhelming that in numbers lay power, even possibly the power that had created the universe. Numbers were the key to vast knowledge – the sort of knowledge that would raise one’s soul to a higher level of immortality, where it would rejoin the divine.

However revolutionary, one of the most significant insights in the history of knowledge had to be worked out, at the start, in the context of an ancient community, ancient superstitions, ancient religious perceptions, without any of the tools or assumptions of later mathematics, geometry, or science, without any scientific precedent or a ‘scientific method’. How would one begin? The Pythagoreans turned to the world itself and followed up on the suspicion that there was something special about the numbers 1, 2, 3, and 4 that appeared in the musical ratios. Those numbers were popping up in another line of investigation they were pursuing.

They had at their fingertips a simple but productive way of working with numbers. Maybe at first it was a game, setting out pebbles in pleasing arrangements. Most of the information about ‘pebble figures’ and the connections with the cosmos and music that the Pythagoreans found in them comes from Aristotle. He knew about Pythagorean ideas of ‘triangular numbers’, the ‘perfect’ number 10, and the tetractus.

The dots that still appear on dice and dominoes are a vestige of an ancient way of representing natural numbers, the positive integers with which everyone normally counts. Dots and strokes stood for numbers in Linear B, the script the Mycenaeans used for the economic management of their palaces a thousand years before Pythagoras, and also in cuneiform, an even older script. Pebble figures were a related way of visualising arithmetic and numbers, but they seem to have been unique to the Pythagoreans.

By tradition, Pythagoras himself first recognised links between the pebble arrangements and the numbers he and his colleagues had discovered in the ratios of musical harmony. Two of the most basic arrangements worked as follows: Begin with one pebble, then place three, then five, then seven, etc. – all odd numbers – in carpenter’s angles or ‘gnomons’, to form a square arrangement.[4]

Or, begin with two pebbles and then set out four, then six, then eight, etc. – all even numbers – and the result is a rectangle.

That is easier to understand visually than verbally, one reason to use pebbles.

Pythagoras and his associates were alert for hidden connections. The pebble figures of the square and rectangle dictated a division of the world of numbers into two categories, odd and even, and this struck them as significant. It was a link with what they were thinking of as the two basic principles of the universe, ‘limiting’ and ‘limitless’. ‘Odd’ they associated with ‘limiting’; ‘even’ with ‘limitless’.

Another way of manipulating the pebbles was to cut a triangle from either the square or the rectangular figure.

In the line of pebbles that then forms the diagonal or hypotenuse of the triangle, the pebbles are not the same distances from one another as they are in the other two sides, nor are they touching one another. Having all the pebbles in all three sides of a triangle at equal distances from their immediate neighbours, or all touching one another, requires a new figure: Set down one pebble, then two, then three, then four, with all the pebbles touching their neighbours. The result is a triangle in which all three sides have the same length, an equilateral triangle.

Notice that the four numbers in this triangle are the same as the numbers in the basic musical ratios, 1, 2, 3, and 4, and the ratios themselves are all here: Beginning at a corner, 2:1 (second line as compared with first), then 3:2, then 4:3. The numbers in these ratios add up to 10. The Pythagoreans decided 10 was the perfect number. They also concluded that there was something extraordinary about this equilateral triangle, which they called the tetractus, meaning ‘fourness’. The tetractus was, in a nutshell, the musical-numerical order of the cosmos, so significant that when a Pythagorean took an oath, he or she swore ‘by him who gave to our soul the tetractus’.

Most scholars think it was after Pythagoras’ death that the Pythagoreans found they could construct a tetrahedron (or pyramid) – a four-sided solid – out of four equilateral triangles, and they probably knew this by the time Philolaus wrote the first Pythagorean book in the second half of the fifth century.[5] The word tetractus, however, was in use during Pythagoras’ lifetime. It hints that there was more ‘fourness’ to the idea than the fact that 4 was the largest number in the ratios. The tetrahedron or pyramid is a solid in which each face is a tetractus, but which also uses the number 4 in other manners – 4 faces, 4 points.

When Aristotle, in the fourth century B.C., was researching the Pythagoreans, he found a list of connections they made between numbers and abstract concepts. He apparently could not discover what they connected with the numbers 6 and 8.

1. Mind
2. Opinion
3. The number of the whole
4. Justice
5. Marriage
6. ?
7. Right time, due season, or opportunity
8. ?
9. Justice
10. Perfect

It is not difficult to understand how Mind might be 1 and Opinion 2. Justice appears twice because of an association with squareness. The Greeks did not think of 1 as a number. ‘Number’ meant plurality, more than 1. So, for them, the smallest number that is the square of any whole number was 4.[6] The first number that is the square of an odd number is 9, and that, too, they associated with justice. The idea that ‘square’ means an evened score – with all need for retaliation at an end – still shows up in the colloquial phrase ‘That makes us square’. Marriage (5) was the sum of the first odd and even numbers (2 and 3). The link between 7 and ‘right time’ or ‘due season’ reflected wider Greek thought. Life happened in multiples of 7. A child could be born after 7 months in the womb, cut teeth 7 months later, reach puberty at 14, and (if a boy) grow a beard at 21.

The Pythagoreans followed one line of thought that seems particularly odd today, accustomed as most of us are to thinking of squares and cubes of numbers but not of other geometric shapes possibly connected with them in a similar manner. The ‘square’ of 4 was 16, but the ‘triangle’ of 4 was 10, the perfect number. Both ideas were equally picturable with pebbles.

        

Stacking the pebbles so as to discover that the ‘cube’ of 4 was 64, you might just as easily pile them up another way so that the ‘pyramid’ of 4 was 20. Montessori teaching exploits the delight of playing games like this with little objects like pebbles – in the case of Montessori, beads.

Having come to the conclusion not only that numbers, but the specific numbers 1, 2, 3, and 4 and the ratios between them were the primordial organising principle of the universe, Pythagorean thinking moved in other directions, some of which seem strange and primitive, but it is not surprising that they overestimated the simplicity of the rationality they had glimpsed and were too expectant of immediate applications and results. They were not unlike the earliest followers of Jesus, coming away from what was for them a transforming experience and trying to apply it to the everyday world, thinking all would be resolved soon. The Pythagoreans had discovered a new road to ‘truth’. Great thinkers thought about truth and proposed answers. Only a shaman – and many regarded Pythagoras as what we today would call a shaman – was sure he had the answer. In fact, Pythagoras and his followers did, but they travelled their new road weighted down with ancient baggage. Still in the age of oracles, divination, and mystic utterances, with its preconceptions about the universe and nature, their naive conception of the world carried over into a naive conception of the power of numbers.

The halcyon days in Croton lasted thirty years. Iamblichus’ biography included long lists of names, which he probably got from Aristoxenus, of Pythagoras’ first followers, who sat at his feet, heard his teaching, argued points and worked out problems with him, played with the pebbles, and experimented with the kanon and with hanging disks. Was the young physician ‘Alcmaeon’ really one of them? Was there actually a ‘Brontinus’ who was husband and/or father of Theano? Were ‘Leo’ and ‘Bathyllus’ real people? And what of the ‘Pythagorean women’, about whom nothing is known but their names on these lists? Frustratingly, there is no specific surviving information about how the new coinage affected the economy or, except the story of Milo’s defeat of Sibaris, about Pythagorean leadership in Croton and the surrounding territory, what offices the Pythagoreans held, or exactly in what capacity they wielded their power – only that they did wield it and that the results were by most accounts beneficial to the region. What is clear is that in about 500 B.C., three decades after Pythagoras arrived in Croton, hostility among the populace and perhaps a coup within the ranks of his followers brought it all to an end. The information is confused and contradictory, with common themes being others’ suspicion that Pythagoras and his followers were either becoming too powerful politically or aspiring to too much power – and, oddly, an unusual respect for beans.

According to Diogenes Laertius, Pythagoras was visiting with friends in Milo’s home when someone deliberately set fire to the house. The arsonists were either Crotonians who feared that Pythagoras might ‘aspire to the tyranny’ or envious, disgruntled people who thought they should have been included in this gathering but had not been deemed ‘worthy of admission’. Pythagoras escaped but was captured and killed when he avoided crossing a bean field and took a longer way around. He must have decided, Diogenes Laertius said, that death was preferable to trampling on beans or speaking with his pursuers. About forty of his companions died as well.

Diogenes Laertius was interested in conflicting accounts, so he also reported a story he got from Hermippus, portraying Pythagoras and his ‘usual companions’ in a militaristic light. They had joined the Agrigentine army to fight the army of Syracuse. The Syracusans put them to flight and captured and killed Pythagoras as he was making a detour around a bean field. Being less squeamish about trampling on beans did not help his companions. About thirty-five were caught and burned at the stake in Tarentum, accused of trying to set up a rival government in opposition to the prevailing magistrates.

Diogenes Laertius showed he had a rather macabre sense of humour by casting part of this story into verse in another of his ‘jesting epigrams’.

Alas! alas! why did Pythagoras hold

Beans in such wondrous honour? Why, besides,

Did he thus die among his choice companions?

Here was a field of beans; and so the sage,

Died in the common road of Agrigentum,

Rather than trample down his favourite beans.

Two other endings to the story came through Diogenes Laertius from the trustworthy Dicaearchus and Heracleides Ponticus; in these, Pythagoras escaped his pursuers but died soon thereafter in Metapontum of self-imposed starvation.[7]

Porphyry gave a more detailed description of what supposedly happened, based on Aristoxenus, naming names and filling in the gaps in the other stories, and Iamblichus had some of the same specifics. According to this fuller account, the huge success of Pythagoras and his associates, and particularly their role in the administration and reform of the cities, aroused envy, most notably and ominously from one Cylon. He was a wealthy community leader of impeccable breeding, but also of a ‘severe, violent and tyrannical disposition’, and he controlled a large group of loyal supporters. He had a high opinion of himself, ‘esteemed himself worthy of whatever was best’, and assumed he would be welcomed to the Pythagorean fellowship. When Cylon approached Pythagoras, ‘extolled himself’, and tried to converse, Pythagoras peremptorily ‘sent him about his business’. Pythagoras, Porphyry pointed out, ‘was accustomed to read in the nature and manners of human bodies the disposition of the man’. Cylon did not take the rebuff gracefully. He assembled his cronies and instigated a conspiracy against Pythagoras and his followers. According to Iamblichus it took some time for Cylon to bring his plans to fruition because of the Pythagoreans’ power and the trust placed in them by the citizens of the various cities. Accounts more sympathetic to Cylon had him as the leader of a group that opposed the oppressive ultra-conservatism of the Pythagoreans.

Iamblichus suggested that Hippasus – who invented the demonstration of the musical ratios using the disks of different thicknesses – may have played a subversive role. Before Cylon’s attack, Hippasus was, according to Iamblichus, one of a faction, among the insiders, who disagreed with Pythagoras and the more orthodox members of the school. He urged Pythagoreans who were playing prominent roles in governing the cities to adopt more democratic policies. He may have attempted to stir up popular feeling against Pythagoras’ leadership, playing into Cylon’s hands.

W. K. C. Guthrie described, with good understanding of human nature, the complicated political situation that probably contributed to the death or exile of Pythagoras:

This combination of forces seems to have been due on the one hand to popular discontent with the concentration of power in the hands of a few, coupled with the ordinary man’s dislike of what he considers mumbo-jumbo, and on the other to the native aristocracy’s suspicion of the Pythagorean coteries, whose assumption of superiority and esoteric knowledge must at times have been hard to bear.2

Porphyry took the longest and most dramatic version of the story from Dicaearchus. Pythagoras was with his friends in Milo’s house when Cylon’s men set it afire. Pythagoras’ most devoted followers threw themselves into the flames to make a bridge with their bodies for the elderly sage to cross and escape. He and a remnant of survivors then tried to reach the city. Fleeing along the road, the others were gradually picked off by their pursuers, but Pythagoras, protected by them as much as possible during their flight, managed eventually to make his way to the harbour of Caulonia and from there to Locri. The Locrians refused him sanctuary. Perhaps they sensed that the days of Pythagorean preeminence had come to an end and feared retribution from Cylon if they sheltered him. Or perhaps they feared Pythagoras himself, for, as the story goes, their message to him as they turned him away was that they admired his wisdom but liked their present condition and way of life and did not wish to change.[8] In any case, the story has it that they sent some old men to intercept him before he could reach their gates and tell him that the Locrians would give him food and supplies but he must ‘go to some other place’. Pythagoras sailed to Tarentum, then back to Croton. The Crotonians also sent him away. Everywhere, as Porphyry reported Dicaearchus’ words, ‘mobs arose against him, of which even now the inhabitants make mention, calling them the Pythagorean riots.’ In Dicaearchus’ account, Pythagoras eventually found asylum in the temple of the Muses in Metapontum, where he starved himself to death, grieving for the friends who had perished trying to save him.

The people of Metapontum prefer another ending. According to the tradition in that city, after Pythagoras arrived as a refugee from Croton, he settled down and established a school. After his death, his house and school were incorporated into a temple of Hera. Fifteen columns and sections of pavement from that temple still remain today in Metapontum, called the Palatine Tables, because knights (paladins) in the Middle Ages assembled there before setting off on the Crusades. In the first century B.C., when Cicero visited Metapontum, people could still identify the house where they believed Pythagoras had lived. Cicero wrote about how moved he was when he visited it.

Porphyry lamented that most of what Pythagoras taught died with him and his closest followers. ‘With them also died their knowledge,’ he wrote, ‘which till then they had kept secret except for a few obscure things which were commonly repeated by those who did not understand them.’ Iamblichus wrote that the cities hardly mourned Pythagoras at all or took much notice of what had happened, though in truth they had lost ‘those men most qualified to govern’. ‘Then science died in the breasts of its possessors, having by them been preserved as something mystic and incommunicable.’

[1]Think of having the lowest string tuned to C on the piano, the fourth string tuned to F above the C, the next to G a whole step above that, and then the top string tuned at C an octave above the lower C.

[2]Musical instruments and human voices, because of intricate differences in the way their structures resonate and amplify sound, emphasise or ‘bring out’ certain overtones more than others, and that is what causes the great variety of sounds they make. That is how a trumpet ends up sounding like a trumpet while a clarinet sounds like a clarinet.

[3]On the piano, equivalent notes might be, for example, middle C (ground note); c (octave above that ground note); g (fifth above that octave); c (fourth above that g). For a demonstration using the piano: Press down gently on the c above middle C without allowing it to sound (removing the damper from the strings). Strike middle C (the ground note) and you will clearly hear the octave. Press carefully on the g above that octave. Strike middle C and you will hear that fifth above the octave. A piano is not tuned to the Pythagorean system, but it is close enough for you to hear these overtones.

[4]A gnomon is an instrument for measuring right angles, like the device used by carpenters called a ‘carpenter’s square’.

[5]Not all pyramids have only four sides. The Great Pyramid that Pythagoras may have seen in Egypt is not a pyramid of this sort. It has five sides: a square base and four triangular sides.

[6]In some later ancient mathematics, whose roots can be traced to the ‘Pythagorean’ tradition and which by some scholars’ interpretation existed separately and in parallel with the Euclidean tradition, the number 2 also had no status as a ‘number’. It was not considered even or odd or prime. Like ‘1’, it was not a number at all, but the ‘first principle of number’.

[7]Heracleides Ponticus is not to be confused with the earlier Heraclitus who so severely criticised Pythagoras. Heracleides Ponticus lived in the fourth century B.C. and was a pupil of Plato.

[8]Part of their ‘present condition’ was an economy that was more primitive than Croton’s. They used no coinage, and would not until more than a century later. See W. K. C. Guthrie (2003), p. 178 n.