Pythagorus (10 page)

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.
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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.
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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.
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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.'
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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.
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