6
In Act V, Scene I of Shakespeare’s The Merchant of Venice, the dashing lover Lorenzo sits his darling, Jessica, down on the ground. The night is silent, but Lorenzo says he hears music:
How sweet the moonlight sleeps upon this bank!
Here will we sit and let the sounds of music
Creep in our ears: soft stillness and the night
Become the touches of sweet harmony.
Sit, Jessica. Look how the floor of heaven
Is thick inlaid with patines of bright gold:
There’s not the smallest orb which thou behold’st
But in his motion like an angel sings,
Still quiring to the young-eyed cherubins;
Such harmony is in immortal souls;
But whilst this muddy vesture of decay
Doth grossly close it in, we cannot hear it.
There’s no audible music playing – only ‘soft stillness.’ But, says Lorenzo, an attentive observer can pick up on a silent kind of harmony that emanates from even the ‘smallest orb,’ from every planet in the sky as it moves along its orbit. Human beings, because they are made of flesh and blood (a ‘muddy vesture of decay’), can’t use their sluggish ears to hear this divine sort of music. It’s beyond our form of hearing – we have to use our minds to intuit it.
In just twelve lines, Shakespeare has given a pretty decent summary of a central idea behind Greek musical philosophy. This idea is sometimes called the ‘music of the spheres,’ and Shakespeare could take it for granted because it survived long after ancient Athens was no more than a memory. The basic concept is that planetary movement and the structure of the universe are ‘tuned’ like a musical composition: the distances between planets are proportional to the distances between notes on a scale, and their movements follow the same patterns as the movements of air which generate melodic sound. In fact, for some ancient theorists, this inaudible cosmic motion was the more fundamental, the more truly musical kind of music. Our audible songs, generated by human hands and mouths, were just a pale fleshly echo in the material world of a deeper, more pure cosmic music.
Shakespeare can put all this in the mouth of a suave young lover because, in the sixteenth-century Renaissance culture of Elizabethan England, Classical ideas were gaining a great deal of traction. The notion that heaven is a harmony was so compelling that even many respectable physicists set out to prove that it was literally true. This may sound silly to us now, and so our question in this chapter will be: what on earth gave the Greeks this idea? Why did it seem plausible to people like Plato and the Pythagoreans that human music was a reflection of some grand universal design? And what was it about that belief that stuck?
Numbers in the sky: Pythagoreanism and mathematical cosmology
We should start with Pythagoras. Or at least, we should start with the Pythagoreans. Pythagoras himself, like Socrates, never wrote anything down in his efforts to contemplate the secrets of the cosmos. But based on the reports of his followers and the ideas which came to be associated with him, it seems fairly plausible that he believed numbers are, in some capacity, the building blocks of the universe.
At the very least, many of Pythagoras’ followers spent a great deal of time trying to work out the basic mathematics that undergird the shape and structure of the material world. Philolaus, who was a famous member of the Pythagorean school in the fifth century BC, claimed that all things in the universe are known through numerical understanding.1 For Pythagorean theorists like Philolaus, numbers weren’t just basic tools for counting things in the world like oranges and horses. They were concepts that made the whole universe possible.
This is less outlandish than it may initially sound. Even today, when physicists study how the universe works, they almost always do so by trying to calculate the movement of planets and molecules in a way that makes mathematical sense. Newton’s second law of motion (force equals mass times acceleration, or f = ma), Einstein’s theory of relativity (energy equals mass times the speed of light squared, or E = mc2), and the various elaborate theories that make up the ever-evolving science of quantum physics, are all attempts to describe the world using numbers. That’s because what we usually think of as numbers – the standard figures like 1 and 2 – are really stand-ins for much deeper patterns and concepts which describe fundamental truths about the universe. 1 doesn’t equal 2, because one thing can only be itself, not something else: if these sorts of things sound basic in the extreme, that’s because they are. They’re the essential realities which mathematics describes and which make the world what it is.
There is a famous story – apocryphal, but still telling – that Pythagoras recognized the mathematical relationship between harmonious notes when he heard anvils being pounded in a blacksmith’s shop. By the end of the fifth century, Pythagoreanism had become associated with the idea that musical notes and the relationship between them were just audible versions of those foundational mathematical truths that also govern the rest of the physical universe. As a result, some Pythagoreans – such as Hippasus (early fifth century), Philolaus (c. 470–c. 385), and Archytas of Tarentum (active in the first half of the fourth century) – became interested in discovering the exact ratios which produced various harmonies in the musical scale. Some of these philosophers – most notably Archytas – demonstrated an interest in the actual practices of real musicians. But for the most part, Pythagoreans weren’t running the numbers because they wanted to get the tuning on their kitharai just right – they weren’t known, in other words, for their stellar concert performances. What they were interested in was music as a sensory way of accessing deeper truths that undergirded the whole universe.
Compare this with Aristoxenus, who, though he began his career as a Pythagorean, eventually rejected the mathematical vision of music in favour of a more intuitive, empirical approach. Aristoxenus’ treatizes on music theory are hugely important for understanding the Greek scale today because they are devoted to music as it is – the notes and harmonies that actual musicians really used. Or at least, the ones they should have used according to Aristoxenus – he still had his own very definite ideas, after all, about the right way to tune a lyre. But those ideas were definitively based on what Aristoxenus thought worked in practice – on the experience of a listener appreciating or disliking a particular set of intervals. If the ideal interval sounded good, but didn’t correspond exactly to a neat mathematical ratio, so be it: Aristoxenus was not one to let maths get in the way of beauty. A somewhat enigmatic group of earlier theorists whom Aristoxenus calls ‘harmonicists’ (harmonikoi) were also known for their empirical interest in musical harmony and rhythm as actually heard ‘in the wild’ at live performances.2
Not so the stricter Pythagoreans. For them, beauty was maths: they believed that everyday melodies should conform to the purest abstract mathematics that theorists can use to describe it. That’s because the mathematics, not the music, came first and foremost as a basic feature of reality. This could be put another way by saying that what we think of as music – the audible sounds of notes and harmonies – was only one small part, and not even a very important part, of the broad science of mousikē that interested the mathematical mystics of the Pythagorean tradition. That science included tunes and songs, but it also included numbers and equations, which in the last analysis were more central as a way of understanding the world.
This Pythagorean version of mathematical music made it easy to draw links between songs and the stars. Because if what we hear when we hear music is a pale human reflection of the most basic mathematical relationships, then intervals like the fifth sound beautiful to us precisely because they are arranged according to principles that govern the rest of the material world as well. Planets in their orbits, according to this theory, move in ways that are governed by the same basic principles as govern the rules of harmony. It would be only natural, then, that really well-tuned music would conform to the structure of the heavens, because both are patterned after one set of pure mathematical truths.
A short history of the entire universe: Pythagoreanism according to Plato
This vision of the universe actually made its way into the work of Plato, who wasn’t a Pythagorean but who apparently took a profound interest in the mathematical and musical researches of Pythagoras’ followers (especially Archytas, who once sent a ship to rescue Plato when he was being detained for political reasons by the king of Syracuse). One of Plato’s dialogues, the Timaeus, contains an origin-myth that tells the story of the whole world in Pythagorean terms. It’s worth summarizing that myth briefly, because it can afford some insight into how the Pythagorean tradition could be used to build one big coherent theory of music, maths, and cosmological physics.
The Timaeus is presented as a sequel to the Republic, Plato’s massive dialogue in which Socrates and his friends try to draw up a blueprint for the perfect political organization. Supposedly taking place on the day after that massive conversation unfolds, the Timaeus features a speech by the title character on the nature and origins of the universe. To hear Timaeus tell it, the whole cosmos was created by a single god. That god – he was called the ‘demiurge,’ which is a Greek word for ‘craftsman’ – made the universe as a single, perfect, living organism. From the model of that vast, living cosmos, all other living beings were created in imitation.
This massive, animate universe was constructed perfectly by the demiurge in the shape of a sphere, with planetary orbits rotating within one another at determined intervals. The distances between those intervals were picked by the demiurge very precisely, according to specific ratios. And so according to Timaeus, the whole universe is fashioned with reference to the same mathematical proportions as a musical scale (the Dorian scale, to be precise). In fact, the correspondence between musical and planetary intervals is no accident. The notes of a well-tuned scale sound good to us because we instinctively recognize within them the same mathematical ratios that dictate the perfect structure of the universe.
This elaborate myth is a way of giving narrative shape to the Pythagorean traditions of musical cosmology that were developing in Plato’s day. Timaeus’ story is an artistic way of representing two major ideas which were current by then: that the whole cosmos is structured according to mathematical principles, and that music could be analysed in terms of the ratios underlying various harmonies. If maths governs the universe, and music is just audible maths, then music must be governed by the same deep numerical truths that guided the construction of the cosmos itself.
It turns out that this pair of ideas helps support the political and ethical theories of music from Chapter 5. Because Timaeus goes on to explain that all human souls are little carbon copies of the universal soul in miniature – our own inner lives are like pocket-sized versions of the heavens. And the movements of our souls – those impulses towards and away from things which are the rudiments of our desires and emotions – are finely calibrated according to the same set of ratios that order the solar system, the universe, and the notes of a well-tuned scale.3
That’s if all goes well. If our souls and our desires are perfectly ordered, Timaeus says, then we’ll act reasonably and want all the right sorts of things. Everybody will live in harmony just like instruments in a perfectly unified orchestra, playing along with the vast and precise music of truth that governs the universe.
This, of course, is not what happens. The world in which we actually live is fraught with wars, cheating, petty theft, and all manner of human misbehaviour. If the universe is music, it sometimes seems horribly out of tune. This, according to Timaeus, is a problem that arises from the fact that our souls have to operate within physical bodies. The process of embodiment was a disorienting affair, and the result of it was that we don’t always act the way we should. Sometimes our desires get away from us and we hurt one another out of greed, lust, or pure spite. These kinds of disorderly desires and feelings come to us because our soul’s divine motion has been thrown out of whack by the fleshly vicissitudes of our earthly body.
That’s where music comes in. Music helps us to realign the motions of our soul with those of the cosmos, because according to Timaeus, it ‘has movements akin to the revolutions of the soul within us.’ Good, well-tuned music can re-calibrate our psychological movements. So, for example, a bracing song in the Dorian mode could spark the courage in the heart of a soldier who has gotten tired and just wants to run home to bed. The proper musical motions would bump his soul back into the proper psychological motion, towards glory and away from shame.
The Unified Field Theory of everything
There are some scholars today who think that the whole ethical tradition of music comes from this set of Pythagorean ideas. Certainly there were lots of legends about Pythagoras using music to calm down rowdy drunks when their souls and desires got out of hand. Damon, the rhythmic ethicist whom we met in the last chapter, seems like he may have collaborated a bit with an aulete named Pythoclides, who some people think might have been inspired by Pythagoras. But this is to reach far back into the shadows of speculation, probably further back than can realistically be trusted. What seems clear is that these two major strands of Greek thought about music – the cosmic music of the spheres and the ethical music of the soul – formed two halves of what would eventually become one giant ‘Unified Field Theory’ of musical mysticism: a Greek-inspired vision of the world and everyone in it as governed by a mathematical, musical logic.
That ‘Unified Field Theory,’ though, was still developing and changing long after Pythagoras, Damon, Plato, Archytas, and the rest were all long dead. The philosophers we’ve been following so far really just laid the groundwork for the beginning of something much bigger than themselves – a way of thinking about music that would far outlast their own lifetimes.
In the ancient world, the direct inheritors of musical cosmology were the Romans. One high-profile representative of Roman musical cosmology was Cicero, the polymathic and often insufferable genius whose hardnosed statesmanship helped shape the era just before the Roman republic ended. Cicero had a lot of side-hobbies outside of politics, and philosophy was chief among them. Plato and the Stoics both interested him immensely, and both of them believed in the ethical, cosmological powers of music. Cicero wrote his own Republic, only fragments of which are still extant. In one of those fragments, a main character (the republican war hero Scipio Africanus) says that the best kind of sound is
that which, using tones separated by unequal but nevertheless carefully proportional intervals, is caused by the rapid motion of the spheres themselves … Learned men imitating this harmony on strings and with their singing have earned their way back to this region [of the heavenly spheres], as have those who have cultivated their talent at searching for divine truths.4
We can hear Timaeus echoing in the background there, and we can also hear the rumblings of what will become Shakespeare’s poetry: ‘such harmony is in immortal souls.’
Elite Roman intellectuals didn’t just bring musical cosmology to Rome. They also eventually, as their empire expanded, exported Greek ideas about music to a vast portion of the known world. Boethius, who lived during the fifth and sixth centuries AD just after the deposition of the last Roman emperor, wrote down what he saw as the core principles of Greek musical philosophy in a book called On the Fundamentals of Music. Right at the outset, this book presents a vision of three interlocking planes of reality, all of them governed by the same musical and mathematical rules. There’s the cosmos, with its planets moving along their orbits – these emit a kind of harmonized sound too profound for the human ear to hear. Then there’s the musical union between a human body and soul, which if properly tuned keeps both together in a peaceful and righteous whole. And finally there’s the kind of audible music we make with instruments and on the stage.
These three types of music – human, planetary, and instrumental – are linked together by one set of mathematical ratios for Boethius just as they are for Timaeus. In fact, Boethius quotes Timaeus in his opening chapter. On the Fundamentals of Music was one of the primary vehicles for preserving and popularizing Pythagorean and Platonic musical philosophy in the centuries after Rome fell. It was so influential that for generations after him, medieval schools taught a curriculum called the quadrivium, inspired in part by Boethius’ description of the Greek arts. Quadrivium means ‘four paths’ or ‘four studies,’ and it refers to the four ways of studying mathematics in the Pythagorean tradition: arithmetic, geometry, astronomy, and music.
Proclus, who lived in the fifth century AD shortly before Boethius, wrote that music was the branch of mathematics which taught ‘the relationship between quantities.’ In other words, the discipline of music was about measuring and understanding the proper way to bring disparate elements into order. In a sense, that summarizes perfectly all the different kinds of mousikē that have emerged over the course of this book. Cosmic music is what makes the planets move in a well-orchestrated pattern, at a perfect distance from one another. Human music is what makes the relationship between a soul and a body work, teaching the soul how to manage its pleasures and emotion in the right proportion and moderation. And when we sing and dance, we use movements and tones to form a picture of these deeper, more sophisticated sorts of music – audible songs are structured in a kind of artistic imitation of the profound logic that binds the whole universe, and our own inner lives, together.
Planetary sheet music
This vision of a cosmos hung perfectly together in delicate, musical balance was tremendously compelling in the medieval period as a way of looking at the world and our place in it. It remained so compelling, for such a long time, that many scientists whom we still regard highly today thought the music of the spheres was a perfectly valid astronomical idea. In the late 1500s and early 1600s, the German visionary Johannes Kepler set out to map the motions of the heavens and understand their logic more perfectly than had yet been done. He was so successful that his three major observations, the three ‘laws of planetary motion,’ are alive and well today as important rules of cosmological physics.
The third of these laws describes a mathematical relationship between a planet’s distance from the sun and the length of its orbit. That relationship is articulated at the very end of a book Kepler wrote called Harmonices Mundi – the Harmonics of the World. The whole of this book is dedicated to proving the basic idea of planetary music: Kepler wanted to show that there are basic relationships inherent in the structure of our solar system which conform to the basic relationships that govern musical scales.5 Even his use of the word ‘harmonics’ is a kind of use that would only be possible for someone inspired by Greek philosophy: Kepler meant to describe not just audible harmony but the foundational resonances and tensional relationships that give rise to the pleasant sounds we hear in everyday music. The Harmonics of the World is a majestic sketch of the universe as a choir of planets, kept in balance and harmony by the mathematical rules that govern their relationships. The third law of planetary motion is the fruit of Kepler’s work in that area, and it is therefore a piece of scientific knowledge that we still owe, in some sense, to the Greeks and their music of the spheres.
And it’s possible we owe more than just that. When we began this chapter, much of its contents may have seemed ridiculous. Certainly it is possible that many modern readers will find it difficult to believe in a demiurge or anything like one – the idea of an actual architect shaping the world with his own two hands may come across as a little primitive at first. But then, a literally physical demiurge of the kind you may have begun by imagining – a flesh-and-blood creator god of the kind one could see and touch with one’s own two hands – would have seemed like a primitive idea to Plato too. Or to most of the Greek intellectuals who thought seriously about music and philosophy. In another one of his dialogues, the Phaedrus, Plato has Socrates say this about the task of describing something eternal and spiritual: ‘to describe it just as it is would be a work of immense and divine explication. But to say what it is like would be a smaller task, fit for a human being.’6
Socrates was talking in the Phaedrus about the ‘Form’ of the soul, the essential reality of what it truly is. But it’s safe to say that his comments apply to any case in which a Platonic dialogue tries to say what things are like beneath the surface – what the essential realities that underlie the material world are like. Mathematics is one such reality: it’s a set of intangible ideas that undergirds the entire tangible, visible, physical universe. Maths is something you can’t see or touch, but it’s threaded through the whole structure of space-time in a profound way.
The real nature of these deep truths is something beyond the human mind’s capacity to grasp. Einstein said once that ‘behind anything that can be experienced there is a something that our mind cannot grasp and whose beauty and sublimity reaches us only indirectly and as a feeble reflection.’ Almost by definition, we can’t directly perceive the ideas underlying the universe, because we’re part of the universe. It’s like a fish trying to see the water. That’s why Plato used myths like the one in the Timaeus: they are intuitive, human ways of getting at truths behind and beyond all human perception.
Among those truths are the mathematical rules that govern the composition of the universe. And it is not so crazy to believe that these rules also govern the patterns that we hear and delight in when we listen to music. In some sense, this must be true, since music is a part of the physical world just like anything else. We still observe, all these years after Pythagoras, that the tones which sound harmonious to us are generated by vibrations in the air whose wavelengths are mathematically related to one another in predictable ways. Modern scientific studies show how these musical wavelengths, when properly tuned, resonate with emotional centres in our brains that direct our feelings and actions on a deep, pre-conscious level. Even unconscious people, as was discovered in a 2015 study in Lyon, France, can have brain activity stimulated and improved by music. And another study from Harvard University recently found that even people who do not understand the language of a given song can often intuit the subject matter and emotional intent just by grasping the song’s musical structure.
At the very beginning of this book, I wrote that music is a human constant. To be sure, there are all sorts of variant patterns in the way we create music across the world and across time: the ancient Greeks themselves found some intervals harmonious that we would consider extremely strange, if not downright ugly. But anthropologists do observe that some things stay the same across cultures – things like a regular, rhythmic beat and a sense of melodic contour can be found pretty much everywhere humans live.
The point of all this is to say that, though the details of musical mathematics have evolved quite a bit since the days of Pythagoras, many of the basic ideas and observations stick with us, albeit in unrecognisable form. Music does appeal to the human mind on a deep emotional level, one which taps into the fundamental mathematical principles that govern our entire universe. It’s no more crazy now than it was in the fifth century BC to say that music is an audible version of those principles, that it makes perceptible to our ears a set of truths that is otherwize imperceptible but nevertheless ubiquitous.
That’s why philosophers, mathematicians, and poets alike are still captivated by the power of music, which exerts a force on our psyche that seems almost impossible to understand. Why, after all, should a set of notes played one after the other, or together in a harmonious orchestra, bring listeners to heights of joy and elation, depths of fear and anguish? Yet they do, still: from electronica to metallica to classical cantatas to R&B, every form of music that has ever been beloved got popular because it spoke to its fans on some deep, visceral level – because it tapped into the deepest workings of their hearts and minds and made them feel things in a way they couldn’t quite explain. Whether one is more inclined to attribute that effect to brain chemistry, or divine providence, or some combination of both at once, it’s actually quite difficult to articulate what music does without making some sort of reference to how it mobilizes the same forces that govern our hearts and direct the planets in their courses.
‘Without music in its best sense there is chaos,’ said the great twentieth-century Russian composer Igor Stravinsky in a 1948 article for the journal Musical Digest. ‘For my part, music is a force which gives reason to things, a force which creates organization, which attunes things. Music probably attended the creation of the universe.’ A Pythagorean would say, well, yes: in some sense, the creation of the universe was music, of the deepest kind there is. Greek philosophy contains some of the first and most profound attempts to describe music and its power while doing justice to all the different ways in which mathematics and melody seem to be intertwined. We are still in the debt of the Pythagoreans today, and not just because they showed us how to measure the sides of triangles: though our language and our science has developed and advanced, we have not stopped hearing the music of the spheres.
Some further reading
There are a few sources on Pythagoras and his relationship to musical numbers and ethics in the ‘further reading’ section of the previous chapter. For a little more on the modern science behind musical understanding across different cultures, start with Kathleen Higgins’s book, The Music Between Us (Chicago, IL: University of Chicago Press, 2012).
Carl A. Huffman is justly famous for his searching accounts of Pythagorean thought. His books on Philolaus (Philolaus of Croton, Cambridge: Cambridge University Press, 1993), Archytas (Archytas of Tarentum, Cambridge: Cambridge University Press, 2005), and Pythagoreanism in general (A History of Pythagoreanism, Cambridge: Cambridge University Press, 2014) have much to recommend them.
Also on Pythagoreanism, for a highly modern and sceptical approach, see Leonid Zhmud’s Pythagoras and the Early Pythagoreans (Oxford: Oxford University Press, 2012).
A few of the scientific studies which chart neurological reactions to musical stimuli can be found online. The study which showed reactions to music in unconscious patients (based in Lyon, France) – ‘Boosting Cognition with Music in Patients with Disorders of Consciousness’, Neurorehabilitation and Neural Repair (volume 29, number 8, pp. 734–42) – can be found at PubMed (https://www.ncbi.nlm.nih.gov/pubmed/25650390).
The Harvard study got a few write-ups in more popular journals, and so a good place to start reading about it would be in the Harvard Gazette (26 January 2018): https://news.harvard.edu/gazette/story/2018/01/music-may-transcend-cultural-boundaries-to-become-universally-human/.
The Albert Einstein quote is from his book, The World As I See It (Mein Weltbild in the German edition of 1934, but translated into English the next year). It is available in a 2014 edition by CreateSpace Independent Publishing.