Pythagoras Spellbound

Revisiting the myth that forged music theory

The man, the myth, the legend.

Where better to begin than with Pythagoras—everyone’s favorite Pre-Socratic mystic, and the star of the origin myth of music theory.

But fair warning: this post contains a 500 word excerpt from a medieval text that you simply can’t skip over. That’s right, you’ll just have to kick off your sandals, retire to the scriptorium, and enjoy savoring every word like a devoted Pythagorean acolyte.

Pythagoras lived in the 6th century BCE, and the earliest surviving record of this myth appears nearly 600 years later in Nicomachus’s Manual of Harmonics (c. 100 CE). But the juiciest version is from about 400 years after that, in Boethius’s De Institutione Musica (early 6th century CE).

Boethius was responsible for transmitting a large body of Ancient Greek knowledge into the Latin-speaking world, and he had a flair for reinterpreting and expanding on the original ideas so they would resonate more with his contemporaries. I am, of course, a big fan.

Come to think of it, De Institutione Musica wouldn’t have taken on that title until at least the 12th or 13th century. The modern-day equivalent would be something like Music: What You Need To Know. But enough framing—let’s settle into Boethius’s take on the Pythagoras myth (tr. Calvin Bower):

Although basic elements of almost every discipline—and of life itself—are introduced through the impression of the senses, nevertheless there is no certain judgement, no comprehension of truth, in these if the arbitration of reason is lacking. For sensation itself is impaired by excess in greatness and smallness alike. It is possible not to perceive very small things because of the minuteness of the sensible objects themselves, and sense perception is frequently confused by very large objects. In pitches, for example, the hearing grasps with difficulty those that are very soft; but if pitches are very loud, the hearing is deafened by the intensity of the sound itself.

This, then, was primarily the reason why Pythagoras, having abandoned the judgement of hearing, had turned to the weights of rules. He put no credence in human ears, which are subject to change, in part through nature, in part by external circumstance, and undergo changes caused by age. Nor did he devote himself to instruments, in conjunction with which much inconstancy and uncertainty often arise. When you wish to examine strings, for example, more humid air may deaden the pulsation, or drier air may excite it, or the thickness of a string may render a sound lower, or thinness may make it higher, or, by some other means, one alters a state of previous stability. Moreover, the same would be true of other instruments.

Assessing all these instruments as unreliable and granting them a minimum of trust, yet remaining curious for some time, Pythagoras was seeking a way to acquire through reason, unfalteringly and consistently, a full knowledge of the criteria for consonances. In the meantime, by a kind of divine will, while passing the workshop of blacksmiths, he overheard the beating of hammers somehow emit a single consonance from differing sounds. Thus in the presence of what he had long sought, he approached the activity spellbound. Reflecting for a time, he decided that the strength of the men hammering caused the diversity of sounds, and in order to prove this more clearly, he commanded them to exchange hammers among themselves. But the property of sounds did not rest in the muscles of the men; rather, it followed the exchanged hammers. When he had observed this, he examined the weight of the hammers. There happened to be five hammers, and those which sounded together the consonance of the diapason (octave) were found to be double in weight. Pythagoras determined further that the same one, the one that was double of the second, was the sesquitertian (the ratio 4:3) of another, with which it sounded a diatessaron (fourth). Then he found that this same one, the duple of the above pair, formed the sesquialter (3:2) ratio of still another, and that it joined with it in the consonance of the diapente (fifth). These two, to which the first double proved to be sesquitertian and sesquialter, were discovered in turn to hold the sesquioctave (9:8) ratio between themselves. The fifth hammer, which was discordant with all, was discarded.

Although some musical consonances were called “diapason,” some “diapente,” and some “diatesseron” (which is the smallest consonance) before Pythagoras, Pythagoras was the first to ascertain through this means by what ratio the concord of sounds was joined together.

After probably eating a few lizards and staring at the sun for a while, he then went home and performed a bunch of experiments with strings, weights, and glasses of liquid until he was absolutely convinced of his findings. Turns out the scientific validity of these observations is problematic if not outright mythological—hammer weight alone doesn’t determine pitch, string length has a clearer relationship to frequency than tension does, and glass resonance also depends on the material qualities of the glass. But none of that seems to matter much to Pythagoras.

So, then, what is this myth about? One alarmingly common take-away is that music is reducible to mathematical principles. As though Pythagoras was that dude at the cocktail party who says, “See? Music is just math.” It’s made up of intervals, and intervals are ratios. Problem Solved.

But that’s not this Pythagoras, no. This Pythagoras is a pretty complicated dude. For one thing, he has a real mistrust of the physical world. Sensation alone is impaired, instruments are unreliable, and the properties of sound most definitely don’t reside in the muscles of men. It even seems mythologically appropriate that his empirical “proofs” should be so questionable. And yet, hammers aren’t so different from any other physical, sensible, unreliable instrument. Right?

Actually, I can think of at least three pretty solid reasons to choose hammers for your mythological experiments. First and most practically, if you live in the ancient world, hammer weight is just about the easiest thing you can measure. Balance scales had already been a staple in trade for thousands of years before Pythagoras, not to mention their significance as a symbol for balance, order, and even truth. As an added bonus, the very means by which these ancient scales measure hammer weight is through ratio, relationship.

But it wasn’t just ratio that Pythagoras was interested in. It was ratio that could be expressed through number. And not just any number, but numbers that approach as close as possible to one. In fact, the word “consonance” means sounding (sonance) together (con)—as one. This gets even more interesting when we consider that the Ancient Greeks didn't count one (the unit) as a number at all, but rather, the foundation for number. A number is a multitude composed of units. That is, there’s no number until there are at least two units in relation to one another.

Now, ratio itself doesn’t require number. Any magnitude can be in relation to any other magnitude without necessarily being expressible as whole numbers (for example, π : √2 is a ratio, but not of numbers). So what Pythagoras was interested in wasn’t quite ratio, and wasn’t quite number, but a very particular, very sacred place where number and ratio align, as near as possible to oneness, to consonance.

The second argument in favor of hammers is that these special ratios-made-of-numbers emerge from the very material of the hammers themselves, not from something like the pulsation of strings, or the movement of air in a pipe. There’s no motion, no vibration, no time involved here. Only material. Number is in this world, but it is not of this world.

This is perhaps why that fifth hammer, which was “discordant with all,” needed to go. Rational thought, for Pythagoras, seems to be something like the ability to perceive fittingness, harmony, the quality of oneness that can emerge from many. This is different from logic, our ability to determine whether something is valid. And it’s also different from reason, where we combine our sense of harmony with our findings of validity to perceive truth. Rational thought is the mysterious, otherworldly part of the human equation where we sense our connection to a larger whole. It works with the material of the world around us, but it looks right through that material into something beyond.

Pythagoras is often accused of discarding the fifth hammer because he just couldn’t bear the realization that everything in the sensible world didn’t fit together in perfect harmony with his idea of a cosmological whole. Or sometimes the fifth hammer is interpreted as a symbol for the limits of human reason in in infinite world (this seems like a good moment to recommend Daniel Heller-Roazen’s book).

But I stubbornly like to imagine Pythagoras saying, “man, that fifth hammer sounds GREAT… but focus on the cosmos, focus on the cosmos...” And then of course he went back and grabbed it for his dance party later that evening.

Which brings us back to worldly material and a third argument in favor of musical hammers. In the ancient world, the blacksmith’s forge was not only a workshop, but a site of alchemical transformation. Metalworking was among the first technologies that transforms matter from one state to another—from ore into metal, and from metal into various tools—and this was seen as participating in mysterious and divine processes. Blacksmiths transmitted a kind of secret knowledge unknowingly, intuitively, through their craft. And hammers have the distinction of being both a culmination of this craft, and an instrument through which it’s performed.

When Pythagoras wandered by the forge that day, it was certainly not the first time he’d heard sounds like these. Musicians could easily sing the consonances, and they could tune their instruments by ear. They performed their craft intuitively, much like the blacksmiths did. And Pythagoras was already so deeply immersed in this musical world, so surrounded by it in his everyday life that he was able to recognize it at the forge instantly and without thinking. But his experience that day was completely different from the countless times he’d passed a musician playing an aulos, or plucking a lyre, or singing some Homeric verse. This time he sensed something different in the sound, and he was spellbound. Pythagoras himself had entered a process of transformation.

I like to imagine we all have a taste of what Pythagoras experienced on that day every time something unexpectedly captures our interest and attention. We’re led, “by some divine will,” to experience our everyday world in a different way, to sense that suddenly and mysteriously, we’re in the presence of something we have long sought. And in this way, we might begin to understand this story less as an origin myth for the fundamentals of music theory, and more as a parable for how we might participate in the activity of music theory.

Or even in some greater activity that we only glimpse through music. Pythagoras had sensed a sacred place where number and ratio align, but in this moment he also entered a sacred place where rationality and irrationality align. He needed his frustrated experience of the imperfect sensible world, and his ideal of the perfection of the cosmos, and his encounter with the mysterious, swirling, sparking forge to bring about this moment.

What he forged from that experience was by no means the first musical tool, but an entirely different kind of tool that operated by a means other than intuition and alchemy, the way musical instruments always had. It was the first musical rule, or measure. And it was the rule of rules—a measure of perfection itself, and of its ultimate limitation.

Although I can’t help but wonder if the tool we’re really meant to notice here might not be Pythagoras’s rule, but his story—of the perfectly alchemical conditions that aligned that day to open an entirely unprecedented way of working with a craft.

Maybe, like the blacksmith’s hammer, it is both a culmination of an activity that never really culminates, and also an instrument for performing this activity ourselves.

Maybe Pythagoras has not left us the perfect rule, but rather, the perfect invitation.