The Mathematical Principles of Music…
Of Everything…
The Pythagorean Foundation of Modern Science…
The (Macrocosmic) Harmony of the Spheres…
The (Microcosmic) Harmony of the Soul…
Even more momentous for Western thought was Pythagoras’ discovery of the mathematical basis of musical intervals.
It was made, as Diogenes Laertius explains, by the experimental use of the “monochord”–an instrument, as its name implies, in which a single string is stretched across a soundboard which supports a movable bridge. By stopping the string at a certain point along its span, Pythagoras realized that the pitch that it produced was directly related to the length of the vibrating portion of the string.
Stop it at the exact centre, pluck the string, and it produces a note precisely an octave above. Stop it so that the ratio of the vibrating to the non-vibrating portion is 3:2, and you produce a major fifth. Stop it so the ratio of the vibrating to non-vibrating lengths of string is 4:3, and the result is a major fourth.
What Pythagoras realized, then, was that the octave, fourth, and fifth–the “major consonances” of Greek music—were produced as a function of certain fixed numerical proportions, and that the sciences of acoustics in particular, and music in general, have an invisible mathematical structure.
It was but a small step for Pythagoras to infer that the physical laws of not only sound but of the entire natural world were informed by hidden mathematical proportions and principles. As Aristotle relates,
The Pythagoreans…thought that the principles of mathematics were the principles of all things….Seeing that the properties and ratios of the musical consonances were expressible in numbers, and that indeed all other things seemed to be wholly modeled in their nature upon numbers, they took numbers to be the whole of reality, the elements of numbers to be the elements of all existing things, and the whole heaven to be a musical scale and a number.
Rather than seeking a unifying and unchanging physis or nature of the cosmos in the material elements—water, air, fire, or “the infinite”, as the Ionian philosophers had done—Pythagoras maintained that the elements themselves, material or quasi-material as they were, were secondary phenomena, necessarily reducible to the primordially elemental, immaterial concepts of mathematics.
One can hardly overestimate the significance of Pythagoras’ insistence upon mathematics as the foundational principle of cosmic order for later science. Within two hundred years it gave rise, in the work of Archimedes, to the science of mechanics. Even Galileo, at the dawn of the modern age, took it as the starting point of his own investigations:
Philosophy is written in the great book which is ever before our eyes—I mean the universe–; but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles, and other geometrical figures, without whose help it is impossible to comprehend a single word of it.
Upon this Pythagorean foundation the whole structure of classical physics was erected.
From his conception of “the whole heaven as a musical scale and a number” arises, finally, another of Pythagoras’ celebrated doctrines–and another ongoing topos of Western thought–, that of the so-called “harmony of the spheres”. As Aristotle explains, Pythagoras reasoned that the heavenly bodies, being of immense size and moving at correspondingly vast speeds, must produce sounds in their peregrinations, indeed, much louder sounds, necessarily, than those of the relatively smaller and slower-moving objects we can hear on earth. If the revolutions of the heavens are inaudible to us, it is only because their music has filled our ears since birth, so that we are unable to distinguish it from silence.
The pitch of these celestial sounds, moreover, must be directly proportionate to their velocity, which in turn is a function of their distance from the earth–that is, the radius of their orbit. The fixed stars, on the periphery of the heavens, and revolving at the fastest speed, make the highest sound; the moon, closest to the earth, with the shortest orbit, and moving at the slowest speed, makes the lowest. What’s more, since these distances are in the ratios of the musical consonances, the sounds produced by the planets and stars blend together in a vast cosmic “harmony”.
The grandeur of this conception ensured, as I say, that it would become a perennial theme in Western poetry, theology, and philosophy. It explains, amongst other things, why music was from Plato to Castiglione so central a discipline in the moral education of the philosopher-prince. As Shakespeare’s Lorenzo puts it, two thousand years after Pythagoras, in the Merchant of Venice:
How sweet the moonlight sleeps upon this bank!
Here we will 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 beholds’t
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…The man that hath no music in himself,
Nor is not moved with concord of sweet sounds,
Is fit for treasons, strategems, and spoils;
The motions of his spirit are dull as night,
And his affections dark as Erebus.
For Shakespeare, Plato, and all latter-day Pythagoreans, the harmony of the spheres is in immortal souls because it was the soul’s birth-song, which it heard when it still lived in the celestial aether, before its unfortunate fall into the body and the world. If ordinary men can’t hear it, it’s not for the reason Aristotle imagines, but because the noise of the world—the distractive din that floods in through the body’s physical organs of sensation—drowns it out. Such men are deaf to the harmony of the spheres, because, as Plutarch explains, “The ears of most souls are blocked and stopped up, not with wax, but with carnal obstructions and passions.”
To hear that harmony requires, on the contrary, that the soul stop up her outer ears; that she mortify the senses by turning her attention away from external earthly and material objects and desires, and cultivating instead the inner incorporeal senses of reason and intelligence through which we alone apprehend reality and truth—truths that like those of mathematics are invariably “immaterial and conceptual”.
By pursuing a life of contemplation and introversion, and through the ascetical regimes already mentioned, the Pythagorean wise man creates the stillness that allows him to hear the celestial music resonating faintly in the interior depths of the soul; hearing it, he is reminded of his celestial birth and reawakened to his own divinity.
With the Pythagorean teaching in the background, later writers and thinkers referred to the wise or virtuous soul as “musical”, “harmonious”, or tonos, attuned. In the musical soul, the discordant and cacophonous carnal appetites and passions have been composed by reason in order and harmony, and so composed, the microcosmic psyche imitates and participates in the greater order and harmony of the macrocosm.