When we hear Johann Sebastian Bach's music, we are most characteristically moved by its emotional substance. It may be deep, exuberant, melancholy, or exciting. But underlying the lovely face of melody and harmony, there is an invisible world of architecture, a hidden order of mathematical loveliness and geometric balance.
Bach was not just a gut-spilling composer; he was a master craftsman who employed numbers, patterns, and symmetry as the secret blueprints for music. To really appreciate his genius, we need to learn to listen, not only with our hearts, but with our minds, to listen for the intelligent and sophisticated architecture that bestows his music with immortality.
The best-known demonstration of Bach's mathematical inclinations is the way he handled canon and fugue. A canon is a type of round, in which one melody line is repeated by another voice entering a little later, such as in "Row, Row, Row Your Boat."
But Bach developed this simple concept to dizzying levels of complexity. He composed canons wherein the following voice sings the melody backwards (retrograde), or upside down (inversion), or twice as quickly, or half as quickly. It is music geometry. It is akin to building a beautiful tapestry wherein every string is a replica or an exact spin of something else. His fugues are even more complex architectural pieces. A fugue is initiated with one musical idea called a subject.
And then the additional voices turn up individually, each chasing this theme in its turn as additional voices entwine counter-melodies to it. The whole work is constructed from this one tiny idea, polished, manipulated, and re-erected on top of itself with intellectual charm, like a mathematician solving a hard problem by studying one axiom from all sides.
Bach's passion for patterns even carried over into his composition. He was passionately fascinated with numerology, the magical strength of the numbers. Alphabetical letters had numerals attributed to them in his day. If we replace B=2, A=1, C=3, and H=8, we get BACH = 14 (2+1+3+8). J.S. BACH = 41 (I=9, S=19). Number 14 occurs ubiquitously in his compositions as an unseen signature. In his last, gigantic composition, The Art of Fugue, the third subject has 14 notes. He could not have inserted anything else into the warp and weft of the music itself, a robust geometric monogram of himself. Not superstition, but structural preference, an autosuggested regulation which provided his imagination with one, idiosyncratic shape.
This mathematical mind is best seen perhaps in his employment of symmetry. Bach's music rings eternally well-proportioned because it is. A fine illustration is his well-known Crab Canon from The Musical Offering.
It is composed as a single unbroken line of sound to be played through from start to finish, but it can just as effectively be played from end to beginning at the same time. The two lines of sound, one ahead and one behind, harmoniously cross each other, crossing exactly in the middle before diverging again. It is a musical palindrome. It is a geometric wonder, a piece of music that is an ideal line of symmetry in time. It displays a degree of organization and intellectual mastery that is positively marvelous. Each note had to have been selected in such a way that it would sound good locally and also in its reflected, inverted state. We can even notice geometry in the way that Bach organized harmony.
Harmony is music's vertical aspect, the chords all sounding together at once. Bach's chord progressions are not random; they have rational paths and frequently trace geometric paths around the circle of fifths, a circular chart of musical keys' relation. Circumnavigation of the circle is what creates the sense of rational movement and direction.
Furthermore, the general shape of his works, from small dance suites to huge passions, is typically derived from the "Golden Ratio" or other proportional divisions. The inner peak of a piece, the emotional high point, will usually be found not in the center, but at 61.8% of the way through the piece, a ratio which occurs in seashells, galaxies, and Renaissance painting, and is well known for its natural beauty. Bach hoped that this ratio would provide a naturally sweet and lovely experience for the listener. Why, then, did Bach hide this hard mathematics in his music?
He was a man of his epoch, the Baroque era, living amidst the Enlightenment. It was an era that took it for granted that there was an orderly world, designed by a divine mathematician. The order of nature—sea shell spirals, orbits of planets—was evidence of God's orderly intellect.
Musical imagination for Bach was a vocation for religion. In producing music of complete order and intellectual beauty, he was not bragging. He was repeating the cosmic harmony of the universe itself. He was constructing a beautiful cathedral, within which each note, each counterpoint, each structural ratio was a stone in its appointed place to serve the cosmos's builder. Geometry was his prayer. The secret architecture never toughens up the music and makes it unfeeling. Rather, it is the very nature of its profound emotional force.
Imagine a beautiful Gothic cathedral. Its beauty is superficial and overpowering. But its greatness is in the concealed flying buttresses, the mathematical proportions of its nave, and the deliberate alignment of its pillars that enable it to rise skyward without collapsing.
You will enjoy the beauty without knowing about the engineering, but with knowledge of the design, you enjoy it better. Bach's music is based on the same principle. The rigid rules of counterpoint and mathematical limitations did not confine him; they set him free. In this world of rules, he discovered boundless freedom of expression. The tension and resolution which we experience in his music is not so much emotional, but a musical equivalent of an ideally solved equation, providing us with a sense of profound balance, solution, and cosmic correctness. Lastly, to hear Bach is to see a miracle of the human mind. It is where the left brain and the right brain meet, where emotion and logic, passion and mathematics are one.
The melodies touch us, but the geometry grounds the movement as perpetual and eternal. His music is timeless because it is grounded in the principles of number, pattern, and symmetry—principles which are true no matter the time or place.
The secret geometry is not thinking; it is the secret animating purpose of its beauty, the reason that his music never fails to sound so inevitable, so perfect, and so deeply moving. It is a reminder that out of the deepest order there can grow the loveliest things.