Bach Hymns Carry Different Information Loads

When physicist Suman Kulkarni fed 337 Bach compositions into a computer network in 2024, she wasn't trying to appreciate the music. She was treating each piece as a mathematical object—a web of nodes and edges, where notes became data points and transitions became probabilities. What she discovered was that Bach's hymns and his toccatas didn't just sound different. They contained measurably different amounts of information, quantifiable through entropy calculations borrowed from computer science. The music that moved listeners for three centuries turned out to have a structure you could graph.

The Math Hiding in Plain Hearing

Take an ordinary piano octave. Count the keys: 8 white, 5 black, 13 total. Now count the notes in a major scale: 8. The foundation of a basic chord rests on the 3rd and 5th notes. These aren't arbitrary choices. They're all Fibonacci numbers—each one the sum of the two before it (1, 1, 2, 3, 5, 8, 13). Divide 8 by 13 and you get roughly 0.615, which approximates the inverse of the golden ratio, that peculiar number (approximately 1.618) that shows up in nautilus shells, sunflower spirals, and Renaissance paintings.

Western music theory didn't develop because composers sat around calculating ratios. It emerged from the physical properties of vibrating strings and the human voice, from what sounded consonant to the ear. But the ear, it turns out, has preferences that align with mathematical relationships. Middle C vibrates at 256 Hz in well-tempered tuning—a power of two that makes the math of intervals work out cleanly.

Mozart's Architectural Precision

Mozart's Piano Sonata No. 1 in C Major contains 100 bars in its first movement. The exposition—where he introduces his main themes—runs for 38 bars. The development and recapitulation—where those themes get explored and restated—take up 62 bars. Divide 62 by 38 and you get 1.63, nearly identical to the golden ratio of 1.618.

This wasn't coincidence. Mozart structured many of his piano sonatas this way, positioning the pivot point between sections at the golden ratio. The effect is subtle but pervasive: the music feels balanced without feeling symmetrical, proportioned without feeling calculated. Beethoven, Bartók, Debussy, and Schubert all did similar things in their sonatas, though no one can quite explain why this particular ratio produces such satisfying results.

The honest answer is that we don't know why 1.618 works. We just know that it does, consistently, across cultures and centuries. Violin maker Antonio Stradivari used the golden ratio throughout his instruments—in the relationship between different component lengths—and those violins now sell for millions of pounds, prized for a sound quality that may owe something to their mathematical proportions.

Bach's Information Architecture

The 2024 study of Bach's work, published in Physical Review Research, approached composition from an unexpected angle. Instead of analyzing harmony or melody, Kulkarni and her colleague Dani Bassett used information entropy—a measure developed by mathematician Claude Shannon in 1948 to quantify how surprising or predictable a message is.

Bach's chorales, designed to be sung in church, showed relatively low entropy. The note progressions were more predictable, easier for congregations to follow and remember. His toccatas and preludes for keyboard, meanwhile, contained much higher entropy—more surprising transitions, denser information, greater complexity. As Bassett put it: "These two sorts of pieces feel different in my bones, and I was interested to see that distinction manifest in the compositional information."

But here's what makes Bach's approach sophisticated: his complex pieces didn't just pile on random surprises. They contained clusters of related note transitions, recurring patterns that gave listeners something to hold onto even as the music ranged widely. The brain could learn the music's internal logic without the composition sacrificing informational richness. Mathematical structure made emotional complexity possible.

The Paradox of Predictability

Cognitive scientist Marcus Pearce studies how we process music through what he calls "probabilistic learning." Our brains constantly predict what sound comes next based on patterns we've absorbed. Too predictable and music becomes boring—no information, no surprise. Too random and it becomes noise—all surprise, no coherence.

The sweet spot lies in a paradox: unity of effect combined with density of change. A composition needs an underlying structure, a "thought-object" that listeners grasp as a whole once the final notes resolve. But it also needs complexity, variations that unfold across time. The mathematical patterns classical composers used—Fibonacci sequences, golden ratios, carefully calibrated information entropy—let them walk this tightrope.

When Lady Gaga's producers placed a key change at the 111-second mark of her 179-second song "Perfect Illusion," they were using the same principle Mozart used. The ratio of 111 to 179 approximates the golden ratio, creating a structural pivot that feels natural even though most listeners would never consciously notice the math.

Why Patterns Feel Like Feelings

The relationship between mathematical structure and emotional response remains mysterious. We can measure the patterns—graph Bach's networks, calculate Mozart's proportions, analyze entropy in Debussy's compositions—but we can't fully explain why these particular arrangements move us.

Perhaps the answer lies in how our brains work: pattern-recognition machines that find pleasure in detecting structure while still craving surprise. Or perhaps it's about nature itself, which seems to favor the golden ratio in its own designs, making music that echoes those proportions feel somehow organic, inevitable.