The Rhythm of Numbers

The number in the bar plays here, with one voice per level. Hits carry the melody, and the starts of level 2, 3 and 4 blocks play bass, bell and gong. In “gap = interval” mode each hit moves the melody up by its gap in half steps, so pi’s gaps of 7 walk the circle of fifths and each 8 slips by a half step. “Jump to the rarest block” goes straight to pi’s short phrase after the 292 long ones.

Stop the continued fraction after each term and you get the number’s best fractions, called convergents. For pi they are 3, 22/7, 333/106, 355/113, 103993/33102, …. A large term means the fraction before it was unusually good, which is why the 292 makes 355/113 so accurate. Prime numerators and denominators are marked ◆. Switch the bar’s number to compare pi with other constants.

A pure fifth is the frequency ratio 3:2, which is log₂(3/2) = 0.58496 of an octave. Its best fractions run 1/2, 3/5, 7/12, 24/41, 31/53. “A fifth is 7 of 12 half steps” is the 7/12 fraction, found by the same rule that finds 22/7 for pi. The 12-note fifth is 1.96 cents flat, and over 12 fifths that small miss adds up to the Pythagorean comma, 23.46 cents. The next good stops are 41 and 53 notes per octave, and 53 is prime.

Euclid’s algorithm divides and keeps the remainder, again and again. Run it on 355 and 113 and the quotients are 3, 7, 16: pi’s continued fraction, cut off. It ends at the greatest common divisor, the oldest test for whether two numbers share a prime. In 2003 the physicist Eric Bjorklund used the same steps to spread pulses evenly in a particle accelerator, and in 2005 Godfried Toussaint showed that the results are rhythms found in music around the world.

Things that swing or spin and nudge each other fall into step at simple ratios. Christiaan Huygens saw two pendulum clocks on one beam fall into step in 1665. Mercury spins exactly 3 times for every 2 trips around the Sun. Heart cells follow the beat of a pacemaker. A simple model, the circle map, shows all of these: as the coupling K grows, “tongues” open at each fraction p/q, and inside a tongue the motion snaps to that fraction.

Number the 12 notes of the octave 0 to 11, starting at C. Every prime above 3 leaves a remainder of 1, 5, 7 or 11 when divided by 12, so it lands on one of four notes: C♯, F, G or B. Play the primes in order and let each gap to the next prime set the note’s length. In 2016 Robert Lemke Oliver and Kannan Soundararajan found that a prime tends not to repeat the remainder of the prime before it. With no memory, repeats would be 25% of pairs. Among the primes up to a million they are 15%, and up to ten million 16.4%. You hear it as fewer repeated notes than chance would give.

Stand at the corner of an orchard planted on a square grid. The trees you can see, with no tree in front of them, are the points (a, b) where a and b share no factor. They fill 6/π² ≈ 60.79% of the grid.

Hydrogen’s energy levels go as −1/n², and its visible lines, the Balmer series, come from the differences 1/4 − 1/n². In wavelengths that is λ = 364.56 · n²/(n² − 4) nanometers, the formula Johann Balmer found in 1885. Electron shells hold 2n² electrons: 2, 8, 18, 32.

Place seed number k at k × (the bar’s number) turns around a circle, at a distance √k from the center. For π − 3 the seeds fall into 7 spokes that curl into 113 arms, because those are pi’s block lengths. Now light up only the seeds whose numbers are prime.

The sunflower, straightened out. Lay the numbers in rows of a chosen width. When the width matches a hidden period, the hits line up in straight columns. When it is slightly off, the columns lean, and the lean shows the error. Pi’s rhythm stands up at its block lengths 7, 106 and 113. The primes line up in the columns that share no factor with the width.

In 1982 Dan Shechtman found a crystal that was ordered but never repeated, a quasicrystal. Many scientists doubted it at first; it won him the 2011 Nobel Prize in Chemistry. Crystallographers read order from diffraction, which is the Fourier transform of where the atoms sit. Pi’s rhythm is a one-dimensional quasicrystal: its diffraction has sharp spikes at multiples of π − 3, wrapped around 1.

Bernhard Riemann’s explicit formula rebuilds the prime staircase, which climbs by log p at every prime power, from waves whose frequencies are the zeros of the zeta function: 14.13, 21.02, 25.01, …. Add more zeros and the curve snaps to the steps. In 1972 Hugh Montgomery and Freeman Dyson noticed that the spacings between the zeros follow the same statistics as the energy levels of heavy atomic nuclei. No one yet knows why.

In 1975 Michael Morrison and John Brillhart published how they used continued fractions to factor 2¹²⁸ + 1. Run the continued fraction of √N, the same steps as pi’s ladder. Each step gives a small number Q that matches a perfect square, counting remainders after dividing by N. When Q is itself a perfect square, one gcd splits N. This page runs the simple version that D. H. Lehmer and R. E. Powers described in 1931.

A times-table circle joins each point x to 2x. Use 2^p − 1 points and doubling does something exact: write x in binary with p digits, and doubling just rotates the digits one place. So each orbit is one p-beat rhythm and all of its rotations, a necklace. When p is prime, every orbit has exactly p members, so p divides 2^p − 2. That is Fermat’s little theorem, proved with rhythms. It does not make 2^p − 1 prime, though: try p = 11, where 2047 = 23 × 89. The real test is the Lucas–Lehmer test. The largest prime known, 2^136,279,841 − 1, with 41,024,320 digits, was found with it in October 2024 by the Great Internet Mersenne Prime Search.

A rhythm of n beats is also an n-digit binary number, with a hit as 1 and a rest as 0. The readout tests it for being prime, lists its rotations that are prime, finds the continued fraction of its density (hits per beat), and checks whether it is spread as evenly as possible. A rhythm needs a hit on its last beat to be an odd number.

Read the first n beats of the bar’s number as one long binary number, and test it for being prime as n grows. Compare with what chance predicts: an odd number with n binary digits is prime about 2 times in n·ln 2.

Each ring is one track: a convergent p/q played as p hits spread evenly over q slots, all rings taking the same time to go around. Stack pi’s 1/7, 15/106 and 16/113 and you hear three versions of the same beat drift and meet. Or layer different constants. Each track has its own voice, a mute and a solo.

Pick how many hits (pulses) and how many slots each drum gets, and the machine spreads the hits as evenly as it can with Bjorklund’s method from section 6. Rotate a track to start it on a different hit. Tracks of different lengths drift against each other and line up again after their least common multiple. Each track shows the continued fraction of slots/pulses and whether the two numbers are coprime.

This is Teddy Rook’s idea grown into a small composing machine. The number’s rhythm gives the beats. Its levels give the form: the shortest blocks are bars, the next level up gives phrases, and the next gives sections. (A level that only adds one beat is skipped, so for pi the bars are 7 beats, the phrases 113, and the sections 33,102.) Each level gets a voice: melody on the hits, a bass note at each bar, a chord at each phrase, and a cymbal and key change at each section. Choose the number, the scale, the voices and the length, then play it, look at the score, and download it as a MIDI file for any music program, or as a WAV sound file.

Teddy Rook used the ideas on this page to compose original music based on continued fractions: pieces whose bars, phrases and sections follow the levels of pi and other numbers, the way the Composer and “Compose a piece” do. The page came first, as the workshop, and the music grew out of it.

Play it yourself. The score of all six movements is free: download the score (PDF, 16 pages, 1.7 MB). Each movement has its own piano lesson in the Music section: the Continued-Fraction Suite, from Golden and Silver to Pi and the rare block at bar 32,996.

Threads to pull

Questions this page raises about pi and the primes. Some are open, some are known, and some are experiments to try. Treat them as starting points, not results.