An Odd Kind of Sympathy

In February 1665, Christiaan Huygens — inventor of the pendulum clock — lay sick in bed and noticed something odd about two of his clocks hanging from the same wooden beam. No matter how they started, within half an hour their pendulums were swinging in perfect opposition — each swinging left just as the other swung right, the ticks interleaved. He wrote to his father about an "odd kind of sympathy" between the clocks.

You can recreate his discovery with a few metronomes and a board resting on two cans. Wind them up, start them out of step, and set them on the board. Here is the real thing, filmed by the UCLA physics department:

The same experiment is easy to simulate, and a simulation lets us slow the process down, speed it up, and take it apart. Below, five simulated metronomes start at random points in their swing. Press play and watch what they do:

They always end up ticking together, although nothing connects them but the board they stand on. Curiously, they agree in step, where Huygens's clocks agreed in opposition — we'll come back to that. In the rest of this article we'll build the experiment up from its parts and see how it happens. One suggestion before we start: the speaker button in the corner of the page adds ticking sounds to every demo, and with it on you can hear the moment agreement arrives.

A metronome

At its heart a metronome is a pendulum: a rod swinging on a pivot, with a small weight that slides along the rod to set the tempo. Moving the weight away from the pivot makes the swing slower, and moving it closer makes it faster. Each tick — each beat — is one half of a full swing.

The dial on a real metronome runs from about 40 beats per minute to 208. Everything in this article works the same way at any tempo, so we'll mostly leave it near 144.

Keeping time

A pendulum on its own winds down. Friction at the pivot and air resistance take a little energy from every swing, and after ten or twenty seconds it hangs still. You can see this happen by switching the escapement off:

The escapement is the mechanism that keeps this from happening. A wound spring stores energy, and a toothed wheel passes a small push to the pendulum at the same point in every swing — the tick you hear is that hand-off. The push replaces what friction took, so the pendulum settles into a steady swing of fixed width, marked above by the dashed lines.

One property of this arrangement matters more than any other for what follows. Press the nudge button above and watch what recovers. The shove throws the swing off its usual width, and within a couple of beats the escapement has trimmed it back — that is the escapement doing its job. But the faint gray pendulum behind the colored one keeps the timing the swing would have kept, and the nudged pendulum never drifts back to it. The escapement polices how wide the swing is; nothing polices when it happens. After a disturbance the metronome swings exactly as before, permanently a little earlier or later than it would have been. That freedom in timing is what the rest of the story turns on.

Throwing its weight around

Now place the metronome on a board, and rest the board on two cans so that it can roll. The pendulum's weight accelerates left and right on every swing, and by Newton's third law the case — and the board beneath it — is pushed the opposite way. The arrow below the ground line shows the sideways force the metronome exerts on the board at each instant; the pale band behind it shows how large that force has recently been.

Try making the base heavier with the slider: the recoil shrinks, because the same force now has more mass to move. The board's motion is drawn at twice its true size here and throughout — in reality the sway is a centimeter or two at most.

The board pushes back

So each metronome shakes the board. The influence also runs the other way: a pendulum whose pivot accelerates feels an extra push, the same push you feel standing in a bus as it brakes. Through the board, every metronome is gently pushing on every other.

What can that push actually change? Not the width of the swing — the escapement defends that. What nothing defends is timing. To see the effect in isolation, let's take the other metronomes away and shake the board ourselves, in a steady rhythm at the metronome's own tempo. On the circle at the top right, the gray dot keeps the rhythm of the shake and the colored dot follows the pendulum:

Start the pendulum behind the shake and it is hurried along until it catches up; start it ahead and it is held back until the shake catches it. The reason is in the timing of the pushes. Running ahead, the pendulum's weight tends to meet the board's push head-on, and loses a little speed to it on every swing; running behind, the pushes arrive at its back and add a little. Either way the pendulum slides into step and stays. Each beat's correction is tiny — with a gentle shake, closing the gap takes dozens of beats — but it always points the same way, beat after beat, for as long as it takes.

Now put two metronomes on a free board, and each plays both parts at once: each shakes the board with its swing, and each has its timing pulled by the shaking of the other.

The two dots on the circle mark each metronome's phase, its position in its cycle. Start the demo and speed time up, and the dots drift toward each other until they merge. Then make the base heavier and randomize again: the pushes weaken and agreement takes much longer. On a base too heavy to move, it never comes.

This is also the place to settle our debt to Huygens, whose clocks agreed in opposition where our metronomes agree in step. Both arrangements are stable ways for two pendulums to share a support, and the support decides between them. A board that is light and free to roll is moved most by pendulums swinging together, and its motion recruits them further — moving in step is self-reinforcing. Huygens's clocks instead hung from a heavy beam that barely moved and mostly drained away whatever energy reached it; there, the arrangement that survives is the one that shakes the support least, and two pendulums in opposition cancel each other's recoil exactly. Our metronomes can be talked into Huygens's choice too: in the playground further down, two metronomes on the lightest base with the roller friction turned all the way up will lock in opposition instead of step.

Measuring togetherness

With more metronomes it helps to have a single number for how synchronized the group is. Place every phase on a circle, as we did above, and find the center of mass of the dots. The distance of that center from the middle of the circle is called the order parameter, written r. When the phases are scattered, the dots surround the center and r is near 0. When they agree, the dots gather at one point and r reaches 1.

Get a feel for it by hand — drag the dots around the circle and watch the arrow, whose length is r:

Notice that r doesn't care where on the circle the dots gather, only how tightly they agree. Now watch the same number measured live, on four metronomes:

The graph records r over time. A typical run wanders for a while, climbs as some of the metronomes catch each other, and then jumps to 1. The net arrow under the board explains the jump. While the phases are scattered, the individual forces largely cancel and the board moves little. As agreement spreads, the forces begin to add, the board swings harder, and its influence on the remaining stragglers grows. The closer the group comes to synchrony, the stronger its pull on whoever is left out.

Different drummers

What happens if one metronome is set differently? Below, the middle one runs at half tempo: 72 beats per minute against 144 on either side. Its pendulum is longer — that is how a metronome slows down — and it pushes on the board exactly as hard as the others: half the tempo means a pendulum four times as long, so its weight covers four times the arc at half the pace, which works out to the very same acceleration and the very same force. You can check this against its force band, which is the same width as its neighbors'. All that has changed is that its pushes no longer arrive in rhythm. Before you press play, commit to a guess: will the board's rhythm drag the slow one along? Will the slow one spoil the fast pair's agreement?

The two outer metronomes find each other and lock. The middle one never joins: the board's rhythm pushes it forward during half of its cycle and backward during the other half, and over a full cycle the corrections cancel. In the graph, r no longer settles at 1 but oscillates as the slow metronome drifts past the others again and again, like a runner being lapped on a track. Set it back to 1× and the group locks within seconds. The 2× and ¼× settings are worth trying too.

Playground

Here is everything at once: up to seven metronomes, any mixture of tempos, an adjustable board, and time running up to a hundred times faster than life. The speaker button in the corner of the page turns on ticking sounds — each metronome gets its own click, spread across the stereo field.

A few challenges: sync the lightest base at 208 beats per minute, which comes to agreement in seconds; find the heaviest base that still gets there before your patience runs out; set one metronome against six; give the odd tempo some company with two at ½×; and — the promised Huygens experiment — put two metronomes on the lightest base, turn the roller friction all the way up, and watch them lock in opposition instead of step.

The same story on a bridge

The metronomes are one costume that this phenomenon wears. For another, consider the Millennium Bridge in London, which opened in June 2000 and closed again two days later. With thousands of people crossing, the deck began to sway from side to side — gently at first, then hard enough that people spread their stances and held the rails. Video from that day shows the crowd doing something familiar: to keep balance on a moving deck, they were timing their steps to the sway, and so pushing on the bridge in unison.

The parts map directly onto our experiment. The deck is the board, except that it hangs on springs and would rather stay still. The pedestrians are the metronomes: each footfall pushes the deck sideways about once per second, and the deck's motion adjusts the timing of their steps. The one new ingredient is damping. A bridge constantly drains energy out of its own swaying, so a small crowd, pushing incoherently, excites nothing. Synchrony has to pay the damping bill before it can grow. Before you touch the crowd slider below, make a prediction: as the crowd grows one walker at a time, does the sway grow with it in proportion?

The answer is no — the sway does not grow, until suddenly it does. Below roughly eighty walkers the deck murmurs and nothing builds, no matter how long you wait. Past the threshold the sway ignites: a chance alignment excites the deck, the moving deck recruits more walkers into step, and the loop closes. The real bridge misbehaved with about a hundred and sixty people on a span, and the engineering fix — dozens of added dampers — amounted to raising the price of the wobble until no crowd could pay it.

Fireflies

Nothing in the story so far actually requires a mechanical connection. In parts of Southeast Asia, fireflies gather in riverside trees by the thousand, and after nightfall a whole tree will flash in unison. There is no board and no deck; what each firefly shares with the others is only the sight of their flashes.

The classic model of this, due to Mirollo and Strogatz, works like a charging capacitor. Each firefly's internal clock charges toward a threshold; on reaching it, the insect flashes and the clock empties. Seeing a neighbor's flash bumps a firefly's own charge upward — a small, discrete nudge in place of the board's continuous push. A firefly close to firing is bumped over the edge and flashes early; a freshly reset one is barely moved. Repeated over many cycles, this gathers the flashes together.

Watch for avalanches: one flash sets off fireflies that were near threshold, and their flashes set off more. With the coupling at zero the field twinkles at random forever; even a weak coupling gathers it into a common beat within a few dozen flashes. The same mathematics was first written down for the heart. The cells of the sinoatrial node — the heart's natural pacemaker — are electrical oscillators of exactly this charge-and-fire kind, and several thousand of them synchronize this way to produce every beat you have ever had. The firefly model began as a model of them.

Applause

Synchronization can also travel by sound. In many concert halls, an audience's applause will settle from a roar into a single unified clap — clap — clap. Physicists who recorded this found something curious: audiences get there by slowing down. Clapping quickly, people's natural tempos are spread too widely for any common beat to take hold. Clapping at half the speed, the spread of tempos narrows, and each person's slight drift toward the rhythm they hear around them becomes enough.

The demo starts at an eager tempo, too scattered to converge. Drag the tempo down and the room finds its rhythm; drag it back up and the rhythm dissolves. With the speaker button on you can hear both regimes. Here the shared, movable thing is not an object at all — it is the summed sound of the room.

What does the pulling?

In the equation, every one of these looks the same: a term that nudges each oscillator's timing toward the timing of the others. But that term is a placeholder. It says that one oscillator influences another without saying how, and the "how" is a different thing in every case — sometimes not physics at all.

Physically these have nothing in common: a force, a photon, an ionic current, a heard beat. What they share is subtler. In each, the disturbance's effect depends on when in the cycle it arrives — a nudge that lands just before an oscillator would have fired moves it more than one arriving just after. Synchrony grows whenever that timing-sensitivity, averaged out, tells the ones running ahead to wait and the ones running behind to hurry. The carrier can be almost anything. This one property is what has to be shared.

Only phases

Strip away the boards, decks, flashes and claps, and the same skeleton has been underneath every demonstration on this page: a collection of oscillators, each described by nothing but where it is in its cycle, each pulled a little toward the rhythm of the crowd. In 1975 Yoshiki Kuramoto wrote that skeleton down as an equation and solved it. His model has two knobs — how strongly the oscillators pull on one another, and how widely their natural tempos are spread — and it makes a sharp prediction: order appears only when the pull exceeds a critical value set by the spread, and when it appears, it appears suddenly.

Each dot is an oscillator moving around the circle, colored by its natural tempo from slow blue to fast red; the gray arrow is the crowd's average, whose length is r. At low coupling the dots spread evenly and the arrow stays short. Raise the coupling past the critical point and a clump condenses, gathering the middle tempos first while the fastest and slowest keep orbiting alone. The readout under the demo computes the critical coupling for the current spread — carry the coupling slider across it in both directions, or hold the coupling still and widen the spread until the order you built dissolves. The metronomes, the bridge, the fireflies and the applause are this picture wearing four different bodies.

Closing words

Huygens watched two clocks agree on a beam and called it sympathy. Three and a half centuries later, the same arrangement has been found in bridges and trees, in theatres and in every beating heart — and on this page we have watched it work through wood, steel, light and sound. The requirements are few: many things keeping a rhythm, each holding its rhythm loosely, and some way, almost any way, for each to feel the others keeping theirs. Given that, and time, they end up keeping it together.