Coupled Metronomes
Twelve metronomes started at random on a platform that can roll. Within a few seconds they are beating together. Knock one out of step and the others pull it back. This is the one card in the column that is not refused, and it is here so the refusals mean something.
A column of eleven refusals never shows its test succeeding, which makes the test look like a way of saying no. So here is the same apparatus shape with the opposite verdict. Every other card fails on one question: does anything read the system's own state and feed it back? The Galton board's pile is never read by anything. The painted bar never reads the grid. Here the platform reads all twelve metronomes at once, by being pushed by them, and hands what it reads straight back. That loop is the whole difference, and it can be switched off with a button.
The physical claims are quoted from the paper listed at the bottom, with the date read. The panel is a simulation of the model that paper describes, and every number quoted from it was measured from this code before the page was written.
On this site a basin is the set of conditions a system comes back from. Push a marble around the inside of a bowl and it rolls back to the bottom; the bowl is its basin, and the rim is its edge, the push beyond which it does not come back. Finding basins, and their edges, is the point of the Laboratory. This bench has one basin, all twelve metronomes beating in step, and it is the one card in the column whose return the system makes for itself. The test for that is a button that switches the basin off.
| basin | what it returns to | what pushes it | its edge |
|---|---|---|---|
| 1. In step | all twelve swinging together, phase lock 1.000 | Shove one pendulum: phase lock drops to 0.98 | Clamp the platform: the pull is gone and phase lock sits near 0.14; the weight slider never reached the edge |
1. In step. Started at random, the twelve gather within a few seconds and stay together. The panel's phase lock readout measures how nearly they swing as one, where 1.000 means every rod is in step. Press Shove one pendulum and it drops to 0.98; the others pull the knocked metronome back and it returns to 1.000. The return is real, and the metronomes supply it themselves. Each one's swing pushes the rolling platform, and the platform's motion pushes every metronome on it toward a shared beat. The energy comes from each metronome's own escapement, the mechanism that feeds it from the wound spring. The pull toward the shared beat comes from the twelve, through the board, and nothing outside sets the beat.
Sliding the weight up metronome 1's rod is the second of the two disturbances, and it is a different kind. It changes that metronome's parts, slowing its natural tempo to 0.56 of nominal at the top of the rod. The bench absorbed it at every position: phase lock never fell below 0.959 (at 86% up), and it held with the board made five times heavier.
The edge: Clamp the platform. Holding the board still removes the only path between the metronomes. Each keeps its own time, and phase lock sits near 0.14 and never rises. Release the board and they gather again. The weight slider does not reach the edge in this model. A weaker coupling would fail somewhere; that point was not measured. A real bench also loses its lock when the springs wind down, which the panel does not model.
What the bench is
Pantaleone's version is as plain as apparatus gets: "Our system consists of two metronomes resting on a light wooden board that sits on two empty soda cans". The board can roll. That is the entire coupling. Each pendulum's swing pushes the board a little, the board moves, and every metronome on it feels that motion as a shove at its own pivot.
His statement of the mechanism is one sentence: "The small motion of the base couples the pendulums causing synchronization." And the result: "Synchronization was attained on a time scale of order tens of seconds."
What keeps each metronome swinging at all is the escapement, which "controls the speed and regularity of the pendulum" by feeding it energy from the wound spring. Pantaleone models that term as being of "the van der Pol type", pumping the swing while it is small and bleeding it while it is large, which is what puts each metronome on a limit cycle of its own before any coupling is considered.
each pendulum pushes the platform → the platform moves → its motion shoves every pivot on it → each pendulum's swing is altered → each pendulum pushes the platform
Compare that loop with the one on the Galton board page, where the pile appears once, at the end, and nothing downstream of it comes back around. Here the state of the ensemble is an input to the ensemble, continuously, and there is no way to describe the apparatus without saying so.
Why this one passes
The discriminator the column turns on is recursion lock (SOV Condition 1): does
the structure close a feedback loop on its own prior state? Every other card fails because nothing
reads the state. This one passes because the platform reads it mechanically, and the proof is that
the reading can be stopped.
Clamp the platformHold the board still and nothing else changes. Every metronome keeps its own perfect time, driven by its own escapement, and they share nothing. In the panel below, clamped, the order parameter sits near 0.14 and never rises. Release the board and they gather again. One button, one path, and the synchrony appears and disappears with it.
That is a cleaner demonstration than any argument, because it isolates the loop instead of describing it. A false basin has no such button. There is nothing to clamp on a Galton board, because there is nothing carrying state between one ball and the next.
The two disturbances
This is the part worth the whole page, and it is the distinction the column's other cards cannot even pose, because they have no state to disturb.
Shove a pendulumKnock one metronome out of step. Nothing about the apparatus has changed; the system is the same system, sitting somewhere else. The others pull it back. Measured here, a shove drops the order parameter to 0.98 and it returns to 1.000. That is basin return, and it is the thing every other card in this column only imitates.
Slide the weight up the rodNow change one metronome's parts. Raising the weight lifts its centre of mass and adds inertia, so it wants to swing slower. This is not a displacement of the state, it is a change to the equations. The ensemble is not being asked to restore something, it is being asked to cope with a different member.
The two look alike on the bench. One is a shove, the other is a slide. They belong to different categories, and a framework that cannot tell them apart will call both of them a perturbation and expect both to recover. What this bench does with the second one is measured in the section below, and the answer was not the one this page originally gave.
The bench, run here
Twelve metronomes, started at random, natural periods spread by under one percent. The first one is marked, and its weight is the one that moves. Let it lock, then try the three interventions in any order.
How to read the graph. One number over time: how nearly the twelve are swinging together. The line at the top means every rod is in step. The line at the bottom means they are scattered. Time runs left to right over the last ninety seconds. A dip that climbs back is the ensemble absorbing something; a drop that stays down means the lock is gone. Blue ticks mark a shove, red ticks a move of the weight, white ticks the platform being clamped or released.
What the weight slider actually does
The first version of this panel showed the lock collapsing once the weight passed about four-fifths of the way up the rod, and this section said so, with a table. Then a reader clamped the platform while the lock was broken and found metronome 1 frozen solid, and still frozen after the clamp came off. Chasing that turned up three faults in the model, all of them mine:
- A rod knocked to a standstill could never restart. The escapement term is proportional to the pendulum's own speed, so a stopped rod is an equilibrium of it. With the board clamped there was nothing left to disturb it. From a small seed it needed 33 seconds to climb back into view.
- The rods had no end-stops and the platform was driving them to eighty degrees, which no metronome does.
- The board had no centring force and drifted half a metre, straight off the canvas.
All three are fixed above. The rods hit the case at 1.15 radians, the board stays on its rollers, and a rod that has not moved for three seconds gets the impulse a wound spring would give it.
And with those fixed, the collapse went away. The corrected bench absorbs the detuned metronome at every position of the slider, and it still does with the board made five times heavier:
| weight position | natural tempo | phase lock |
|---|---|---|
| bottom | 1.00 | 1.000 |
| 40% up | 0.82 | 1.000 |
| 70% up | 0.69 | 0.999 |
| 80% up | 0.65 | 0.998 |
| 86% up | 0.62 | 0.959 |
| 90% up | 0.60 | 0.967 |
| 95% up | 0.58 | 0.974 |
| top | 0.56 | 0.973 |
So the honest reading is the opposite of the one this page first gave. The coupling through the board is strong enough to entrain a metronome that wants to run nearly twice as slow as the rest, and the spectacular failure shown before was an artifact of unbounded swings and a runaway platform. The distinction between shoving a pendulum and rebuilding one still holds, because it is a distinction about what kind of change each is. What does not hold is the claim that this bench can be pushed past its limit by the slider. Within its range, it cannot.
Recorded here rather than quietly corrected, because a page that only ever confirms its own first draft is not measuring anything.
What would show this reading wrong
Synchrony that survives clamping the platform. If the twelve gathered with the coupling path removed, something other than the platform would be carrying the state, and the reading of the loop given here would be wrong about what reads what.
A shove that does not come back. If a displaced metronome stayed displaced while the apparatus was unchanged, there would be no return to measure and this card would belong with the others.
A demonstration that the panel's escapement model, rather than the coupling, is what produces the lock. The clamp test is the guard against that, and it is in the panel rather than in an argument.
Honest limits
Nothing was built or measured on a bench. The numbers on this page come from the panel's own model, run offline before publication, and they describe that model. Pantaleone's paper is the source for the apparatus, the coupling mechanism, the escapement and the timescale.
The paper studies two metronomes and discusses more; the panel runs twelve, which is a choice about what reads well rather than a replication of the published experiment.
The spread of natural periods, the platform mass, its damping and its centring are chosen so the bench locks in a few seconds and sways visibly. They are not measured from hardware. The finding that detuning is absorbed across the slider's range is a finding about this model at these settings, and a weaker coupling would fail somewhere.
The case walls and the let-off are modelling decisions, not measurements. Both were added to fix faults found by running the panel, and both change what it does: the walls in particular are why the lock no longer breaks at the top of the rod.
The order parameter reports phase agreement at one-to-one. It reads low when the ensemble is locked at another ratio, which is why the table above names the ratio separately.
Sources
Pantaleone, J. (2002). Synchronization of metronomes. American Journal of Physics, 70(10), 992–1000. Source of the apparatus, the coupling through the base, the escapement description, the van der Pol form of the driving term, the timescale of synchronization, and the spring running down. Read from the author's copy at the University of Alaska Anchorage. uaa.alaska.edu, metro.pdf
Related on this site: False Basins, the column this is the control for; the Galton Board and the Clock-Driven Pattern, which look like this one and are refused; and the taxonomy for the axis this card sits on.