# You are getting sleeeeeeepy…..

By Julianne Dalcanton | October 1, 2012 1:33 pm

Ikeguchi Laboratories has posted one of the most fantastic “physics in action” videos I’ve seen in a long time:

The concept is simple — 32 metronomes on a table, all set to the same tempo, but started at slightly different times. But here’s the fun bit — although they begin “out of phase“, after about 2 minutes, they all lock onto the same phase and synchronize! (Well, almost all — there’s a rebel on the far right that takes an extra minute to get with the program).

So what’s going on? The key is that the metronomes are not on a solid table, but instead are on a slightly flexible platform hanging from a string. Thus, as a metronome’s pendulum rod changes direction, it imparts a small force to the platform, which leads to small motions in the platform. The moving platform then gives small nudges back to the metronomes. These forces will tend to push the other metronomes to speed up or slow down to match the timing of the original metronome, bringing the metronomes “in phase”.

Now the really fun bit (for me at least), was watching exactly how this played out in practice. If you watch the video closely, you’ll see that the synchronization does not happen all at once, nor does it happen randomly. Instead, the synchronization tends to take place first in pairs, with adjacent metronomes locking onto the same phase. This behavior makes a lot of sense, because the strongest forces on a metronome will initially be from its nearest neighbors, at least until enough metronomes are in phase that you start getting a large scale coherent swaying of the platform (which starts to happen about a minute in, becoming increasingly strong during the next minute). The pairs are also more likely to be oriented along the rows (side-by-side), reflecting the direction in which the metronomes cause the platform to move.

The other phenomena you can notice is that adjacent pairs will frequently spend quite a bit of time “180 degrees out of phase” (i.e., showing totally opposite behavior from the neighboring pair — going “tick” exactly when the other goes “tock”). If you watch one of these sets, after happily going along for a while, the equilibrium will shift, and the pairs start to change their tempo. The relative phases will shift, and will gradually drag the pair that’s 180 degrees out of phase back in line with the rest of them, before reverting back to the natural frequency of the metronome. This behavior is probably clearest in the “rebel on the right”, which is in a quasi-stable equilibrium, and spends an extra minute beating a syncopated tempo.

So, lots of interesting stuff to see in a remarkably simple set-up!

(Ed: In comments, Andy Rundquist linked to a post of his analyzing an earlier metronome video, along with bonus Mathematica code, along with a link to a much older publication about modeling the system.)

(Ed: Aaaaand now Paul Gribble (in comments) has just checked relevant Python code into Github.  Y’all are nuts.  My kind of nuts, but nuts.)

CATEGORIZED UNDER: Science
• HP

I’ve often wondered how this phenomenon would play out if the metronomes were set to integer ratios of one another instead of all at the same tempo. For example, what if one-third of the metronomes were set to 60 bpm, one-third to 120 bpm, and one-third to 180 bpm?

• Julianne Dalcanton

Ooooooh! That would be fun.

How much is a metronome?

• T Monroe

If metronomes don’t make you sleepy, try Pendulums

• HP

Julianne, Sam Ash has a Wittner mini mechanical metronome (similar to the one in the video) for 38 bucks. (Not a commercial endorsement; just the first relevant link I could find to a respectable music retailer.) So you’re looking at over 12oo dollars worth of metronome to replicate the video.

Like everything else, most metronomes are digital these days (or iOS/Android apps), and the few remaining mechanical metronomes still being made tend to be prestige items.

• Pieter

This is very cool. But I find there is something vaguely fascist about them all going in lock step…

• Brett

The principle of superposition!

• http://arundquist.wordpress.com Andy Rundquist

I don’t think the argument about nearest neighbors is necessary. When I’ve modeled this, I have the oscillators all on a solid substrate that can shift left and right. The motion seems very similar to this movie. For this movie, the fact that the surface can move vertically as well, indeed, by different amount depending on your location, I would suspect there could be a nearest neighbor effect.

• Julianne Dalcanton

Thanks for pointing out that link, Andy. I definitely envisioned that the up-and-down flexibility (bouncing) was the only feasible driver of the nearest neighbor effect given that the platform isn’t stretchy, but I wasn’t explicit.

• Fermi-Walker Public Transport

Wow, that is what I call “peer pressure”.

• Beth Drissell

Reminds me of humans walking on a swinging bridge!

• GoddessAnonymous

Virgos will love this, LOL!

• DB

In actual fact this video has little to do with physics per se. I am reliably informed by an acquaintance who works at Ikeguchi Labs that investigators videoed over 3000 trillion iterations of the ‘experiment’. Of those, they then posted video of the iteration selected to maximize the number of you tube hits.

• Richard M

Pieter, I had almost the same thought. I think this is an analogy to how political and religious movements form.

On a smaller scale, I thought of bunches of neurons firing — chaotically at first, but gaining coherence over time. So perhaps this reflects how our brains work?

DB, that’s hilarious. Maybe your acquaintance is right. At least, there’s a Colbertesque truthiness to his or her statement, so it must be true!

• http://www.originalenclosure.net Ben Marshall-Corser
• tt

DB, i am reliably informed by math that 3000 trillion iterations would take
22831050228 years

• Georg

but started at slightly different times.

Wrong!
The only relevant time scale in this experiment is the
duration of one period, compared to that the starting
time diffences were huge!
In fact, all phase relations up to 2pi were happened at start.
This phenomenon is called “coupled oscillators”, well known
since centuries. If the oscillators are precise enough
(=have small bandwith, eg good pendulum clocks)
just fastening them on the same wall is enough to synchronize
them.
Georg

• http://x-sections.blogspot.com Rhys

I thought I understood this, and then I didn’t, and then I read Andy’s blog post…
So it’s not really just coupled oscillators, but relies on a driving term (i.e. internal energy source of each metronome) which tries to keep each oscillator at a fixed amplitude?

• http://arundquist.wordpress.com Andy Rundquist

@Rhys: the driving term certainly does the trick, and, according to the AJP article, is how metronomes actually work. I wouldn’t be surprised, though, to find that other mechanisms can accomplish the same synchronization.

• nato

Socialism works!

• OMF

And thus their came in the midst of the Global Rescession, and sudden surge in demand for metronomes from Physics departments throughout the world.

• http://swingthebat.net Phil P

For tonight’s homework, set up the Lagrange equations for 32 coupled oscillators …

• James Gallagher

This obviously explains why sleep evolved in animals, otherwise, if everyone in China jumps up and down at the same time…

• http://www.gribblelab.org Paul Gribble

After looking at various papers on the topic I found some equations of motion, did some algebra and wrote some python code. Here is the result

video of simulation results animated:

python code:
https://github.com/paulgribble/metronomes﻿

• James Gallagher

Basically, the system has a dynamical attractor which is an eigenvector of H^2, where H is the Hamiltonian (so a period-2 attractor wrt H)

Or is there energy loss in the system?

• Ian Danforth

If you think this is cool you will love Sync: How Order Emerges From Chaos In the Universe, Nature, and Daily Life by Steven H. Strogatz (http://www.amazon.com/Sync-Emerges-Universe-Nature-ebook/dp/B0072M0X2Y/)

For example after reading that I can answer HP’s question and say ‘No’ the sync of coupled oscillators will never happen if the members of a population are too far out of sync to begin with.

He goes into how Fireflies flash in unison, how this metronome effect works, and how all of this is modeled by crazy high dimensional math in which you discover ‘strange attractors.’

I highly recommend it!

• http://skepticsplay.blogspot.com miller

Andy, I like your explanation that all but one mode dies out, while the last is driven by energy from the metronomes. However, the first explanation that came to my mind was that the pendulums are nonlinear oscillators. If a pendulum is behind the others, the motion of the board would drive it higher, causing it to have a longer period. If a pendulum is ahead of the others, the motion of the board would drive it lower, causing it to have a shorter period.

But I’m not sure if this works as an explanation without going through the math or making a simulation. Did your first simulation account for pendulum nonlinearity?

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• http://arundquist.wordpress.com Andy Rundquist

@miller: The “normal modes” thought that I had with coupled pendula (with the full nonlinear equation of pendulum motion) didn’t do the trick, though I suspect at tiny amplitudes you might see it. The van der pol type term that I added into the frictional force (add energy below a threshold amplitude, take it out above it) seems to model metronomes quite well and when I put that in the coupled, nonlinear pendula, I get the synchronization.

• James Gallagher

(previous post wasn’t formatted correctly it can be deleted)

I had trouble getting Paul’s script to run in ipython so I rearranged to code to enable it to run from a command line (“python metronomes.py”), download here. You need NumPy, SciPy etc packages, in Fedora just do “yum install numpy scipy pylab python-matplotlib”, in Ubuntu do “sudo apt-get install python-scipy python-matplotlib”. The metronome animation runs first for 60 seconds and then the graphs are displayed.

Also, I couldn’t get Andy’s mathematica code to work in Mathematica 6.0, if anyone wants to try on a newer version the notebook is here , or maybe explain what I did wrong?

• http://bkinsman1Aaol.com Bill Kinsman

OK: There are 31 followers but only one leader. Figure out which single metronome is calling the shots. Take as much time as you need.

Something that I saw, which no one seems to have mentioned, is what happens at long times after the initial synchronization. You can see that a few metronomes now and then try to “escape.” They start to wander away, but are then brought back into lock. It must be a quadratic minimum of some sort with a pretty flat bottom, but I’m trying to understand how an escape is allowed to get started and then grow to a certain size before it’s yanked back…

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• James

I note that before all coming into synch, a lot of the metronomes became exactly half a cycle out of phase with the bulk (in fact, the last one was exactly out of phase with all the others for a while). This is no doubt explainable by the physics of the situation – I suspect that those not in or exactly out of phase are dragged towards one of those two extremes by the motion of the table, with the motion of the majority eventually pulling those exactly out of phase into line. It would be interesting to see whether or not this would happen if the experiment was set up with half of the metronomes in phase with each other and the other half in phase with each other but exactly out of phase with the first group. In some cases, it might result in all of the metronomes ending up in phase, in others (depending on their location within the pattern and therefore the dynamics of the table’s motion) it might remain a 50/50 split…

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• Mike B

Just a small correction: the article states that “These forces will tend to push the other metronomes to speed up or slow down to match the timing of the original metronome”

This is incorrect – the metronomes will ALL, including the first, be pushed to speed up or slow down to match some combined or average timing of ALL the metronomes (ie, the metronomes all impart a motion on the table, and each metronome is affected by the motion of the table). Not to match the timing of the original metronome, as this implies that only the first metronome has any effect on the table.

• Matti Lamprhey

What you are not told is that this video is actually being played in reverse. Simple when you know how.

• theagent57

just a shame that you cant see the edges of the board,and the help it receives at about 90 seconds in.

• Mike Wray

This all reminds me of Rutherford’s theory explaining why grandfather clocks tend to stop on Thursdays. Such clocks, driven by a falling weight on a cord, are generally wound up on Sundays. By Thursday the weight has fallen by a distance that makes the length of its cord equal to the length of the pendulum. You then have two oscillators of equal frequency coupled through the slightly flexible frame of the clock. Add in a few losses into the system, and, lo, the pendulum feeds energy into the weight on its cord – and stops.

Does anyone have 32 grandfather clocks?

Oh look, they are all synchronised, what are the chances of all the metronomes being synchronised? I therefore say that god must have put them there!

• http://None Fred Vaughan

Movement of the air surrounding the metronomes must make a considerable contribution, even if the platform were rigid and fixed they might still come into synch; nonconformists would be slowed by turbulence until they were “saved”. How about trying the original setup in a vacuum to see if they take longer to get together but please don’t ask me to start them off.

• http://new-savanna.blogspot.com/ Bill Benzon

I make synchrony the centerpiece of my book on music, Beethoven’s Anvil: Music in Mind and Culture, where I discuss Strogatz on fireflies and Barbasi on synchronized clapping. I argue that, when people make music together they individually give up so many degrees of freedom that the overall neural-state space for the group is no larger than that for any one independent individual. And that’s a good thing, otherwise no one in the audience would be able to make sense of the performance as no one has more than their own brain available to make sense of the sound. I’ve got a fair number of posts on this and related subjects at my blog, New Savanna. Start with this one, Cooperation, Coupling, Music, and Soccer and then look at onther posts on coupling.

• Peter Lund

“Movement of the air surrounding the metronomes must make a considerable contribution, even if the platform were rigid and fixed they might still come into synch;”

Didn’t Huygens actually perform this experiment with pendulum clocks hanging on a wall with or without an air barrier between the pendulums?

He sure did!
http://en.wikipedia.org/wiki/Odd_sympathy

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