“Pythagoras” is two identical sets of eight pipes that could be struck by seven different mallets each. The mallets are controlled by a bar that could be swung back and forth by an attached handle which the user controls on the platform. Before the repairs, I had never paid attention to its intricacies, partly because there was not much time to play with them in the time before the next train arrived, and partly because the old rusty version didn’t make great sounds and I thought they were just some randomly sized pipes. Plus, the handle lacked fine control, and the best one could do was to hopefully transfer as much energy as possible to even get the thing going.

When it came back new, it was looking much like a real instrument and now I wondered what else you could do with it besides swinging the handle back and forth like most people do. Surely you could play an actual melody, right?

Granted, I don’t think that was even an intent by the designer, as the mallet lengths are somewhat too similar. I even thought there were only two lengths, long and short, but upon closer inspection each mallet does have a unique length. So a little signal processing thought came to me that swinging the handle a certain way should allow individual pipes to be struck at designated times. So far I’ve only successfully separated the swinging of long and short mallet sets; that one is easy: they respond to two fairly different oscillating frequencies.

What really bothered me though was not knowing what notes were played by the pipes. It’s always hard to tell since multiple pipes are struck and then other multiple pipes are struck soon after, then the train comes and people talk. But I had some ideas… there was clearly a minor chord and a major chord in there, and nothing sounded chromatic. One day I took a photo of the installation thinking how hard could it be to figure out the frequencies from the pipe lengths…

So glad that the two sets of pipes are identical so I could correctly draw perspective lines, and roughly measuring the lengths with some arbitrary units I get: 14.0, 17.25, 18.0, 21.75, 19.75, 18.5, 16.0, and 14.5. Yet doing \(12 \log(L_0 / \mathbf{L}) / \log(2)\) gives consecutive oddball fractional intervals in the quartertone range and no way at all to reconcile with the diatonic scale even considering rounding error.

Turns out pipes are “metallophones,” and there is no column of resonating air inside like in “aerophones” such as organs, woodwinds, and brasses. Indeed, the timbre is different and the metal pipe itself bends back and forth to make sound. Fine, so it should be closer to struck and plucked strings, but those too have frequencies \(\propto 1/L\), what gives? So I looked up this book^* referenced in this paper and it says that metal pipes (and also stiff plates and stiff strings) have completely different physics than the more familiar tensioned strings, the restoring forces being internal elasticity and shear, some kind of fourth-order differential equation results that makes frequencies \(\propto 1/L^2\).^† Ok, I didn’t even know there was this third class of resonant behavior.

Now things could fall into place. Doing \(12 \log(L_0^2 / \mathbf{L}^2) / \log(2)\) gives: -6.55, -3.21, -0.95, 0, 1.47, 4.08, 7.49, 8.70 once the pipe lengths are sorted from long to short. There would be a plausible fit to the diatonic scale, if these semitones are rounded this way: -7, -3, -1, 0, 2, 4, 7, 9. The pipes are actually arranged like 9, 2, 0, -7, -3, -1, 4, 7, so the even and odd pipes (as the mallets swing back and forth between them) would indeed make major and minor chords, if only the fourth pipe were not a -7, but a -5 (the next closest rounding). So I went back to the station to listen to just the fourth pipe (hard to do), and I just don’t hear a -7, in fact, I hear a -5, so there is probably an unfortunate measurement error in the lengths of the longest two pipes. But correcting for that first interval, and taking into account the highest note sounds like B3 (easy to hear), the eight pipes probably sound the notes: B3 E3 D3 A2 B2 C♯3 F♯3 A3, and this result I actually believe.

† \(\frac{d^4Y}{dx^4} = \frac{\rho \omega^2}{EK^4} Y = \frac{\omega^4}{v^4} Y\) where \(v^2 = \omega K \sqrt{E/\rho}\). I think \(E\) is Young’s modulus or something and \(\rho\) is material density, \(\omega\) is radian frequency, and \(K\) is the radius of gyration (an RMS radius over a cross section, basically). \(Y\) is the phasor of the transverse displacement. Apparently the solution is: \(y(x, t) = \cos(\omega t + \phi) [A \cosh kx + B \sinh kx + C \cos kx + D \sin kx]\). For a bar free at both ends (our case), allowed frequency “modes” are: \(f_n \approx \frac{\pi K}{8L^2} \sqrt{E/\rho} (2n+1)\), where \(n=1,2,…\).

Could just use a docking station.

This is one of the nicer stations. Still looks like a 19th century dungeon, though; which of course, it is.


One of the nicest things about the subway stations is the porcelain-tile wall art. Since the trains are always late, one can spend a lot of time observing these oddities.

But have you noticed that it’s not trivial to make these pieces all line up and look nice — because the letter strokes make non-right angles? See how the tile alignments are fudged, near the bend of the letter Y on the left side? There is a long side of a triangular piece aligning with a side of a square piece, where the hypotenuse of the triangle has to be a little bit longer. So they just jam it in there. It sticks out a little bit.

And here is a letter V. Clearly when they do the tiles, they make each line of tiles for the \ strokes before the corresponding / strokes, because the \ tiles run longer.

These were all taken at the station called 23rd and Ely. Of course the station names don’t correspond to where the exits are. One stop at 49th Street actually produces exits mostly on the 47th Street. Another at 42nd Street actually opens onto 40th Street. Go figure. Actually, station names in Manhattan itself are almost consistently “wrong” in this way, which leads me to believe that Manhattan streets have been renumbered at some point.

The porting process was as follows:

• Immediately, random number stops working, new Sprint-assigned MSID works as phone number!
• Within minutes, AT&T cuts off connection and accounts access to old number, which ceases to work.
• Hours later, MSID stops working as phone number, old number now rings new phone.

The fact that the MSID works as a phone number during porting and remains in the phone’s settings, must mean that it is in fact the true “phone number” identifier for the phone. The phone just gives out its stored “phone number” for the purpose of outgoing calls. I’m guessing every carrier has a pool of phone numbers in each area code to give out, and there is a static allocation database somewhere. During porting, Sprint assigns a new number from its own pool, which becomes the MSID. Then AT&T (who owns the old number) changes its databse to forward calls to the old number on to Sprint, instead of processing them internally. Finally, Sprint changes its database to take those incoming calls and forwards them to the assigned MSID.

When you get a new line, a number from the carrier’s pool is assigned so no forwarding is needed, and that must be why the MSID matches the phone number in that case.

Now, if I were to port again to a third carrier, what would they do? Maybe they’ll look up the number and discover they should talk to AT&T?

Here’s some real information which I haven’t read but maybe corroborates or discounts what I wrote. http://www.syniverse.com/pdfs/GuidetoWNP6thedition.pdf