“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 \(n=1,2,…\).

Kendall goes on to question the utility of free speech as understood by Mill and lists a number of arguments based on practicality and human nature against the idea of an “open society” actually working as Mill intended, all of which are probably valid but ultimately unsatisfying. However, one extended philosophical point stood out:

Third, Mill denies the existence … not only of a public truth [my note: for the purpose of lubricating free debate], but of any truth whatever… whenever and wherever men disagree about a teaching, a doctrine, an opinion, an idea, we have no way of knowing which party is correct; the man (or group) who moves to silence a teaching on the ground that it is incorrect attributes to himself a kind of knowledge (Mill says an “infallibility”) that no one is ever entitled to claim short of (if then) the very case where the question is sure not to arise — that is, where there is unanimity, and so no temptation to silence to begin with.”

….

The proposition that all opinions are equally — and hence infinitely — valuable, said to be the unavoidable inference from the proposition that all opinions are equal, is only one — and perhaps the less likely — of two possible inferences, the other being: all opinions are equally — and hence infinitely — without value, so what difference does it make if one, particularly one not our own, gets suppressed? This we may fairly call the central paradox of the theory of freedom of speech. In order to practice tolerance on behalf of the pursuit of truth, you have first to value and believe in not merely the pursuit of truth but Truth itself, with all its accumulated riches to date. The all-questions-are-open-questions society cannot do that; it cannot, therefore, practice tolerance towards those who disagree with it. It must persecute — and so, on its very own showing, arrest the pursuit of truth.

There is something interesting here, although first the wrong parts must be excised. For one, I don’t like the infinitely valuable or infinitely without value sentence. That’s stupid. The only conclusion is obviously just what the original says, that all opinions are equally valuable, so as long as the value is positive, then at least some argument can be made to not suppress them. As a side note, I’m not even sure that Mill says all opinions are equal or whether just all opinions have positive (but possibly unequal) value. Kendall’s scaling relationship between the value of an opinion and how much it should not be suppressed may not be Mill’s idea at all. Either way, we can treat the phrase equality of opinions as either what it says or as positive valuation of opinions, and read it as equal treatment of opinions.

Now then the good part. The meta-tolerance paradox itself is a bit contrived (even if it does come up almost daily) since once the “no truth” hammer is found, it can hit pretty much anything. But, it caused me to think of a related and much more relevant paradox. It isn’t about believing in the Truth a priori. It is about believing that there is some truth to be found — that the pursuit has an end, that there is a purpose to the debate. Otherwise free speech for that purpose would have no point, either. Now, if the purpose is to seek convergence from open debate, then there is certainly nothing to guarantee that the process of debate will ever converge (setting aside what it converges to) — not because there is not a truth. But even worse is if every opinion is to be equally valuable for all time, for it prohibits convergence! For convergence to happen, by definition, some opinions will need to be reduced and others bolstered, perhaps on account of reason. The only way out of this is to say opinions are equal only initially. Unfortunately, society has no time origin. Even if it did, we are way past time 0 and hence must be in a state biased toward certain possibilities of truth and therefore inequality of various opinions. Therefore, it is the inequality of opinions on the way to convergence to truth — the ostensible goal of free speech — that strikes down the non-suppression of some speech on the grounds that all opinions are equal. This is the central paradox of Mill’s thesis that I see.

I guess the point is that non-suppression must be based on something less absolute than what Mill says, as it is in practice.