It’s a bit more complicated than that

A proposal:

At around age 14, set aside a week away from the standard education curriculum for kids to work, in groups, in a very focused way: on a project with a simple brief, but a complex reality.

What does that mean?

Well, it might be to design a way of giving everyone in the UK £100 (as an alternative quantitative easing approach). It might be to identify all public buildings to find better ways of using them. Or to model what would happen if jails were abolished (or if speeding convictions carried automatic jail sentences). Or to design a rail system that would be ultra-resilient to sudden, massive demand and freak weather conditions.

Anything really.

Or at least anything that would show that a bit more effort is required in reality to do some of the things that really matter in this world. Even though they might sound simple. So that first one: giving everyone £100? Well, you’d need to work out who “everyone” was…what would qualify as entitlement…how to get the money to them securely and trackably…how to deal with claims that it hadn’t been received (true and fraudulent)…how to deal with those who didn’t want to be on any state registers but still wanted their cash… You get the picture. Putting real world details around a nice, simple concept.

You’d cover analysis, planning, teamwork, logistics, consequences (seen and unforeseen). And probably a whole lot more. You’d learn about edge cases, the ability of a small number of difficult situations to eat up disproportionate resources, and how you have to design for the awkward, not just implement for the easy.

And out of all this, there might, just might, be a tiny chance that statements like “well, I don’t see why they can’t just…” or “how hard can it be to…” would be cast around just a little less lightly. And questioned a little harder when falling from the mouths of politicians.

Because the problem is this: when we’re small, our world is small. And simple. Decisions are clear, motives unambiguous, morality absolute. Things are, or they are not. Laws are clear, enforceable and enforced.

The King says “make it so!” and the Knights make it so. The Princess makes her choice, and the losing suitor slinks away, never to play a part in this or any other story.

And so it goes.

And then things change. Our world gets bigger, and more difficult. We realise that society is a loose patchwork of consents, of unwritten codes, of behaviours.

And do we change, too? Or are we content to carry on with an increasing pretence that the world is monochrome, that things can be made to happen by dictat, and that anything involving sixty million people need not be any more complicated than something involving a handful?

Do we continue with these childish fantasies, and follow leaders who–even if they believe in their hearts that what they propose is at most only partly achievable–must dance the dance of the simple: spouting policies that can never be delivered, just so they continue to look…like what? Like leaders. Right.

Left-wing and right-wing, we dance up and down the same spectrum of choices: of levers that can be pulled in various directions, of societal mores running from the brutal to the soft, the feudal to the flattened. We might choose different starting positions, and have certain favourite themes and moves. But if we get stuck with these lame little models of “why don’t they just…”, and come to believe that wickedly difficult problems are actually easy, then we’re all stuffed.

Because we do believe. On a mass scale. Because we were never taught any differently. We weren’t taught to think harder, to go deeper, to challenge rigorously, or to live the reality of what implementation might actually be like.

And so things like social and economic policy get very broken. Preposterous, simplistic “solutions” float around: hoodies marched to cashpoints, rioters’ families evicted, a single ID number, watertight borders, cities scoured of benefit claimants, a single central health record for everyone…the list goes on. (That last one would make a great school project, by the way. Starter: think who might need to view and/or change that record–including the patient–what their interests and motivations might be, and how all those agendas stack up against the benefits.)

It’s an awful lot easier to believe in simple magic than to work through hard science. And very much easier to whip a crowd up behind you, too. Asking those hard questions has become the antithesis of leadership. What a splendidly vicious circle!

And yet, it can be broken. With so many other large scale problems of capability or understanding, we try to fix things at source. Through education, for example. It beats me why we’ve never seriously attempted this route.

Would a more aware, canny, and yes perhaps cynical population really be that frightening? Or would we sniff out the stupid and actually become far more tolerant as a result?

I think it’s worth a try. Be a hell of an interesting week, anyway.

(It took me until I was 17, and half-way through A-level Economics, before the reality pennies started to drop. That transactions mostly have two sides to them. That someone’s good deal is often another’s bad one. That public finance doesn’t begin with some kind of magic money tree in the Treasury courtyard. That expectation can be as powerful an influence as factual evidence. That people don’t always do what you expect them to, even when we have laws to force them to. It felt like a real privilege to have my mind stretched like that, and I’ve always felt a bit more of it could go a long, long way.)

Imperfect harmony – continued

It’s over a year since I wrote the first part of this. I was trying to show how there was something very strange going on in the way we’re generally told that music works; something that when I found about it became a powerful metaphor for hidden complexity, fakery and the acceptability of approximations.

Where had we got to? That using various perfect mathematical ratios to divide a vibrating string we could build up a series of musical notes. Playing some of these together produces pleasing harmonies, because of the way the waveforms behave together.

Pythagoras got onto this pretty quickly. You can read a thorough account of his theory here. Cutting to the chase, he described a bunch of different notes using nice whole number ratios. We can order these notes into something we’d call a scale. C, D, E, F, G, A, B, C – that sort of thing.

The great thing about an ordering like this is that you can keep on building… Scales can start on different notes, you can introduce half-way steps between the notes to help make it all work out. C# becomes essential to help make the key of A major work properly, and so on.

I used to think it was amazing how this maths all worked out: how there could be 12 perfectly spaced semitones on a piano keyboard that could make up any scale or key you wanted.

And then I realised that it didn’t work out. Almost exactly, but not quite. And the fact that the maths so nearly works out perfectly is, for me, the most fascinating part of all this. It means that music theory can get away with a bit of cheating and approximation.

How so? Well, take the Pythagorean scale at face value. Each time you get to the fifth note in a scale (so that would be G in the scale beginning with C) you have a “perfect fifth” with a harmonic relationship 3:2 times the frequency of the C you started with.

Take the fifth (G), and build another perfect fifth on that (D). Go from D to A, and so on… eventually, after working through all the 12 semitones in an octave (see what I meant in the previous post about “oct-” being a bit of a red herring?) you get back to the C you started with. It all neatly works.

But it doesn’t. And it can’t. Here’s why.

Go up 7 Cs (each an octave higher than its predecessor). You end up with a pretty high note of course, but because you’ve doubled the frequency each time it’s fairly obvious you are 2^7 times higher in frequency than your starting point. That’s 128 times the frequency.

Now, do it the other way – going from perfect fifth to perfect fifth. You have to go through a few more cycles because you’re not jumping as much as an octave each time. In fact, you need to go through 12 cycles, which sort of makes sense as you’ve covered all the 12 semitones in turn as starting points. The eventual frequency you reach is (3/2)^12. Got a calculator? Well, it’s awfully close to 128. In my humble view, weirdly, bizarrely close. It comes out at about 129.746. That’s only 1.36% out from the C you would have arrived at from just doubling each time.

You’ve gone up 7 octaves and you’re only just over 1% out from where those neat harmonic intervals would have taken you? Doesn’t that strike you as rather interesting?

This tiny difference is called the Pythagorean Comma. It has to exist because the Fundamental Theorem of Arithmetic says you can only make up a number by multiplying one set of smaller prime numbers together. So you couldn’t get to 128 by multiplying anything other than 2 by itself 7 times… So those neat Pythagorean ratios were always going to run into trouble at some point.

So what? The big ‘what’ is that because the Comma (= ‘gap’) is so small, it only takes a bit of tweaking to the Pythagorean ratios to get 12 intervals that DO fit exactly into a repeating pattern. You just have to make some of the notes very slightly lower, and some slightly higher. But eerily, the adjustments are so small that one hardly ever notices.

That’s how a piano could be built. A modern (post c17th or so) piano usually has these adjustments (called temperament) made to its tuning so that each interval is exactly the same. The same goes for most instruments where the notes are ‘fixed’ in its construction (like a guitar, with fixed frets, but unlike a violin where the player has discretion about where precisely the strings are stopped with the fingers).

Because of this mathematical curiosity – that tiny Pythagorean Comma – a totally flexible muscial system can be built. So we can have jazz, infinitely complex harmony, tunes that modulate smoothly from key to key, and far greater flexibility than if we were stuck with Pythagoras’ precise spaces between the notes.

The trade-off is that most things we hear as harmonies in music actually aren’t. Very, very slightly, the whole structure of music involves ‘cheating’ to make it work.

And I think that’s a fantastic story-behind-a-story. Tiny compromises, and acceptance of approximation, produce a far more beautiful (if slightly ‘wrong’) end product than could rigid adherence to the rules.

Lift your spirits

There’s a lot to a lift.

If you fancy yourself as a bit of an analytical thinker, go and get a piece of paper and a pencil. Think of a lift you know. It doesn’t matter whether you love it, hate it, or have no particular feelings and just think of it as a means of changing floors in a building.

Now, for exactly 10 minutes – time yourself – make as many notes as you can about the factors involved in answering these questions:

“What has made this lift like it is?”


“How should this lift be?”.



Turn off the screen. Go on. Really – Turn It Off.
















Anything interesting emerge? Maybe, maybe not. How much did you generate? Anything there about logical behaviour, customer service standards, risk, accessibility, aesthetics, lifespan, safety, efficiency, queuing theory, optimisation, heuristics, geographical location, environmental impact, user health, cost, policy, procurement, politics?

What I love about lifts is that the basic premise is immensely simple and transparent. If you have two floors in a building, a box on a wire can take you between them. With the exception of odd twists such as hydraulic mountings this is essentially all they do. In a sense. But as soon as you start to introduce additional variables: number of floors, number of lifts, uneven distribution of users over time, etc. things can get very complex very quickly. Which makes “what has made this lift like it is?” a pretty interesting exercise in analysis. Finding the story-behind-the-story.

Oh, before going back to the paradigm which will be adopted from here onwards of “a box on a wire” it’s worth briefly remembering the first lift that really caught my attention. My faculty building at university had a splendid creation called a Paternoster. (Pub mythology has it that they’re now illegal. I wouldn’t be surprised.) Every floor of an eight-storey building had two floor-to-ceiling open hatches next to each other. In the two shafts that lay behind them circulated perhaps 20 platforms on a belt stretching the entire height of the building and back again, naturally. At any point in time eight platforms would be ascending, eight descending, and I guess two each at top and bottom going round the wheels in the basement and 9th floor machine room. For the avoidance of doubt, the platforms were on gimbals – should one ever accidentally travel the voids at top and bottom (and who didn’t?) it wasn’t a question of being hurled from floor to ceiling as the ‘virtual lift box’ inverted. Of course there were nods to safety: pivoting boards built into ceiling, floor and lift platform (think of these as the ‘cutting edges’ and you’ll get the picture) which if disturbed would freeze the whole thing and prevent amusing toe-severing scenes on the way to Dr Brady’s lectures. (While I know many who would do the whole 16-floor ’round trip’ just to see what it was like, I don’t know of a single soul who tried to test the safety boards to see if their fingers/toes remained intact.)

Back to ordinary lifts. Think again of a set of lifts you know. Are they designed around the user? First thing in the morning – when everyone’s coming in – do they ‘rest’ on the ground floor, ready to receive their cargo, and returning after they’ve dropped each load off on higher floors? What about the end of the day? Would you expect them to take up resting positions distributed over the higher floors to increase the chances of one being ready and waiting as users arrive? Would you skew this to favour the higher floors? That might help more people to take the stairs… Already it’s possible to see how different objectives might be met by tweaking the way the lifts work.

There’s a nine-floor building in London with what seems like a fairly generous selection of eight lifts. Until you watch their behaviour closely. Then you see that, even if all eight are resting on the ground floor, your request for upward travel will only result in the same one opening and re-opening its doors until it’s full. (It probably knows it’s full via a weight sensor in the floor.) You certainly won’t get another one opening its doors for you until the first has left full. This means that some poor souls will have two or three minutes to endure with the doors opening and closing before finally leaving. And others jab frustratedly at lifts they know are there, but can’t use. What on earth’s going on? Then you remember that this used to be the Department of the Environment. Was it a procurement coup to pick the tender with the lowest energy use? Or is a political point being made by tweaking the lift logic to ‘maximum efficiency’ even if this results in a less-than-delightful user experience?

The mathematicians have some fancy algorithms to optimise the basic logistics of covering ground efficiently. There’s even a branch of queuing theory known as elevator theory. Amongst other things it can be used to design how the magnetic heads that sweep over hard disk drives work. (If you have several heads, and need to collect data from different portions of the disk, then you face very similar planning challenges to moving people to different floors of a building.) But, as we’ve seen, even the purity of logistical efficiency can take second place to health, politics or other objectives.

A closing thought: next time you’re waiting for a lift, or indeed, in a lift, and you don’t quite get the service you want – you might well be getting the service that some else wants you to have. And that goes for pretty much every other customer experience you encounter…

Imperfect harmony

(Or why the Fundamental Theorem of Arithmetic means we shouldn’t get so stressed when big projects go tits-up. A journey via the physics of music, the ingenious Bach, and a whole lot of faking.)

Take a string. Tie it tightly between two fixed points. If you have a guitar to hand, even better, but it’s not essential. In fact, you should be able to grasp all of what follows without any props at all, providing you’re prepared to take on trust some of the descriptions.

Please now forget everything you know, assume, or half-remember being taught about music theory.

Pluck the string and listen to the sound. That note – the clear, strong one you can hear? This is our starting point. Our fundamental. Hold your finger lightly against the string, half-way along. Pluck again, in the centre of one of the halves. Hear that new note?

If the string’s tight enough, it should be higher than the open-string note, but in a way, its twin. Sounding very similar in character – in perfect harmony in fact – just higher. (“Twice” as high in a sense. The string is vibrating exactly twice as quickly as it did when it was twice the length.)

If you laid the sound waves of the open-string note side-by-side with those of the higher note, every other ‘peak’ of the waves would be in exactly the same place. That makes for great harmony.

(All going swimmingly so far; let’s mix it up by finding another note).

This time hold your finger lightly against the string, but a third of the way along (doesn’t matter from which end). Pluck the shorter part of the string with your other hand. Now this is quite different. You get a note that’s even higher than the ‘half-way’ note. Play the ‘half-way’ note again. Then the “third-the-way-along” again.

Listen to the difference. If they were played together at the same time (you’d obviously need two identical strings) they’d also sound in pleasant harmony. The higher note’s sound-waves are packed together a little more tightly, but laying the two side-by-side, the peaks would again coincide regularly, this time in a 3-to-2 ratio, rather than 2-to-1 as before. Let’s introduce some labels here to cut down on the hyphens… Call the ‘half-way’ note the ‘octave’.

(Try and wipe from your mind the oct– prefix as being anything to do with the number eight. All that can come later, as I hope will become clear). Call the ‘third-the-way’ note the ‘dominant’. What we’ve done so far, without using any musical theory to speak of, is build a relationship that mathematically links two notes together. Two notes that are in perfect harmony.

This simple description will be the building block of what follows. Which should get quite a bit more weird soon – bear with me. (By the way, the demonstration with the tied string and so on couldn’t have been done using notes you’d find on a piano. Not an ordinary piano anyway. Because a piano has to ‘fake’ its notes, just a tiny bit. All will become clear, I hope.)