increpare games

A copied/pasted selection of a thread from tigsource. There is certain amount of extra discussion there, but all the actual analyses are copied below. It’s a little messy (especially the opening description, which might anger some specialists immensely, should they be unfortunate enough to stumble across this page), for which I apologise.

After having played about a little bit today with things relating to structuralism, I thought it might be fun to try to apply Levi-Strauss‘s canonical formula of mythology to some games. (the closest I could find to a discussion of this nature on the web was this rather elementary discussion on gamedev.net).

The canonical formula looks like:

It’s supposed to depict some sort of transformation, with the fraction on the left representing some sort of relationship between the numerator and the demoninator, the arrow in the middle representing the transformation, and the fraction on the right a relationship between the permuted contents of its numerator and denominator. Basically, you can fill it out however you want. a^-1 is supposed to be some sort of opposite of a. Also, generally either a and b represent characters, and x and y represent some properties, or vice versa. And generally f doesn’t mean anything. (I take that back. f indicates that there’s a functional relationship between its two arguments. ‘functional relationship’ means that one of its arguments is a property of the other, or is an action performed on/by the other. Basically by ‘meaningless’ I mean ‘not a variable’).

It all is a bit arbitrary, but there’s certainly a knack to describing things using it. Actual analyses follow below the fold

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Hmm. This is going to be harder than I had initially thought now that I think of it. Basically, given three melodies that work in counterpoint they are written on three staves, one above the other. And we have the identity that the interval between the lower and middle voice added to the interval between the middle and upper voice will give the interval between the lower and upper voice. And we have rules that relate how these voices should interact with eachother.

However, one can still apply most of the rules quite well if we make things a little bit abstract and no longer require the above relationship [a,b]+[b,c]=[a,c] to hold, but rather that it hold only up to a certain constant interval I.

And what’s the sense in this? Well with this you still have three melodies, only now instead of all three being contrapuntally amenable, we have that any two of them are. I haven’t seen this expounded elsewhere, and given how practical it seems I thought I’d mention it here.

And why the devil did I want to mix up Fuchs with cohomology? Well, I was trying to figure out an easy example of a Gerbe :) (I failed, as it happened, but it’s quite teasingly close!).

And why this rambley ramble here as opposed to something more deliberate? Because I’ve been meaning to post this since November last year, that’s why. So this means I get to relax about it now. Chill, you know? And if anybody should wish for any explication I would be Only Too Happy to provide it.

Hmm, so in the interest of subscribing to this mathematical carnival what’s doing the rounds now, I’m writing something specifically mathematical in nature, my favourite elementary derivation. I’m trying to make it understandable and brief – if anyone is having trouble following, I can help in the comments.

Okay, so lets say we’re hearing a signal given by f(t); let’s assume it’s periodic. Now, to monitor what we hear, we have to view this as a sum of sine waves

f=a*sin(t)+b*sin(2t)+c*sin(3t)+ …

so a,b,c represent the frequencies we hear of frequency 1,2,3, etc, and the bigger the coefficient the bigger the amplitude.

Lets look at two really simple sounds, pure sine waves of different frequencies sin(at), and sin(bt). SO, they each have exactly one frequency present.

Now, so what if there were some non-linearity of our hearing system. That is what if, when someone plays f=sin(at)+sin(bt), we don’t actually hear this, but rather something more complicated.

So, the simplest way of such a thing being non-trivial is to introduce a quadratic non-linearity (ignoring coefficients…we’re thinking that because “all” functions can be taylor expanded as f(t)+f(t)²/2+f(t)³/3!+…, the next best thing to having just f(t) is to having the first two terms).

So, anyway, now when someone plays a signal f(t), we don’t hear f(t), but rather f(t)+f(t)²

So, what results from this? Well, we have to break down f(t)+f(t)² to being a sum of sine waves first.

f(t)+f(t)²=sin(at)+sin(bt)+ (sin(at)+sin(bt))²
=sin(at)+sin(bt)+sin(at)²+sin(bt)²+2sin(at)sin(bt)

looking up trig tables and decomposing further we get (ignoring coefficients)

=sin(at)+sin(bt)+cos(2a)+cos(2b)+sin(a+b)+sin(a-b)

Ah, so look at this. We might be led to deduce from this that, if non-linearities were present, when someone plays two frequencies at the time, we will also perceive sounds playing at double either frequency, their sum, and their difference.

Now, the multiples of a and b can be reasonably expected to be masked by overtones (though it is possible to bring them out), but the difference (and, to a lesser extent, the sum), on the other hand, can be controlled very easily, just by bringing the two source sounds closer together or further apart. And, indeed, we can quite easily hear them.* Which is darnedly neat.

I like this calculation so very, very much because it’s surprisingly fruitful; whenever I feel like I’m loosing my faith in the power of Taylor expansions, I go through this derivation again.

The phenomenon was first noted by Tartini, the derivation was by Helmholtz, and these extra tones are sometimes called Tartini tones or, more commonly combination tones.

What’s also interesting, by the by, is that these effects are actually quadratic: I remember, the first time I heard an example (they can be found on the interweb quite easily, here for instance), I was listening to them with earphones, and the strength of the combination tone totally overwhelmed the two base ones. But, when I played it on speakers, it was much weaker relative to these tones, and if I went too far away, I couldn’t hear it at all.

*okay; this is a lie, actually the third order terms are the easiest to hear, corresponding to things like 2a-b…and there’s an explanation for this (check out Dave Benson’s notes if you want to know more).

In a dream again, I found myself in a labyrinth. This was not a labyrinth of walls, though, it was a labyrinth of space and, instead of feeling claustrophobic confinement, I felt the oppression of a space too rich, too rich and dense, to meaningfully comprehend.

This was a desert, with occasional rocks, trees, allowing me some orientation with respect to my world and, yet, I couldn’t shake off a feeling of general disorientation.

I set my sight on a particular tree, which was standing next to a sharply jutting piece of sandstone several paces away. I made my way towards it, or rather, tried to – I found after one or two paces these to be on the periphery of my vision, though a little closer – I turned to face them again, and stepped forward: walking towards this tree was like keeping one’s balance on a bicycle – it took constant corrections.

I picked up a small stone, some black, rough stone, from the sand, and tossed it forwards a short distance; I found that I was getting the hang of this desert, and was able, with some small exertion of effort, to walk to it and pick it up again, though I could only move but slowly.

I picked up two stones, and made an effort to toss them both t the same point a few metres away. Indeed, they looked to land rather closely together. And, yet, as I walked towards one, I found the other getting further and further away, until, by the time I had reached my destination, the other one was almost as far from me as I was now from the point at which I had thrown it.

The geometry of this place was…thick; it seemed that even the slightest error in orientation, on a walk of but a few paces, could result in you being further away than when you started.

I then knew, as one does in dreams, that my destination – the place where I needed to go, was many miles away, that I could spend a lifetime of lifetimes trying to find it and, even if someone could point me in its rough direction from where I was, I would have little chance of ever finding it.

The following night I had another dream: I awoke inside the dream, to find myself breathing deeply, calmly, in some spacious, geometrical environment. I looked about, trying to take in in my surroundings, only a fear began to grab at me: the space was uninhabited by anything else, it just went on forever in all directions, and yet there was more: as it dawned upon me that I was in a space of dimensions many more than three, that space was going to infinity not merely in three directions, but twenty – this frightened me, I found myself unable to move, praying for some relief to the immense extent of this world.

My prayers were suddenly answered: I found myself trapped in a cramped prison cell; trapped between forty different walls on forty different sides.