OK.  Does my first pass at the theory and why I've put it there make sense to someone that knows zero about signal processing?

I think people that don't already know the subject will be lost. What's the purpose of bandpass filtering? Isn't the goal of certain noise algorithms to achieve a subjective aesthetic effect?

Well talking in a hand waving kinda way about signal processing is tricky.  I guess the more important question is if it's even worth talking about at all? 

I'm trying to get better acquainted with using noise in textures right now, grappling with it conceptually, having just had my first working experience with calling Simplex noise to assemble a cloudy texture.

I'm not at all clear that getting into Fourier analysis is helpful. I can see where using 'harmonics' can make the coding neater, and it seems to work well with the mathematics of fractals, but it doesn't seem to be entirely necessary. One can add noise that has energy at frequencies that are unrelated to the base frequency with no problem. Visual textures are not like sound waves, where one deals with the prevalence nodes and anti-nodes and standing waves in the "real world," and the ear and hearing portions of the brain has evolved to make use of data in this form.

So, is it simpler, instead, to describe noise as having components that are at various periods, and not worry about the Fourier analysis, at least, at the "beginner" level? Or are techniques to analyze textures to determine their strongest component frequencies in use and an important part of creating textures?

I'm thinking, for a dimension, given a length L and a value "n" along that length, a "basic" unit of noise might be of length n/L. Since n can go from 0 to L, the result of this fraction is 0 to 1. One can multiply this value by different factors to get different degrees of scaling. Obtaining noise with (n/L * K) will be K times more detailed than noise at n/L.

But we are free to make K whatever we want. K can be a float or double. It doesn't have to be an integer or a progression of integers.

(We might also talk about how to "relate" the periodicity of one type of noise to another via a scaling factor that is applied to the n/L (O to 1) results of the different noise generation techniques? Maybe this is already done?)

Then, there is total latitude with what we do with the output noise values (which range from -1 to 1), whether to sum them or lerp them, or use them in trig functions. It seems wide open, as long as the function results in a legal Color value for a pixel.

(It occurs to me, one could also talk about a more concrete value the periodic nature of noise by finding a number of pixels that corresponds to an average of one swing in the random number. But I think I am getting into fuzzy thinking, as I don't know how to describe a "period" of randomness, and the way in which we relate the numbers to pixels on the screen is so fluid.)

As I said, I'm a beginner with using noise, and am happy to be corrected on any point.

Hmm, when generating textures like clouds and terrain it helps to know a little about the effects of noise colouring, distortion and how it can be fun to mess around with to produce different results.

And thinking back to clouds, isn't the very reason you see clouds down to the low frequency bias, with the overtones creating the fuzziness.

I think it would help to show the differences visually though, or even better yet, produce an app to demonstrate it. I learned a lot just by tweaking my 2d noise generator. This could be the difference between generating really mundane levels or elaborate worlds that feel like they've been really well designed  IMO.

I agree with what philfrel said about having components at various periods, and experimentation is key. When I was at uni I remember toying around rending series upon series of sine waves, some directional, some radial, until (*I swear*) it looked like plasma. Of course that wasn't the memorable part, the memorable part was when I made a few small modifications and ... err ... it stopped looking awesome and had to investigate and figure out what on Earth was going on Still don't know how I did it

In terms of good, I think it's key to remember that these random values can be feeding into game behaviour. This could be what gives flavour to your map generation algorithm. The more applicable and relative to games the examples are the more that people will digest and be able to use it.

OK. I made a first pass at a second pass.  Any better or still wankery?

Fixed some typo's.  Completed "brief" and added a sketch for gradient noise.

This is almost always a very good idea! Do this, (I'd be intrested too but mind licensing!) 

Doh! Looky there. A "modify" button on the first post.