I’m doing this music project, and I wanted to add math to it, and I also got really into Panthu Du Prance’s “Black Noise” – it’s such a deep musical journey if you listen through it with an open mind; for me, the album felt like my first listens of Boards of Canada’s “Music Has the Right to Children”.

My fourier analysis is a bit rusty right now. Anybody more in tune with the psychoacoustic physics want to lend a hand and drop some tips on some psychoacoustic maps for the input and output signals? I’m coding a vintage-sounding lofi matrix for chiptune uses (so like signal discretification + coloration).

Thinking here for a model of what I’m trying to do – to be honest I’m really just making up terms as we go along based on things I remember: a digital signal would be discrete, so this matter would have to be handled discretely. The discrete signal may be represented mathematically as a continuous function $f:\mathbb{Z} \rightarrow [-1, 1]$ where Z has the discrete topology and [-1, 1] the usual. The codomain, when I code this, will be represented by floats, but the math for floats is similar enough to math for real numbers on a bounded interval that the sound will pretty much be reproduced up to human perception.

Or to say that it’s with hi-fi signals I want to code. I want my shit to sound like vinyl signals being fed through NES hardware. Aw yeah.

Of course, I’ll also have to implement output for the discrete case, since the vst host may not do floats and do 24-bit or 16-bit integers instead (which is kinda weird now since we have 32-bit processors, but whatever, audio industry standards lol).

So let’s think about signal maps. A signal map is a map $\phi:[-1, 1]^\mathbb{Z} \rightarrow [-1, 1]^\mathbb{Z}$, where $[-1, 1]^\mathbb{Z}$ is the function space for signals as modeled above. We may attempt to fit this function space with some properties, then.

Afterwards, I guess I should try to find a topology on the thing. Or maybe even a Hilbert space. Actually, it could very well be a Hilbert space given the right choices for the norm, addition and scalar multiplication. Though, that’s stuff that I would do just for the hell of doing it.

What I would need to do is to figure out specific signal maps that would make interesting effects for this project, really. Like for example the lofi matrix + analog coloration dsp I wanted to make would probably include discretification at some point. That would be some sort of discontinuous map f(x) = some chopped bits in the float representation of x.

Cool, I have a plan of attack now.

tl;dr: have a lollipop. I’m going to do some research on some shit hell yeah.

edit:

Thinking a bit more on the subject, I think I have an algo for making signals more discrete, given by $f(x) = [\rho x]/\rho$, where $\rho$ is the granularity of the discretification.

moar edit:

Audio information preservation in the face of discretification is probably closely related to the fact that simple functions are dense in $L_0$ spaces and the algo I listed above is invariably a simple function (so in a sense density in $L_0$ suggests that all continuous functions can be approximated this way, which isn’t surprising, but nice and intuitive).