How to encode ordinal variables when null value is valid?

Question

Tl;dr how can I encode a feature that has multiple distinct states each with different numbers of parameters. Am I going to have to break this into multiple models?

I'm sorry, I'm sure this has been asked before, but there's a vocab word I don't know, so I've been researching in circles.

Specifically, I want to encode music data, where pitches are different distances from each other, but all pitches are the same distance from silence.

I don't need silence to literally be a null value, but I can't just encode it as zero either. Silence is a valid category, but it's not on the same dimension as pitch.

What is the vocab word I need to research this further? Everything I'm finding says how to fill in null values with no consideration as to whether or not null is a valid answer. And it looks like the other people doing machine learning on music haven't found a good answer either.

Edit: So far this seems to be the closest answer, but it doesn't work because null represents an unknown in that example which does not add information, whereas in my case, null is a known value, that does add information

Answer 1

Okay, it's really looking like there's no way to avoid dummy variables, as suggested by whuber. Since silence is a different dimension than pitch, there isn't a way to get around adding another dimension to account for it. There's a lot of weird tricks you can try, but that extra dimension has to happen somewhere.

As for the other side of the question, if google sends anyone working on music in machine learning here, the keyword that helped my searches was "tonnetz". https://arxiv.org/pdf/1709.00375.pdf

And I'll hopefully save you a lot of heartache trying to break down the traditional tonnetz that you'll see, where rightward motion represents going up a major fifth and going up and to the right represents a major third:

That whole thing simplifies down to

pitch = 3x + y

I'll probably post a more in-depth explanation, but for now, just look at this grid

2  5  8 11
1  4  7 10
0  3  6  9

until you see how it's the same as the tonnetz. You can still draw all the same lines between everything, but now we can just give our algorithms a vector (x, y) that satisfies the equation pitch = 3x + y

Since it's not possible to invert this process without choosing a subset with no duplicates, I figure we can use

x = integer(pitch / 3)
y = pitch % 3

I'm sure this can be extended to other kinds of tonnetz, but this concludes my contribution for now.