To me it looks like a harmless fungus from continuous moisture. How far up the plant does it go? Have you sprayed a fungicide? I had some black spots on mine from too much rain this spring.

The black spotted leaves are sporadically around the plant, on the bottom, middle, and high up. The new leaves growing are fine though.

We haven't had any rain and the leaves don't get wet when I water the plant.
Humidity around 30%.

I haven't sprayed fungicide only BT, which made the leaves wet.

What's a good all-purpose fungicide to use?

Take a look at this link and compare the pics to your leaves. Looks right to me. Best fungicide is Daconil but many others are available. many discussions here about them the search will pull up for you.

Dave

Here is a link that might be useful: TAMU - Septoria Leaf Spot

spray leaves with a copper fungicide. Much better for you and the environment then Daconil.