I'm kind of a snob about color mixing. I make no apologies, because color is so important to artists, and vibrancy is essential. The world is excruciatingly vibrant, after all.
So why talk about black? Isn't that the absence of color? Well, that is science, and true from a logical viewpoint, but artists will not often benefit from logic.
There is no such thing as absence of color on a surface using any normal art media. The darkest black reflects some light, and has an impact. The way you use black is very important, as both a solid black area, and for mixing into other colors.
I don’t use black paint tubes - and I don't allow my students to have any black in their case. The pigments in black paints are made of dead, burned things, like bones. It's basically charcoal, and I don't like the idea of mixing charcoal into my colors. I really don't like the gray you get mixing black pigments into white paint. It is a lifeless and unattractive gray. Black pigments suck the life right out of your paintings. Shadows are actually full of intense colors, and black can’t make these.
Instead, I use what I refer to as perfect pigment opposites: two pigments that work perfectly together. These combinations are complementary colors, but they're more than that, they really do complete each other - to the point that they completely cancel each other out, creating very little reflected color if mixed to a balanced state. The result looks black. But it's not dead; it has no black pigment. Instead it is full of color, so when you mix these blacks into other colors this can really help them stay vibrant. But another benefit is that it also helps a painting have an amazing continuity of color, when you use the prominent colors in the painting to also make your blacks.
Perfect matches are easy to find.
There are 4 perfect pigment opposites in my palette of 10 pigments, that will make a neutral black or dark gray. Two of these matches are single-pigment-to-single-pigment, meaning you only need two tubes of paint to make your black. These matches are:
Burnt Umber with Ultramarine Blue
Pyrrol Red with Phthalo Green
The second match is extremely difficult to balance and creates a lighter black than the first one. The pigments are also super-strong, so adding just a bit of one can tip the mix over too far. It's also hard to see if you've hit it, because changing a dark red to almost black, suddenly looks greenish, and vice versa. I only recommend black from these pigments if there is a lot of reds and greens already in the painting.
The Blue and the Brown pigments are much easier. They're already dark, and they're not as brilliant. It's also easier to remember, because they both begin with a B, just like Black. Make sure to use a decent brand of paint. There are some modern student grade brands that just mimic colors, so they may not work very well. Amsterdam is one of my favorite student grade brands.
In most brands, the brown will be a bit stronger, and so you'll need a bit less than the Ultramarine Blue. Experimentation is always needed when mixing any color, so just keep after it until it doesn't look too brown or too blue.
When mixing a black, the easiest way to test your result is to use a bit of white and make a light gray. Any color imbalance will show up more clearly. Of course, a blue gray and a warm gray are easy to push towards, by simply adding more of one pigment.
That's it. That's the best way to make and use black in your paintings.
Cheat sheet: If I need to use a lot of solid black, such as the edge of a canvas, a large area, or an underpainting, I will sometimes use a black pigment. If you want a dirty look, black pigment will work very well for that. Rules are not sacrosanct when it comes to creating!
As always find out what works best for you. The world needs happy artists.
Addendum - my 10 pigments
Light Azo Yellow (or Hansa Opaque)
Light brilliant red such as Napthol or Pyrrol
Magenta - Quinacridone is great
Dioxazene Purple (or dark violet)
Ultramarine Blue
Cyan Blue (or Cerulean, not Cobalt)
Phthalo Green
Burnt Umber
Raw Sienna (or a dark yellow)
Titanium White - never use a student grade white