RED
To get from pink to Burgundy you need to add a little red and some blue.
add blue
add a lot of white paint and you will have a lovely pale violet
Add some blue color in it..
You would add some purple to the peony pink.
You mix a little of bit of dark blue and brown and you get black. If it looks gray-ish add some green, it should help.
Red and Blue would form purple - as you add more blue into the mixture the colour created would be a darker purple, you can also add TINY amounts of yellow to create a brown/purple outcome, too much yellow to the mix would create a dirty looking green.
BLACK
if you're mixing paint, i would start with a warm red, add a bit of warm, deep blue, and then, if you need to darken, add a tiny bit of true black.......that's what i would do.
To make black from purple, you would add a dark shade of blue, such as navy or indigo, along with a small amount of black paint. This will deepen the color and darken it to create black.
You can add the color blue to make purple more blue.