Hello, I am not a native english speaker, so sorry for my grammar.
Why does Mark recommend permanent alizarin crimson instead of a warmer red? This confuses me a bit because he uses ultramarine blue (a tertiary color). It would be more consistent if he uses a cerulean blue like phtalo. A limited palette should be a triad like (ultramarine, yellow and red) or (cyan, yellow and magenta). Of course you can mix a wide color range with alizarin but you can also mix more colors with cerulean instead of ultramarine. At the moment I am trying out some mixtures (Cad Red light, Ultramarine and Cad Yellow light) and (Permanent Alizarin Crimson, Cad Yellow light and Ultramarine). I can mix bright oranges and violets with both ones. Perhaps the question is only why ultramarine? Cerulean would perfect for a clean primary palette.
I also feel unclear about burnt umber. You can mix it easily by burnt siena and ultramarine. Is it right, that umber is used by mark because it is just more neutral and easier to handle?