I often take the easy way out and work with white roux, usually mixing it 1:10 for a sauce, and using the warm liquid into hot roux method. Tonight I needed a more toasted taste. I was afraid to try dark roux outright, and went for brown roux. "The professional chef" gives a bechamel recipe with 1:10 (white roux to milk) and an espagnole recipe at 1:8.5 (white roux to stock), so I thought I'd be on the safe side with 1:7.2 (brown roux to stock). The result was much thinner than I expected, I had to emergency-bind it with arrowroot.
So, I would like to know, what is the correct ratio for roux? Let's assume that I am thickening the same liquid (e.g. beef stock), and I know how thick a 1:10 mix of white roux and liquid gets. What ratio of blonde, brown and dark roux to liquid creates the same viscosity as 1:10 white roux?