What are the mother sauces?
Five foundational sauces codified in French cuisine, from which a large number of others are derived by addition. The system is a teaching structure more than a rulebook, and its value is in showing that most sauces are variations on a handful of techniques.
The five, as standardised by Escoffier:
Béchamel — milk thickened with a white roux (equal parts butter and flour cooked without colouring). The base for cheese sauce (Mornay), and for lasagne and gratins.
Velouté — a light stock — chicken, veal or fish — thickened with a blond roux. More savoury than béchamel because the liquid carries flavour rather than being neutral.
Espagnole — brown stock thickened with a brown roux, with tomato and browned mirepoix. Reduced further it becomes demi-glace, the base of many classical meat sauces.
Tomato sauce — in the classical version, considerably more involved than a modern tomato sauce, traditionally including pork, stock and sometimes a roux.
Hollandaise — the outlier. Not thickened with roux at all but an emulsion of egg yolk and butter, acidulated with lemon or a vinegar reduction. Related to béarnaise, which adds tarragon and shallot.
Antonin Carême originally proposed four; Escoffier revised the list, replacing one with tomato and adding hollandaise.
Why the structure is useful. It reduces sauce-making to two techniques and a handful of decisions: thicken with roux or emulsify with egg; choose the liquid; adjust the colour of the roux; add flavourings. Understanding that, you can construct sauces without recipes, which is the point.
The practical knowledge that goes with it:
Roux colour matters twice — darker roux tastes nuttier and thickens less, because cooking breaks down the starch. A brown roux needs considerably more of it.
Temperature difference prevents lumps. Adding cold liquid to hot roux, or hot to cold, works; equal temperatures produce lumps.
Emulsified sauces split with heat, which is why hollandaise is made over gentle indirect heat.
Modern kitchens use fewer roux-based sauces, favouring reductions, purées and emulsions — but the underlying principles are the same.