Oils, and what each one contributes
Updated
Every property of a finished bar comes from the fatty acid profile of the oils that made it.
Fatty acids, not oils
Soap makers talk about oils and the bar is made of fatty acids.
Lauric and myristic give hard bars and big bubbly lather, and are cleansing to the point of being stripping at high levels.
Palmitic and stearic give hardness and a stable creamy lather.
Oleic gives mildness and conditioning, and a slow-tracing, soft, long-curing bar.
Linoleic and linolenic are conditioning and are the ones that go rancid.
Which is why two very different oils with similar profiles behave similarly, and why a substitution should be made on profile rather than on name.
The four working oils
Olive, high oleic. Mild, slow to trace, soft at first and excellent after a long cure. A hundred-percent olive bar is the classic and needs months.
Coconut, high lauric. Hard, hugely bubbly, cleansing, and drying above roughly a third of the recipe.
Palm, balanced palmitic and oleic. Hardness and a stable lather, and it carries a sustainability question that is worth a real answer rather than a substitution made quietly.
Castor, ricinoleic. A few percent boosts and stabilizes lather markedly; much more makes a sticky bar.
Most recipes in the craft are some arrangement of those four.
The luxury oils
Shea, cocoa butter, avocado, sweet almond, rice bran, hemp.
They contribute at the margin and they are what the label is really selling, which is a legitimate reason to include them.
Be honest about the proportion: five percent of an oil is a real ingredient and is not what makes the bar work, and saying "made with shea butter" is true while implying more.
Building a formula
Start with a hardness contribution, a lather contribution and a conditioning contribution, then adjust.
Change one oil at a time, recalculate the lye every time, and make a small test batch.
And keep the formula recorded by oil, supplier and percentage, because substituting a supplier can change the fatty acid profile of a nominally identical oil.