Why are secondary amines the most basic?

A commentator on In the Pipeline remarked that in a series of compounds he was looking at, adding a methyl to a primary amide to turn it into a secondary amide surprisingly and "counterintuitively" increased its aqueous solubility.

But is it really that surprising? The remark made me recall the counterintuitive-looking trends in the basicity of amines that we learn about from college chemistry textbooks. Rationalizing these trends by considering tradeoffs is exactly the kind of process chemists revel in. The question is, among primary, secondary and tertiary amines, which class is the most basic?

The aqueous solubility of amines is dominantly governed by their basicity. The basicity is in turn dictated by two important effects: inductive effects from alkyl substituents, and the ability of the protonated amines to get solvated by water.

If you consider the first factor then you would predict tertiary amines to be the most basic since they have the largest number of alkyl substituents. However, they would also be the least favorably solvated because of these bulky, lipophilic alkyl groups hanging off them. In fact the steric hindrance created by the alkyl groups affects the solvation so badly that tertiary amines are usually the least basic among the three classes. By this token you would expect primary amines to be the best solvated and most basic, but in this case the inductive effect is weak so the solvation does not result in salvation.

Not surprisingly, it's the middle ground that wins. Secondary amines are solvated well-enough to be soluble, while also being enough pumped up by inductive effects to be reasonably basic. I suspect a similar argument applies to secondary amides. There could of course be other complicating factors like polar vs apolar solvents, but that's the simplest argument.

Trends in the basicity of amines provide a classic example of why chemistry is so interesting; it's deciphering the delicate trade-offs between various general factors in specific cases that gives chemists a buzz. The miraculous starts looking obvious.

P.S. As an aside, I wonder if someone has recently done a high-level theoretical calculation to dissect and validate these factors.