Getting a quart into a quint pot...

There's no mystery of the terms Quart and Quint.
A Quart guitar is 4 notes up - a Quint is 5.

But going down is a bit confusing ...

In Music Theory, the subdominant is not the note under the dominant, it's actually the note that is as far below the tonic as the dominant is above.
Which explains why the submediant is apparently above the mediant - it isn't - it's as far below the tonic as the mediant is above.

We digress... The Quart is 4 notes above the prime, and the Quart Bass is 4 below.

This means that the Quart is in A and the Quart Bass is in B - two members of the same family, with apparently the same name, and they play with different tunings.

And in the same way, a Quint is in B, and the Quint Bass is in A.

And this is why Dr Niibori decided a simple sizing and naming convention was called for!