19-eq and 31-eq are meantone tunings!

>While the rest of Paul's post accurately describes the
>implications or approximations of 19- and 31-eq to meantone
>tunings, it is important to note that, while these ETs
>are close enough that for all intents and purposes they can
>be assumed to be identical to their respective meantones,
>strictly speaking, by mathematical definition, 19- and 31-eq
>are *not* meantone tunings.

What mathematical definition are you using?
By "respective meantones," you reveal that
you are willing to consider both 1/3-comma and
1/4-comma negative regular temperaments to be
meantone tunings. Clearly any tuning where all
fifths are diminished from JI by an amount around
1/4-1/3 of a comma will function in the same way;
i.e., three fourths make a consonant minor third,
and four fifths make a consonant major third. No
sensible mathematical definition would require the
amount of tempering to be an Egyptian fraction of
a comma, or a rational fraction of a comma, etc.,
unless it is the strict definition that implies
1/4-comma meantone. Since you clearly aren't using
that definition, 19-eq and 31-eq most definitely
are meantone tunings.