Q: The missing link -- affine geometry & exterior algebra?

All right, folks . . . I'm not sure if I missed anything important
since I last posted, but before I catch up . . .

In the 3-limit, there's only one kind of regular TOP temperament:
equal TOP temperament. For any instance of it, the complexity can be
assessed by either

() Measuring the Tenney harmonic distance of the commatic unison
vector

5-equal: log2(256*243) = 15.925, log3(256*243) = 10.047
12-equal: log2(531441*524288) = 38.02, log3(531441*524288) = 23.988

() Calculating the number of notes per pure octave or 'tritave':

5-equal: TOP octave = 1194.3 -> 5.0237 notes per pure octave;
.........TOP tritave = 1910.9 -> 7.9624 notes per pure tritave.
12-equal: TOP octave = 1200.6 -> 11.994 notes per pure octave;
.........TOP tritave = 1901 -> 19.01 notes per pure tritave.

The latter results are precisely the former divided by 2: in
particular, the base-2 Tenney harmonic distance gives 2 times the
number of notes per tritave, and the base-3 Tenney harmonic distance
gives 2 times the number of notes per octave. A funny 'switch' but
agreement (up to a factor of exactly 2) nonetheless. In some way,
both of these methods of course have to correspond to the same
mathematical formula . . .

In the 5-limit, there are both 'linear' and equal TOP temperaments.
For the 'linear' case, we can use the first method above (Tenney
harmonic distance) to calculate complexity. For the equal case, two
commas are involved; if we delete the entries for prime p in the
monzos for each of the commatic unison vectors and calculate the
determinant of the remaining 2-by-2 matrix, we get the number of
notes per tempered p; then we can use the usual TOP formula to get
tempered p in terms of pure p and thus finally, the number of notes
per pure p. Note that there was no need to calculate the angle
or 'straightness' of the commas; change the angles in your lattice
and the number of notes the commas define remains the same, so angles
can't really be relevant here. As I understand it, the determinant
measures *area* not only in Euclidean geometry, but also in 'affine'
geometry, where angles are left undefined . . . Anyhow, since both of
these methods could be used to address a 3-limit TOP temperament, in
5-limit could they be still both be expressible in a single form in a
general enough framework, say exterior algebra?

In the 7-limit, the two methods give us, respectively, the complexity
of a 'planar' temperament as a distance, and the cardinality of a 7-
limit equal temperament as a volume. But 7-limit 'linear'
temperaments get left out in the cold. The appropriate measure would
seem to have to be an *area* of some sort -- from what I understand
from exterior algebra, this is the area of the *bivector* formed by
taking the *wedge product* of any two linearly independent commatic
unison vectors (barring torsion). If the generalization I referred to
above is attainable, all three of the 7-limit cases could be
expressed in a single way. Anyhow, if this is all correct, I want
details, details, details. The goal, of course, is to produce
complexity vs. TOP error graphs for 7-limit linear temperaments,
something I currently don't know how to do. If someone can fill in
the missing links on the above, preferably showing the rigorous
collapse to a single formula in the 3-limit and with some intuitive
guidance on how to visualize the bivector area in affine geometry (or
whatever), I'd be extremely grateful.