Major and minor brats, and linearized temperament

If we are in a 5-limit tuning system with fixed values f, t, and m
for the fifth, major third, and minor third respectively (written
multiplicatively, so that mt = f) then we may define the major brats
as (6*t-5*f)/(4*t-5), (4*t-5)/(2*f-3), (2*f-3)/(6*t-5*f). If we call
the first of these "b", then the others are linear fractional
transformations of b, with product 1: b, 5/(3-2*b), (3-2*b)/(5*b).

Similarly, we may define the minor brats as
(5*m-6)/(5*m-4*f), (2*f-3)/(5*m-6), (5*m-4*f)/(2*f-3); if we call the
first one "c", then the others are linear fractional transformations
of c: c, (c-1)/(2*c), 2/(c-1). If we substitute m=f/t into c, we find
the basic relationship b = f*c between them. To the extent that the
tuning system is accurate, b/c will be nearly 3/2, if if b is
rational with small height, c will be close to such a number. We can
compute the tuning of the system for both values, and it would seem
reasonable to split the difference and take an intermediate value
about half-way between. It turns out this is the linearized value.

If we write del f for f-3/2, del t for t-5/4, and del m for m-6/5,
then we have

b = 3/2 - 5/4 (del f)/(del t)

1/c = 1 - 4/5 (del f)/(del m)

If we write d(f) for the log error of f, log(2f/3) (in any base; for
instance, base the comma of the system, base cents, etc.) and so
forth for d(t) and d(m), then we have

b*(1/c) = 3/2 - 5/4 (del f)/(del t) - 6/5 (del f)/(del m) +
(del f)^2 / del(t) del(m)

This is approximately

3/2 (1 - d(f)/d(t) - d(f)/d(m) + d(f)^2/d(t)d(m))

Since d(f) = d(t) + d(m), this reduces to 3/2; hence when using the
linearizing approximation the first-order error cancels out, and we
may use it to define the average error value we were seeking. It
should be noted that this is a general construction; we aren't just
considering things like 5/19-comma meantone, but also 7/101-vulture
vulture, etc. It seems there is still much to be learned even about
the tuning of 5-limit harmony.