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.