The Brat

If t is the size of major third in some temperament, and f is the
size of the fifth, then the ratio of the beats between the major
third and minor third of a triad in close root position will be the b-
ratio, or *brat*:

brat = (6t - 5f)/(4t - 5)

This is undefined for JI, is a specific number for an equal
temperament, and depends on the tuning of the generator in the case
of a linear temperament. For a 5-limit linear temperament, therefore,
we can choose the tuning so as to make the brat something relatively
simple.

The brat may be negative (as it is for the 34 et) but the real
interest is in the absolute value. It can be small (0.03 for the 19
et) or large (-38.96 for 28 et, 22.48 for 87) depending on the
relative error of major versus minor thirds. It can very a lot over a
rather small range of generator values in a linear temperament, which
gives us possibilities for assigning it to desirable values.

--- In tuning@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"
<gwsmith@s...> wrote:

It can very a lot over a
> rather small range of generator values in a linear temperament,
which
> gives us possibilities for assigning it to desirable values.

One example of this is the range of poptimal meantones, for which the
brat ranges from -3.216 to minus infinity. For 11/41-comma meantone,
we get a value of exactly -4; 3/11-comma meantone is just barely out
of the poptimal range, and has a brat of -3.

Some other interesting values are 7/25-comma meantone, with a brat of
-2, and 5/17-comma meantone, with a brat of -1. Robert of course is
looking at 1/7-comma meantone, with a brat of +2. In case anyone is
interested, the values for fifth and third we get for these exact
rational meantone systems (q-comma meantone, with q a rational
number, which is to say the same thing as the brat being rational or
infinite) are algebraic numbers of degree four. In the case of 5/17
and 1/7 comma meantone, the fifth is an algebraic integer.

--- In tuning@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"
<gwsmith@s...> wrote:
> If t is the size of major third in some temperament, and f is the
> size of the fifth, then the ratio of the beats between the major
> third and minor third of a triad in close root position will be the
b-
> ratio, or *brat*:
>
> brat = (6t - 5f)/(4t - 5)

If z = (f - 3/2)/(t - 5/4) is the ratio of errors of the
approximation to the fifth over the approximation of the major third
(*not* in terms of cents, but as frequency ratios) then

brat = 3/2 - 5/4 z

This should make it clear how it generalizes.

Gene,

Why only look at the ratio of minor third beats to major third beats?
What about the beats in the fifth? And why only look at major triads?
What about minor triads?