The Brat
Given a major triad in close root position, let t is the size of the major third expressed as a ratio of frequencies,
and f the size of the fifth; then the ratio of the beats between the major third and minor third of the triad is
the beat ratio, or brat:
brat = (6t - 5f)/(4t - 5)
This is undefined for a justly tuned triad, but gives certain special values if only one of the chord's intervals
is pure; for a pure fifth, the brat is 3/2, and for a pure minor third, it is 0. For a pure major third, it is
defined as infinity (not positive infinity or negative infinity, simply infinity.) It 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 sometimes choose the tuning so as to make the brat something relatively simple,
and for a well-temperament, we can try to make the various brats simple. We should also note that there is a minor
beat ratio, which is simply brat/f, and that the six ratios of chord frequencies are given by b, 5/(3-2 b), (3-2
b)/(5 b) and their inverses, where b is the brat.
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 vary a lot over a rather small range of generator values in a linear temperament, which gives
us possibilities for assigning it to desirable values.
Here are some circulating (mostly) temperaments, along with a table giving the major third, fifth, and brat above each scale degree.