in mathematics, a harmonic range is the distance between four harmonic conjugates.
let's say our backstreet boys are forming collinear points by standing on a line:
brian and nick are harmonic conjugates with respect to a.j. and howie if:
|brian * a.j.| | = | |brian * howie| |
|a.j.* nick| | |howie * nick| |
further, a.j. and howie are harmonic conjugates with respect to brian and nick.
in short, our boys divide the line in the same ratio. line brian-nick is divided harmonically at a.j. (internal ratio) and at howie (external ratio). if | brian * howie | = 1, then:
|brian * a.j.| | = | a ( 1 - a ) | |brian * nick| | = | 2a | |
1 + a | a + 1 |
all four boys form a harmonic range within our harmonic line segment!
but what if kevin wants to rejoin the group? no problem, he can stand as the midpoint between brian and nick:
then:
kevin * nick^2 = (kevin * a.j.) (kevin * howie)
who doesn't love a good 1998 bsb harmony?
