First time posting on MSE, very nervous. Been lurking for a long time.
I've been looking through Formalized Music (Xenakis) and building software around the ideas in the book. Upon thorough inspection, some of the descriptions in the annex are written in a style that I'm not very used to.
I've tried digging around for more of this, but no one seems to really talk about it with enough rigor.
He writes:
Second Law
$f(j) dj = \frac{2}{a}(1-\frac{j}{a})dj$
Each variable (pitch, intensity, density, etc.) forms an interval (distance) with its predecessor. Each interval is identified with a segment $x$ taken on the axis of the variable. Let there be two points $A$ and $B$ on this axis corresponding to the lower and upper limits of the variable. It is then a matter of drawing at random a segment within $AB$ whose length is included between $j$ and $j + dj$ for $0 \leq j \leq AB$. Then the probability of this event is:
$P_j = f(j)dj = \frac{2}{a}(1-\frac{j}{a})dj$ , for $a = AB$.
I can't really seem to picture how this will be the case, and I'm not really understanding the notation of "_j" in this case for P. I assume dj is an arbitrary point away from point j, and so the function takes an input of a coordinate(?).
I appreciate any help at all in advance! Thank you.