My terminology here is from "Measure Theory" by JL Doob, Springer Graduate Texts in Mathematics, 1994.  Metric and pseudometric spaces are described on pages 3 to 5.

My question is what happens if the triangle inequality is an equality so we have the distance formula d(s, u) = d(s, t) + d(t, u).  This can only be true if every distance is 1, and 1 = 1 + 1 because every a = a2.

Now if the pseudometric distance formula is d(s, s) = 0, does this follow directly from the triangle equality or is there something else required?  Is d(s, s) = 0 the same as the Borel function f(x, y) = 0 where x = y?

What I am hoping is that if there is an algebra of subsets where every set is a singleton, then that by itself makes a pseudometric space with a distribution on the Borel Square [0, 1] x [0, 1].  Does the triangle equality automatically make an L-space?

May be it is not that simple. If the triangle equality metric alone does not make a pseudometric space by itself, what does? A field of zeros? What if I have the algebra formed by the union, intersection, and compliment of an elemental set? What if I have a set, a relation, an agumented frame, and a map?

My goal here is to show that music structures are a pseudometric space where a probability distribution is imposed on pitch values by the set construction operation.  This follows my previous question demonstrating that music theory is a 3-fold structure, not two fold as commonly thought.  In a 2-fold structure there is no way to understand why every note does not have a probability of 1/12..