New hexachordal theorems in metric spaces with a probability measure

The Hexachordal Theorem is a fancy combinatorial property of the sets in Z/12Z discovered and popularized by the musicologist Milton Babbitt (1916-2011). Its has been given several explanations and partial generalizations. Here we complete the comprehension of the phenomenon giving both a geometrical and a probabilistic perspective. 1 An introductive example For describing a set A, one can adopt a statistical method and look at the mean distance between two points picked randomly from A. To fix ideas, assume that A is a subset of the sphere S equipped with the chord distance d and the surface measure μ. By mean distance we meanM1(A) = μ(A)−2 ∫∫ A×A d(x, y) dμ(x)dμ(y) the value corresponding to p = 1 in the range of the power mean distances (Mp(A))p>0 where Mp(A) := ( 1 μ(A)2 ∫∫ A×A d(x, y) dμ(x)dμ(y) )1/p . (1) It is clear that rotating A on S does not modify M1(A). To state the obvious the other power mean distances –as for instance the quadratic mean distance M2(A)– are also conserved after rotation. Finally the (essential) diameter of A is obviously conserved – besides the fact it is limp→∞Mp(A) = suppMp(A). Nothing surprising in all that: the set A is “the same” before and after rotation. As we will prove in this paper, if μ(A) = μ(S)/2 another –this time nontrivial– operation conserves M1(A) and any other power mean Mp(A), namely the complementary map: A 7→ A := S \A. Not only are the power means conserved but also the complete distribution –or law, a central notion from probability theory we recall later– of the random distance between two independent points1. To give a concrete and, we think, surprising example one can consider A to be the set of points with latitude between -30° and 30°, as illustrated in Figure 1. As we just said we have Mp(A) = Mp(A) for every 1Since the diameter of S2 is finite, we face the classical Hausdorff moment problem in which the distribution of the random distance is uniquely characterized by (Mp(A))p∈N. However, we prove the invariance independently of this argument and for non bounded spaces as well.