I've been playing with this idea of position finding with tritones, and beginning to apply it to music that is not strictly diatonic. In particular, it seems as though it should be possible to describe scales as conjoint tritones in order to begin thinking in terms of regions of local diatonicity.
The idea goes back (at least) to Richmond Browne's 1981 article, Tonal Implications of the Diatonic Set. It seems that if tritones are so able to point our ears toward the correct tonal landscape, then it might be that the particular species of tritones involved in scale constructions could be helpful ways to theorize the structure of various scale types. For instance, a diatonic set could be described as the union of an alpha- and a delta-tritone:
For more, here's a brief paper I wrote for David Clampitt's course on scale theory:
It seems that I should go ahead and post these things for people to peruse. More reorganizations imminent!