All-interval structures are subsets of musical spaces that incorporate one and only one interval from every interval class within the space. This study examines the construction and properties of all-interval structures, using mathematical tools and concepts from geometrical and transformational music theories. Further, we investigate conditions under which certain all-interval structures are Z (or GISZ) related to one another. Finally, we make connections between the orbits of all-interval structures under certain interval-groups and the sets of lines and points in finite projective planes. In particular, we conjecture a correspondence that relates to the co-existence of such structures.