On Formal Intervals between Time-Spans

A formal interval system (FIS) is an ordered triple (THINGS, IVLS, int), where THINGS is a set, IVLS is a mathematical group, and int is a function from THINGS × THINGS to IVLS satisfying three conditions: (1) from all r,s, and t in THINGS, int (r,t) = int (r,s)int(s,f) [group product in IVLS]; (2) for all s and t in THINGS, int(t,s) = int(s,t) [group inverse in IVLS]; (3) for every s in THINGS and every i in IVLS, there exists a unique t in THINGS satisfying the equation int(s,t) = i. The FIS is a useful general model for our intuitions about "intervals" between "things" in many specific musical contexts. A "time-span" is an ordered pair (a,x), where a is a real number and x is a positive real. This pair is meant to model a temporal event that begins a units of time after (or – a before) some referential moment and extends x units of time therefrom. A change of referential time-unit and a change of referential moment relabel the events (a, x) and (b, y) as events (au + m,xu) and (bu + m,yu). We seek a FIS with time-spans for its THINGS whose interval function is invariant under such transformations: int((au + m,xu), (bu + m,yu)) = int((a,x),(b,y)). There is in fact exactly one such FIS, up to isomorphism in the pertinent sense. This FIS is discussed and explored.