Free style JI
Lou Harrison invented this term from a technique he applied first in the middle section of his piece "At The Tomb of Charles Ives" (see also Archived page on CD Baby); Instead of working with a set of fixed pitches, his concept was instead to use a set of fixed intervals regardless where this lead one. David Doty realized a midi version of a Symphony in Free Style That Mr. Harrison wrote.
In adding freedom you may be sacrificing a 'safety' of familiarity, tonality, simplicity of materials, etc. It is a dangerous and rewarding world out there. Another consideration is that unless an effort is made to restrict the range of tonalities, the numerators and denominators grow with time and become unwieldy and eventually impossible to manage. If no such effort is made, they grow linearly. One possible solution is nanotempering--using an equal temperament so high it cannot be distinguished from JI.
Toby Twining's Chrysalid Requiem makes local use of subharmonic, harmonic, 3s-and-7s, and other subsets of JI; however its large-scale modulations wander far and never return precisely to the 1/1 begun with.
Notations which are capable of notating the whole of free-JI: