  In collider physics, jet algorithms are a ubiquitous tool for clustering
particles into discrete jet objects. Event shapes offer an alternative way to
characterize jets, and one can define a jet multiplicity event shape, which can
take on fractional values, using the framework of "jets without jets". In this
paper, we perform the first analytic studies of fractional jet multiplicity
$\tilde{N}_{\rm jet}$ in the context of $e^+e^-$ collisions. We use fixed-order
QCD to understand the $\tilde{N}_{\rm jet}$ cross section at order
$\alpha_s^2$, and we introduce a candidate factorization theorem to capture
certain higher-order effects. The resulting distributions have a hybrid jet
algorithm/event shape behavior which agrees with parton shower Monte Carlo
generators. The $\tilde{N}_{\rm jet}$ observable does not satisfy ordinary
soft-collinear factorization, and the $\tilde{N}_{\rm jet}$ cross section
exhibits a number of unique features, including the absence of collinear
logarithms and the presence of soft logarithms that are purely non-global.
Additionally, we find novel divergences connected to the energy sharing between
emissions, which are reminiscent of rapidity divergences encountered in other
applications. Given these interesting properties of fractional jet
multiplicity, we advocate for future measurements and calculations of
$\tilde{N}_{\rm jet}$ at hadron colliders like the LHC.
