  The orbital parameters of binaries at intermediate periods ($10^2$--$10^3$ d)
are difficult to measure with conventional methods and are very incomplete. We
have undertaken a new survey, applying our pulsation timing method to {\it
Kepler} light curves of 2224 main-sequence A/F stars and found 341
non-eclipsing binaries. We calculate the orbital parameters for 317 PB1 systems
(single-pulsator binaries) and 24 PB2s (double-pulsators). The method reaches
down to small mass ratios $q=0.02$ and yields a highly homogeneous sample. We
parametrize the mass-ratio distribution using both inversion and MCMC
forward-modelling techniques, and find it to be skewed towards low-mass
companions, peaking at $q=0.2$. While solar-type primaries show a brown dwarf
desert across intermediate periods, we find a small but significant
(2.6$\sigma$) population of extreme-mass-ratio companions ($q<0.1$) to our
intermediate-mass primaries. Across periods of 100--1500 d and at $q>0.1$, we
measure the binary fraction of current A/F primaries to be 15.4%$\pm$1.4%,
though we find that a large fraction of the companions (21%$\pm$6%) are white
dwarfs in post-mass-transfer systems with primaries that are now blue
stragglers, some of which are the progenitors of Type Ia supernovae, barium
stars, symbiotics, and related phenomena. Excluding these white dwarfs, we find
the binary fraction of original A/F primaries to be 13.9%$\pm$2.1% over the
same parameter space. Combining literature measurements with ours, we find the
binary fraction across these periods is a constant 5% for primaries $M_1<0.8$
M$_{\odot}$, but then increases linearly with $\log M_1$, demonstrating that
natal discs around more massive protostars $M_1>1$ M$_{\odot}$ become
increasingly more prone to fragmentation. Finally, we find the eccentricity
distribution of the main-sequence pairs to be much less eccentric than the
thermal distribution.
