  Inspiralling black hole-neutron star (BH-NS) binaries emit a complicated
gravitational wave signature, produced by multiple harmonics sourced by their
strong local gravitational field and further modulated by the orbital plane's
precession. Some features of this complex signal are easily accessible to
ground-based interferometers (e.g., the rate of change of frequency); others
less so (e.g., the polarization content); and others unavailable (e.g.,
features of the signal out of band). For this reason, an ambiguity function (a
diagnostic of dissimilarity) between two such signals varies on many parameter
scales and ranges. In this paper, we present a method for computing an
approximate, effective Fisher matrix from variations in the ambiguity function
on physically pertinent scales which depend on the relevant signal to noise
ratio. As a concrete example, we explore how higher harmonics improve parameter
measurement accuracy. As previous studies suggest, for our fiducial BH-NS
binaries and for plausible signal amplitudes, we see that higher harmonics at
best marginally improve our ability to measure parameters. For non-precessing
binaries, these Fisher matrices separate into intrinsic (mass, spin) and
extrinsic (geometrical) parameters; higher harmonics principally improve our
knowledge about the line of sight. For the precessing binaries, the extra
information provided by higher harmonics is distributed across several
parameters. We provide concrete estimates for measurement accuracy, using
coordinates adapted to the precession cone in the detector's sensitive band.
