  Asteroseismic large frequency separations possess great diagnostic value.
However, their expressions as scaling relations are predicated on homology
arguments which may not hold in general, resulting in mass- and
temperature-dependent deviations. The first-order asymptotic expressions, which
should in principle account for this structural evolution, also deviate more
from fitted frequency-separation estimates than do the simple scaling
relations, and exhibit qualitatively different behavior. We present a modified
asymptotic estimator, and show that these discrepancies can be accounted for by
the evolution of the acoustic turning points of the asteroseismic mode cavity,
which is typically neglected in first-order asymptotic analysis. This permits
us to use a single expression to accurately estimate the large frequency
separations of main-sequence, ascending red giant branch, and red clump stellar
models, except at transition points between two asymptotic regimes during the
subgiant phase of evolution, where the WKB approach fails. The existence of
such transition points provides theoretical justification for separately
calibrated scaling relations for stars in different evolutionary stages.
