  In the asymptotic parameterisation of mode frequencies, the phase function
$\epsilon(\nu)$ completely specifies the detailed structure of the frequency
eigenvalues. In practice, however, this function of frequency is reduced to a
single scalar $\epsilon$, defined, particularly by observers, as the intercept
of a least-squares fit to the frequencies against radial order, or via the
central value of this function. The procedure by which this is done is not
unique. We derive a few simple expressions relating various observational
estimators of $\epsilon$ for radial modes to each other, and to the underlying
theoretical object. In particular we demonstrate that a ``reduced' functional
parameterisation is both insensitive to mis-estimations of $\Delta\nu$, and
easy to evaluate locally in terms of both observational and theoretical
quantities. It has been shown previously that such a local definition of
$\epsilon$ can distinguish between stars on the ascending part of the red giant
branch and those in the red clump. We find that this sensitivity to
evolutionary stage arises from differences in the local frequency derivative of
the underlying phase function, a consequence of differences in internal
structure. By constructing an HR-like diagram out of purely seismic
observables, we provide a unified view of the \textit{Kepler} asteroseismic
sample, as well as the initial results from \textit{TESS}. We investigate how
various astrophysical quantities and modelling parameters affect the morphology
of isochrones on this seismic diagram. We also show that $\epsilon$ can be used
as an independent input when deriving stellar parameters from global
asteroseismic quantities.

Solar and Stellar Astrophysics