Consider a one-parameter family of circle diffeomorphisms which
unfolds a saddle-node periodic orbit at the edge of an ‘Arnold tongue’.
Recently it has been shown that homoclinic orbits of the saddle-node
periodic points induce a ‘transition map’
which completely describes the smooth conjugacy classes of such maps
and determines the universalities of the bifurcations resulting from
the disappearance of the saddle-node periodic points.
We show that after the bifurcation the relative density (measure)
of parameter values corresponding to irrational rotation numbers is
completely determined by the transition map and give a formula for
this density. It turns out that this density is always less
than 1 and generically greater than 0, with the exceptional cases
having infinite co-dimension.