Consider a one-parameter family of circle diffeomorphisms whichunfolds a saddle-node periodic orbit at the edge of an ‘Arnold tongue’.Recently it has been shown that homoclinic orbits of the saddle-nodeperiodic points induce a ‘transition map’which completely describes the smooth conjugacy classes of such mapsand determines the universalities of the bifurcations resulting fromthe 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 iscompletely determined by the transition map and give a formula forthis density. It turns out that this density is always lessthan 1 and generically greater than 0, with the exceptional caseshaving infinite co-dimension.