  We define new parameters, a zero interval and a dual zero interval, of
subsets in $P$- or $Q$-polynomial schemes. A zero interval of a subset in a
$P$-polynomial scheme is a successive interval index for which the inner
distribution vanishes, and a dual zero interval of a subset in a $Q$-polynomial
scheme is a successive interval index for which the dual inner distribution
vanishes. We derive the bounds of the lengths of a zero interval and a dual
zero interval using the degree and dual degree respectively, and show that a
subset in a $P$-polynomial scheme (resp. a $Q$-polynomial scheme) having a
large length of a zero interval (resp. a dual zero interval) induces a
completely regular code (resp. a $Q$-polynomial scheme). Moreover, we consider
the spherical analogue of a dual zero interval.
