  We define a function, called s-multiplicity, that interpolates between
Hilbert-Samuel multiplicity and Hilbert-Kunz multiplicity by comparing powers
of ideals to the Frobenius powers of ideals. The function is continuous in s,
and its value is equal to Hilbert-Samuel multiplicity for small values of s and
is equal to Hilbert-Kunz multiplicity for large values of s. We prove that it
has an Associativity Formula generalizing the Associativity Formulas for
Hilbert-Samuel and Hilbert-Kunz multiplicity. We also define a family of
closures such that if two ideals have the same s-closure then they have the
same s-multiplicity, and the converse holds under mild conditions. We describe
the s-multiplicity of monomial ideals in toric rings as a certain volume in
real space
