We study the asymptotic behavior of the first eigenvalue and eigenfunctionof a one-dimensional periodic elliptic operator with Neumann boundaryconditions. The second order elliptic equation is not self-adjointand is singularly perturbed since, denoting by $\epsilon$ the period,each derivative is scaled by an $\epsilon$ factor.The main difficulty is that the domain size is not an integer multipleof the period. More precisely, for a domain of size $1$ and a givenfractional part $0\leq\delta<1$, we consider a sequence of periods$\epsilon_n=1/(n+\delta)$ with $n\in \mathbb{N}$. In other words, the domaincontains $n$ entire periodic cells and a fraction $\delta$ of a cellcut by the domain boundary. According to the value of the fractionalpart $\delta$, different asymptotic behaviors are possible: in somecases an homogenized limit is obtained, while in other cases thefirst eigenfunction is exponentially localized at one of theextreme points of the domain.