A society is a geometric space with a collection of subsets that represent voter preferences. We call this space the spectrum and these preference sets approval sets. The agreement proportion is the largest fraction of approval sets that intersect in a common point. Klawe et al. considered linear societies where approval sets are the disjoint union of two intervals, or double intervals. We examine arc-shaped double intervals on circular societies. We consider the case of pairwiseintersecting intervals of equal length and call these double-n circular societies. What is the minimal agreement proportion for double-n societies? We show that the asymptotic agreement proportion is bounded between 0.3333 and 0.3529and conjecture that the proportion approaches 1/3.