Distributed consensus under data rate constraint is an important research topic of multiagent systems. Some results have been obtained for consensus of multiagent systems with integrator dynamics, but it remains challenging for general high-order systems, especially in the presence of unmeasurable states. In this paper, we study the quantized consensus problem for a special kind of high-order critical systems and investigate the corresponding data rate required for achieving consensus. The state matrix of each agent is a 2mth order real Jordan block admitting m identical pairs of conjugate poles on the unit circle; each agent has a single input, and only the first state variable can be measured. The case of harmonic oscillators (m = 1) is first investigated under a directed communication topology which contains a spanning tree, while the general case of m >= 2 is considered for a connected and undirected network. In both cases, the sufficient number of communication bits to guarantee the exponentially fast consensus is shown to be an integer between m and 2 m, depending on the location of the poles.