Distributed consensus with data rate constraint is an important research topic of multi-agent systems. Some results have been obtained for consensus of multi-agent 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 systems and investigate the corresponding data rate required for achieving consensus. The state matrix of each agent is a 2m-th 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 corresponding to 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 it is concluded that the sufficient number of communication bits to guarantee the consensus at an exponential convergence rate is an integer between $m$ and $2m$, depending on the location of the poles.