The propagation of chemical waves through narrow channels has been investigated. Thin layers of excitable Belousov-Zhabotinsky mixtures are connected by precision-bore capillary tubes of different internal diameters. A wave initiated on one side of an otherwise impenetrable barrier enters and travels through the capillary tube, forming a hemisphere of excited solution at the exit. When the tube diameter is greater than a critical value, the excitation serves to initiate a circular wave in the second compartment; otherwise, the hemisphere collapses and no wave is initiated. Electrochemically generated periodic wave trains give rise to resonance patterns characterized by firing numbers 1/n, where n = 1, 2, etc. is the number of waves entering the tube for every wave exiting. These firing numbers correspond to one branch of a Farey tree; higher periodic resonances in modeling calculations indicate that more fully developed Farey sequences may also occur. A one-dimensional mapping procedure is proposed to describe the appearance and ordering of the resonance patterns.