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In this correspondence, we introduce a dual-tree rational-dilation complex wavelet transform for oscillatory signal processing. Like the short-time Fourier transform and the dyadic dual-tree complex wavelet transform, the introduced transform employs quadrature pairs of time-frequency atoms which allow to work with the analytic signal. The introduced wavelet transform is a constant-transform, a property lacked by the short-time Fourier transform, which in turn makes the introduced transform more suitable for models that depend on scale. Also, the frequency resolution can be as high as desired, a property lacked by the dyadic dual-tree complex wavelet transform, which makes the introduced transform more suitable for processing oscillatory signals like speech, audio and various biomedical signals.