This is the final version of the article. Available from the Royal Society via the DOI in this record.Data accessibility: All data related to this research are contained in the manuscript.We show that the spin-wave spectrum in an elliptical helix has a band character. The size of the first band gap calculated using the perturbation theory is shown to scale as square root of the eccentricity. Curved magnonic waveguides of the kind considered here could be used as structural elements of future three-dimensional magnonic architectures.This research has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under Grant Agreement no. 247556 (NoWaPhen), the European Union's Horizon 2020 research and innovation program under Marie Skłodowska-Curie grant Agreement no. 644348 (MagIC)
