An oblique, three-dimensional coordinate system called helicoidal rectangular coordinates, is introduced for which the coordinate surfaces coincide with the walls of a tube, with helical centerline and rectangular cross-section. The wave equation in helicoidal rectangular coordinates is solved in terms of sinusoidal functions in time and in the coordinates in the cross-section; the radial dependence is specified by Bessel functions of integer order in the case of a bent and untwisted tube having only curvature of flexion, which contains cylindrical waves. In the case of a bent and twisted tube, with curvature of torsion as well as of flexion, the Bessel functions have real order, not necessarily an integer. The sound fields are determined in the helical tube of rectangular cross-section, including the application of boundary and initial conditions, for waves propagating along the tube and for standing modes trapped between two sections. The application of wall boundary conditions specifies the eigenvalues and eigenfunctions for the modes in the cross-section; the tables of radial wavenumbers and plots of eigenfunctions are given for modes with lowest radial n=1,2,3, azimuthal m=0,1,2 and vertical l=0,1 orders, for several cases: flat and twisted bends, with smaller or larger curvature of flexion and torsion, and square or rectangular cross-sections.