At least up to Rayleigh numbers of the order 1013, a feature of turbulent convection in confined containers is a self-organized and coherent large-scale motion ("mean wind"). For aspect ratio unity, the mean wind is comparable in scale to the container size. Its magnitude is measured here using short-time temperature correlations in a cylindrical container of aspect ratio unity; the working fluid is cryogenic helium and the Rayleigh numbers span from 106 to 1015. The self-organizing advection of "plumes" by the mean wind leads to periodic temperature oscillations near the sidewall. Comparisons of the observed oscillation frequency to the rotational rate of the mean wind, however, have differed by a factor of 2 in the recent literature. It is argued here that this apparent discrepancy is the result of the evolution of the shape of the mean wind, from a tilted and nearly elliptical shape at low Rayleigh numbers to a squarish shape at high Rayleigh numbers, thereby altering the effective path length from which the rotational rate of the mean wind is deduced.