The vibrating-wire technique has been widely applied in measurements of both viscosity and density. When tensioned by a suspended mass or sinker, the device may be used to measure these properties simultaneously. The sensitivity to density arises mainly from the effect of buoyancy on the tension and hence the resonance frequency of the vibrating wire. Earlier studies have used simplified working equations to relate the resonance frequency and the tension. In this work, we have employed the exact equations describing forced simple harmonic oscillations of a stiff wire under tension and also a second-order analytical solution of those equations. We show that the exact working equation describes the actual resonance frequency of tensioned tungsten wires with improved accuracy. This finding has been validated by means of experimental measurements of the resonance frequency at prescribed tensioning forces and at temperatures in the range (293 to 473) K. One consequence of this is that the true independently measured sinker volume may be used in preference to one obtained by calibration of the assembled sensor. The remaining parameters to be obtained are then the radius of the wire, Young's modulus E for the wire, and an empirical parameter that accounts for the remaining departure from the theoretical expression. The values of wire radius obtained by calibration in a liquid of known viscosity have been compared with those from independent mechanical measurements having a relative uncertainty of 0.2 %. These comparisons have been made for both drawn wires and for a centerless ground rod with a diameter of 0.15 mm, and the differences are discussed in the light of scanning electron microscopy (SEM) studies of the circularity and uniformity of the wire. Finally, we provide experimental evidence that rough drawn wires may yield inaccurate viscosity measurements when applied to fluids having viscosities very different to the one used for calibration; conversely, a centerless-ground rod gave good results even when the viscosity to be measured was 60 times that of the calibration fluid.