Context: In transverse coronal loop oscillations, two periodicities have been measured simultaneously and are interpreted as the fundamental kink mode (with period P_1) and the first harmonic (with period P_2). Deviations of the period ratio P1/2P2 from unity provide information about the extent of longitudinal structuring within the loop. Aims: Here we develop an analytical approximation that describes the shift in P1/2P2 in terms of the ratio L/Λc of the length 2L of a coronal loop and the density scale height Λ_c. Methods: We study the MHD wave equations in a low β plasma using the thin tube approximation. Disturbances are described by a differential equation which may be solved for various equilibrium density profiles, obtaining dispersion relations in terms of Bessel functions. These dispersion relations may be used to obtain analytical approximations to the periods P1 and P2. We also present a variational approach to determining the period ratio and show how the WKB method may be used. Results: Analytical approximations to the period ratio P1/2P2 are used to shed light on the magnitude of longitudinal structuring in a loop, leading to a determination of the density scale height. We apply our formula to the observations in Verwichte et al. (2004) and Van Doorsselaere et al. (2007), obtaining the coronal density scale height. Conclusions: Our simple formula and approximate approaches highlight a useful analytical tool for coronal seismology. We demonstrate that P1/2P2 is linked to the density scale height, with no need for estimates of other external parameters. Given the accuracy of current observations, our formula provides a convenient means of determining density scale heights.