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Aims. With strong evidence of fast and slow magnetoacoustic modes arising in the solar atmosphere there is scope for improved determinations of coronal parameters through coronal seismology. Of particular interest is the ratio $P_{1}/2P_{2}$ between the period  P 1 of the fundamental mode and the period  P 2 of its first harmonic; in an homogeneous medium this ratio is one, but in a more complex configuration it is shifted to lower values. Methods. We consider analytically the effects on the different magnetohydrodynamic modes of structuring and stratification, pointing out that transverse or longitudinal structuring or gravitational stratification modifies the ratio $P_{1}/2P_{2}$. Results. The deviations caused by gravity and structure are studied for the fast and slow modes. Structure along the loop is found to be the dominant effect. Conclusions. The departure of $P_{1}/2P_{2}$ from unity can be used as a seismological tool in the corona. We apply our technique to the observations by Verwichte et al. (2004), deducing the density scale height in a coronal loop.

On the period ratioP$\mathsf{_{1}}$/2P$\mathsf{_{2}}$ in the oscillations of coronal loops