The composition of the Sun is an essential piece of reference data for astronomy, cosmology, astroparticle, space and geo-physics. This article, dealing with the intermediate-mass elements Na to Ca, is the first in a series describing the comprehensive re-determination of the solar composition. In this series we severely scrutinise all ingredients of the analysis across all elements, to obtain the most accurate, homogeneous and reliable results possible. We employ a highly realistic 3D hydrodynamic solar photospheric model, which has successfully passed an arsenal of observational diagnostics. To quantify systematic errors, we repeat the analysis with three 1D hydrostatic model atmospheres (MARCS, MISS and Holweger & M{u}ller 1974) and a horizontally and temporally-averaged version of the 3D model ($langle$3D$rangle$). We account for departures from LTE wherever possible. We have scoured the literature for the best transition probabilities, partition functions, hyperfine and other data, and stringently checked all observed profiles for blends. Our final 3D+NLTE abundances are: $logepsilon_{mathrm{Na}}=6.21pm0.04$, $logepsilon_{mathrm{Mg}}=7.59pm0.04$, $logepsilon_{mathrm{Al}}=6.43pm0.04$, $logepsilon_{mathrm{Si}}=7.51pm0.03$, $logepsilon_{mathrm{P}}=5.41pm0.03$, $log epsilon_{mathrm{S}}=7.13pm0.03$, $logepsilon_{mathrm{K}}=5.04pm0.05$ and $logepsilon_{mathrm{Ca}}=6.32pm0.03$. The uncertainties include both statistical and systematic errors. Our results are systematically smaller than most previous ones with the 1D semi-empirical Holweger & Muller model. The $langle$3D$rangle$ model returns abundances very similar to the full 3D calculations. This analysis provides a complete description and a slight update of the Na to Ca results presented in Asplund, Grevesse, Sauval & Scott (arXiv:0909.0948), with full details of all lines and input data.
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