The occurrence in arcs of high-magnesium andesites with Mg#'s of around 0.7 implies that such rocks are generated in, or equilibrated with, mantle peridotite under conditions which do not pertain beneath ridges. In this paper we use a review of experimental data on anhydrous peridotite melting to show that these primitive high-Mg andesites cannot be generated by anhydrous melting of lherzolite. Addition of H2O to peridotitic compositions has been shown, in numerous studies, to displace partial melts from basaltic towards more andesitic compositions leading to the possibility that high-Mg andesites are "wet" melts of lherzolite. Our review of the experimental data demonstrates however, that addition of H2O alone cannot explain the increases in both SiO2 and MgO contents (on an anhydrous basis) required to shift from basaltic to high-Mg andesitic melts in equilibrium with lherzolite residue. A much more plausible alternative is that these melts are extracted from a harzburgite residue, a model which we develop in more detail. We performed experiments at 0.6 GPa in which high Mg-andesitic melts were equilibrated with an olivine + orthopyroxene residue and find (in the absence of H2O) that MgO and SiO2 contents increase in the ratio 2:1 as the degree of undersaturation in clinopyroxene increases. This is the right sign and magnitude of effect to explain the compositions of primitive high-Mg andesites. Data from earlier studies of the CaO-MgO-SiO2 system at 1 atm and natural compositions at 1.5-2 GPa are in excellent agreement with our observations. When the effects of H2O and clinopyroxene undersaturation are added together we obtain a line in Pressure-H2O space which describes the conditions under which a given high Mg-andesite could be in equilibrium with a harzburgite residue. Application to rocks from White Island (New Zealand), Amphlett Island (Papua New Guinea), Setouchi Belt (Japan), Mt. Shasta (USA), Adak Island and Piip volcano (Aleutians, USA) yield, for crustal thicknesses > 20 km, H2O contents of the melts of 2-7%, in generally good agreement with the available compositions of melt inclusions.