‘Since, then, not even these people have divided the primary genera of the tetrachords in a way that agrees with perception, let us ourselves try, here as well, to preserve what is consistent both with our hupotheseis concerning melodic relations and with the appearances, in accordance with those conceptions of the divisions that are primary and natural’ (33.1–5). So begins 1.15. From a technical point of view this long chapter is the core of the Harmonics, and the analyses that it contains provide the basis for all Ptolemy's later constructions. It will be as well to remark at the outset, however, that they are not his last word on the division of tetrachords. His object is to identify the rational credentials of systems that perception will recognise as perfectly formed. The divisions derived here are perfectly formed from a rational perspective, and in a certain sense from that of perception too; yet it turns out that few of them are acceptable in musical practice precisely as 1.15 describes them. The relation between theoretical perfection and aesthetic acceptability is more complex than has so far emerged. Ptolemy is probably alluding to distinctions of this sort when he describes the conceptions developed here as ‘primary and natural’. They constitute in some way both the mathematical and the aesthetic foundations of the systems used in practical music-making, without being quite identical with them.