Ptolemy's Harmonics is outstanding in its field, and significant in the history of scientific thought for its sophisticated blend of rationalist and empiricist methodology. While rejecting Aristoxenian empiricism (see aristoxenus) outright, insisting with the Pythagoreans that musical structures must be analysed through the mathematics of ratio and shown to conform to ‘rational’ principles, Ptolemy criticizes the Pythagoreans for neglecting perceptual evidence: the credentials of rationally excogitated systems must ultimately be assessed by ear. He pursues this approach with meticulous attention to mathematical detail, to the minutiae of experimental procedures, and to the design and use of the special instruments they demand. Book 1 establishes the ratios of concords and melodic intervals, and divisions of tetrachords in each genus. Here and in book 2 Ptolemy's criticisms of earlier theorists preserve important information, especially about *Archytas and *Didymus (3). Book 2 analyses complete two-octave systems. Perhaps mistakenly, it dismisses as musically insignificant the contemporary conception of τόνοι as ‘keys’, thirteen (or fifteen) transpositions of identical structures: on Ptolemy's view their role is to bring different species of the octave into the same central range, and there can be only seven.