‘It would not be right to attribute these errors to the power of reason, but to those who ground reason in faulty hupotheseis’ (tois mē deontōs auton hupotithemenois, 15.3–4). This statement stands as a preface to Ptolemy's own account of the concords in 1.7. The Pythagoreans' errors have been shown up principally by recourse to the evidence of perception. That strategy implies that the test of perception is to be trusted; but the errors it reveals should not deter us from the quest for rational principles. Hence where propositions derived from supposedly rational hupotheseis are at odds with the perceptual data, neither reason as such nor the senses should be blamed for the conflict, nor should either be dismissed as unreliable in its own sphere of competence. The proper conclusion is that the hupotheseis have been misconceived or wrongly applied.

Ptolemy's exposition of the correct principles begins with a three-fold classification of musical intervals. ‘Preeminent in excellence is the class of homophones, second that of concords, and third that of the melodic. For the octave and the double octave plainly differ from the other concords as do the latter from the melodic, so that it would be more appropriate for them to be called “homophones”.