By the end of Book 1 Ptolemy has completed his analysis of divisions of the tetrachord; but he takes one further step before moving on to a new topic. 11.1 is occupied by an account of an alternative method of confirming the patterns of ratios attributed to the attunements of practical musicians in 1.16. Here Ptolemy reverses his former procedure. Instead of first arguing to the values of the ratios on ‘rational’ grounds and then confirming the results by ear, he now begins by constructing the attunements by ear on the strings of an eight-stringed instrument, and then argues that the ratios of intervals constructed in this way must indeed have the values he has assigned to them. From here Ptolemy is led on to a discussion of certain other instruments that can be used for the same purpose; this occupies 11.2. We shall review the contents of these two passages, among others, in Chapters 10 and 11.

The transition to a new phase of the investigation is clearly signalled at the beginning of 11.3. ‘Let that be our outline of what is scientifically understood (ta theōroumena) concerning the concordant and melodic relations between notes that are established in conformity with the lengths of string plucked, the homophones being included along with the concords. The next topic for discussion after these is that dealing with the systēmata’ (49.4–7).